1.
Regular polygon
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In Euclidean geometry, a regular polygon is a polygon, equiangular and equilateral. Regular polygons may be star. These properties apply to all regular polygons, whether star. A regular n-sided polygon has rotational symmetry of order n. All vertices of a regular polygon lie on a common circle, i.e. they are concyclic points. That is, a regular polygon is a cyclic polygon. Thus a regular polygon is a tangential polygon. A n-sided polygon can be constructed with compass and straightedge if and only if the odd prime factors of n are distinct Fermat primes. See constructible polygon. The symmetry group of an n-sided regular polygon is dihedral group Dn: D2, D3, D4... It consists of the rotations in Cn, together with reflection symmetry in n axes that pass through the center. If n is then half of these axes pass through two opposite vertices, the other half through the midpoint of opposite sides. If n is odd then all axes pass through the midpoint of the opposite side. All simple polygons are convex. Those having the same number of sides are also similar.
Regular polygon
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The zig-zagging side edges of a n - antiprism represent a regular skew 2 n -gon, as shown in this 17-gonal antiprism.
Regular polygon
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Regular convex and star polygons with 3 to 12 vertices labelled with their Schläfli symbols
2.
Edge (geometry)
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In a polygon, an edge is often called a side. In a polyhedron or more generally a polytope, an edge is a segment where two faces meet. A segment joining two vertices while passing through the exterior is not an edge but instead is called a diagonal. Conversely, the graphs that are skeletons of three-dimensional polyhedra can be characterized as being exactly the 3-vertex-connected planar graphs. This equation is known as Euler's formula. Thus the number of edges faces. For example, a cube has 6 faces, hence 12 edges. Similarly, in a polyhedron, exactly two two-dimensional faces meet at every edge, while in higher dimensional polytopes three or more two-dimensional faces meet at every edge. Extended side Olshevsky, George. "Edge". Glossary for Hyperspace. Archived from the original on 4 February 2007. Weisstein, Eric W. "Polygonal edge". MathWorld. Weisstein, Eric W. "Polyhedral edge".
Edge (geometry)
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Three edges AB, BC, and CA, each between two vertices of a triangle.
3.
Vertex (geometry)
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In geometry, a vertex is a point where two or more curves, lines, or edges meet. As a consequence of this definition, the point where two lines meet to form the corners of polygons and polyhedra are vertices. A vertex is a point of a polygon, polyhedron, or other higher-dimensional polytope, formed by the intersection of edges, faces or facets of the object. However, in theory, vertices may have fewer than two incident edges, usually not allowed for geometric vertices. However, a smooth approximation to a polygon will also have additional vertices, at the points where its curvature is minimal. There are two types of principal vertices: mouths. A principal xi of a simple polygon P is called an ear if the diagonal that bridges xi lies entirely in P. According to the two ears theorem, every simple polygon has at least two ears. A principal xi of a simple polygon P is called a mouth if the diagonal lies outside the boundary of P. This equation is known as Euler's formula. Thus the number of vertices is 2 more than the excess of the number of edges over the number of faces. For example, a cube has 6 faces, hence 8 vertices. Weisstein, Eric W. "Polygon Vertex". MathWorld. Weisstein, Eric W. "Polyhedron Vertex".
Vertex (geometry)
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A vertex of an angle is the endpoint where two line segments or rays come together.
4.
Coxeter diagram
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In geometry, a Coxeter–Dynkin diagram is a graph with numerically labeled edges representing the spatial relations between a collection of mirrors. It describes a kaleidoscopic construction: each graph "node" represents the label attached to a branch encodes the dihedral order between two mirrors. An unlabeled branch implicitly represents order-3. Each diagram represents a Coxeter group, Coxeter groups are classified by their associated diagrams. Dynkin diagrams are used to therefore semisimple Lie algebras. Branches of a Coxeter -- Dynkin diagram are labeled with a rational p, representing a dihedral angle of ° / p. When p = 2 the angle is 90° and the mirrors have no interaction, so the branch can be omitted from the diagram. If a branch is unlabeled, it is assumed to have p = 3, representing an angle of 60°. Two parallel mirrors have a branch marked with "∞". In principle, n mirrors can be represented by a complete graph in which all n / 2 branches are drawn. In practice, nearly all interesting configurations of mirrors include a number of right angles, so the corresponding branches are omitted. Diagrams can be labeled by their graph structure. The first forms studied by Ludwig Schläfli are the orthoschemes which have linear graphs that generate regular polytopes and regular honeycombs. Plagioschemes are simplices represented by branching graphs, cycloschemes are simplices represented by cyclic graphs. As a matrix of cosines, it is also called a Gramian matrix after Jørgen Pedersen Gram.
Coxeter diagram
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Coxeter–Dynkin diagrams for the fundamental finite Coxeter groups
5.
List of planar symmetry groups
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This article summarizes the classes of discrete symmetry groups of the Euclidean plane. The symmetry groups are named here by three naming schemes: International notation, Coxeter notation. The two-dimensional line groups, with a direction of periodicity are given with five notational names. The Schönflies notation is given as infinite limits of 7 dihedral groups. The yellow regions represent the fundamental domain in each. The p2 groups, with no reflectional symmetry, are repeated in all classes. The related reflectional Coxeter group are given with all classes except oblique. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6 H.S.M. Coxeter, Regular and Semi Regular Polytopes I, H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, Coxeter, H. S. M. & Moser, W. O. J.. Generators and Relations for Discrete Groups. New York: Springer-Verlag. ISBN 0-387-09212-9.
List of planar symmetry groups
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C 1, [] + (•)
6.
Dihedral symmetry
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In mathematics, a dihedral group is the group of symmetries of a regular polygon, which includes rotations and reflections. Dihedral groups are among the simplest examples of finite groups, they play an important role in group theory, geometry, chemistry. The notation for the dihedral group of order n differs in geometry and abstract algebra. In geometry, Dn or Dihn refers to the symmetries of the n-gon, a group of order 2n. In abstract algebra, Dn refers to the dihedral group of order n. The geometric convention is used in this article. A regular polygon with n sides has 2 n different symmetries: n rotational symmetries and n reflection symmetries. Usually, we take n ≥3 here.The associated rotations and reflections make up the dihedral group D n. If n is odd, each axis of symmetry connects the midpoint of one side to the opposite vertex. If n is even, there are n/2 axes of symmetry connecting the midpoints of opposite sides and n / 2 axes of symmetry connecting opposite vertices. In either case, there are n axes of symmetry and 2 n elements in the symmetry group. Reflecting in one axis of symmetry followed by reflecting in another axis of symmetry produces a rotation through twice the angle between the axes. As with any geometric object, the composition of two symmetries of a regular polygon is again a symmetry of this object. With composition of symmetries to produce another as the binary operation, this gives the symmetries of a polygon the algebraic structure of a finite group. The following Cayley table shows the effect of composition in the group D3.
Dihedral symmetry
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The symmetry group of a snowflake is Dih 6, a dihedral symmetry, the same as for a regular hexagon.
7.
Internal angle
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In geometry, an angle of a polygon is formed by two sides of the polygon that share an endpoint. A polygon has exactly one internal angle per vertex. If every internal angle of a simple polygon is less than 180°, the polygon is called convex. In contrast, an angle is an angle formed by a line extended from an adjacent side. The sum of the external angle on the same vertex is °. The sum of all the internal angles of a simple polygon is 180° where n is the number of sides. The sum of the external angles of non-convex polygon is °. The interior angle concept can be extended in a consistent way to crossed polygons such as star polygons by using the concept of directed angles. In other words, 360k° represents the sum of all the exterior angles.
Internal angle
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Internal and External angles
8.
Degree (angle)
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A degree, usually denoted by °, is a measurement of a plane angle, defined so that a full rotation is 360 degrees. Because a full rotation equals 2π radians, one degree is equivalent to π 180 radians. The original motivation for choosing the degree as a unit of angles is unknown. One theory states that it is related to the fact that 360 is approximately the number of days in a year. Some ancient calendars, such as the Persian calendar, used 360 days for a year. The use of a calendar with 360 days may be related to the use of sexagesimal numbers. The earliest trigonometry, used by their Greek successors, was based on chords of a circle. A chord of length equal to the radius made a natural quantity. One sixtieth of this, using their sexagesimal divisions, was a degree. Aristarchus of Samos and Hipparchus seem to exploit Babylonian astronomical knowledge and techniques systematically. Hipparchus were the first Greeks known to divide the circle in 360 degrees of 60 arc minutes. Eratosthenes used a simpler system dividing a circle into 60 parts. The division of the circle into 360 parts also occurred in ancient India, as evidenced in the Rigveda: one wheel, navels three. Who can comprehend this? On it are placed together sixty like pegs.
Degree (angle)
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One degree (shown in red) and eighty nine (shown in blue)
9.
Dual polygon
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In geometry, polygons are associated into pairs called duals, where the vertices of one correspond to the edges of the other. Regular polygons are self-dual. The dual of an isogonal polygon is an isotoxal polygon. For example, rhombus are duals. In a cyclic polygon, longer sides correspond to shorter sides to smaller angles. Further, congruent sides in the original polygon yields congruent angles in the dual, conversely. For example, the dual of a highly acute isosceles triangle is an isosceles triangle. In the Dorman Luke construction, each face of a dual polyhedron is the dual polygon of the corresponding figure. As an example of the side-angle duality of polygons we compare properties of the tangential quadrilaterals. This duality is perhaps even more clear when comparing an isosceles trapezoid to a kite. New edges are formed between these new vertices. This construction is not reversible. That is, the polygon generated by applying it twice is in general not similar to the original polygon. As with dual polyhedra, one can perform polar reciprocation in it. Then the dual polygon is obtained by simply switching the edges.
Dual polygon
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Dorman Luke construction, showing a rhombus face being dual to a rectangle vertex figure.
10.
Convex polygon
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A convex polygon is a simple polygon in which no line segment between two points on the boundary ever goes outside the polygon. Equivalently, it is a simple polygon whose interior is a convex set. A simple polygon, not convex is called concave. The following properties of a simple polygon are all equivalent to convexity: Every internal angle is equal to 180 degrees. Every point on every segment between two points inside or on the boundary of the polygon remains inside or on the boundary. The polygon is entirely contained in a closed half-plane defined by each of its edges. For each edge, the interior points are all on the same side of the line that the edge defines. The angle at each vertex contains all other vertices in its edges and interior. The polygon is the convex hull of its edges. Additional properties of convex polygons include: The intersection of two convex polygons is a polygon. Krein–Milman theorem: A convex polygon is the convex hull of its vertices. One only needs the corners of the polygon to recover the entire polygon shape. Hyperplane separation theorem: Any two convex polygons with no points in common have a separator line. If at least one of them is compact, then there are even two parallel separator lines. Inscribed triangle property: Of all triangles contained in a convex polygon, there exists a triangle with a maximal area whose vertices are all polygon vertices.
Convex polygon
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An example of a convex polygon: a regular pentagon
11.
Cyclic polygon
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In geometry, the circumscribed circle or circumcircle of a polygon is a circle which passes through all the vertices of the polygon. Its radius is called the circumradius. A polygon which has a circumscribed circle is called a cyclic polygon. All regular simple polygons, all isosceles trapezoids, all rectangles are cyclic. A related notion is the one of a bounding circle, the smallest circle that completely contains the polygon within it. All triangles are cyclic; i.e. every triangle has a circumscribed circle. Since this equation has three parameters only three points' coordinate pairs are required to determine the equation of a circle. Since exactly three points are required to determine a circle, every triangle can be circumscribed. The circumcenter of a triangle can be constructed by drawing any two of the three perpendicular bisectors. The radius is the length to any of the three vertices. In coastal navigation, a triangle's circumcircle is sometimes used as a way of obtaining a line using a sextant when no compass is available. The horizontal angle between two landmarks defines the circumcircle upon which the observer lies. Suppose that A = B = C = are the coordinates of points A, B, C. Using the polarization identity, these equations reduce to the condition that the matrix has a nonzero kernel. Thus the circumcircle may alternatively be described as the locus of zeros of the determinant of this matrix: det = 0.
Cyclic polygon
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Circumscribed circle, C, and circumcenter, O, of a cyclic polygon, P
12.
Equilateral polygon
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If it does then it is a regular polygon. If the number of sides is at least five, an equilateral polygon need not be a polygon: it could be concave or even self-intersecting. All isotoxal polygons are equilateral. An equilateral triangle is a regular triangle with 60 internal angles. An equilateral quadrilateral is called an isotoxal polygon described by an angle α. It includes the square as a special case. A equilateral pentagon can be described by two angles α and β, which together determine the other angles. Equilateral pentagons exist, as do concave equilateral polygons with any larger number of sides. An equilateral polygon, cyclic is a regular polygon. A tangential polygon is only if the alternate angles are equal. Thus if the number of sides n is odd, a tangential polygon is only if it is regular. The principal diagonals of a hexagon each divide the hexagon into quadrilaterals. Triambi are equilateral hexagons with trigonal symmetry: Equilateral triangle With interactive animation A Property of Equiangular Polygons: What Is It About? A discussion of Viviani's theorem at Cut-the-knot.
Equilateral polygon
13.
Isogonal figure
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In geometry, a polytope is isogonal or vertex-transitive if, loosely speaking, all its vertices are equivalent. We say that for any two vertices there exists a symmetry of the polytope mapping the first isometrically onto the second. All vertices of a n-dimensional isogonal figure exist on an - sphere. The isogonal has long been used for polyhedra. Vertex-transitive is a synonym borrowed from modern ideas such as graph theory. All regular polygons, regular star polygons are isogonal. The dual of an isogonal polygon is an isotoxal polygon. Some even-sided apeirogons which alternate two edge lengths, for example a rectangle, are isogonal. All planar 2n-gons have dihedral symmetry with reflection lines across the mid-edge points. 2D tiling has a single kind of vertex. Geometrically distorted variations of uniform polyhedra and tilings can also be given the configuration. Quasi-regular if it is also isotoxal but not isohedral. Semi-regular if every face is a regular polygon but it is not isotoxal. Uniform if every face is a regular polygon, i.e. it is regular, semi-regular. Noble if it is also isohedral.
Isogonal figure
14.
Isotoxal figure
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This article is about geometry. For transitivity in theory, see edge-transitive graph. In geometry, a tiling, is edge-transitive if its symmetries act transitively on its edges. The isotoxal is derived from the τοξον meaning arc. An isotoxal polygon is an equilateral polygon, but not all equilateral polygons are isotoxal. The duals of isotoxal polygons are isogonal polygons. In general, an isotoxal 2n-gon will have Dn dihedral symmetry. A rhombus is an isotoxal polygon with D2 symmetry. All regular polygons are isotoxal, having double the minimum symmetry order: a regular n-gon has Dn dihedral symmetry. A regular 2n-gon is an isotoxal polygon and can be marked with alternately colored vertices, removing the line of reflection through the mid-edges. Tiling must be either isohedral or both. Regular polyhedra are isotoxal. Quasiregular polyhedra are not isohedral; their duals are not isogonal. Not every polyhedron or 2-dimensional tessellation constructed from regular polygons is isotoxal. An isotoxal polyhedron has the same dihedral angle for all edges.
Isotoxal figure
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A rhombic dodecahedron is an isohedral and isotoxal polyhedron
Isotoxal figure
15.
Geometry
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Geometry is a branch of mathematics concerned with questions of shape, size, relative position of figures, the properties of space. A mathematician who works in the field of geometry is called a geometer. Geometry arose independently in a number of early cultures for dealing with lengths, areas, volumes. Geometry began to see elements of mathematical science emerging in the West as early as the 6th century BC. By the 3rd BC, geometry was put into an axiomatic form by Euclid, whose treatment, Euclid's Elements, set a standard for many centuries to follow. Geometry arose independently with texts providing rules for geometric constructions appearing as early as the 3rd century BC. Islamic scientists expanded on them during the Middle Ages. By the 17th century, geometry had been put on a solid analytic footing by mathematicians such as René Descartes and Pierre de Fermat. Since then, into modern times, geometry has expanded into non-Euclidean geometry and manifolds, describing spaces that lie beyond the normal range of human experience. While geometry has evolved significantly throughout the years, there are some general concepts that are less fundamental to geometry. These include the concepts of points, lines, planes, surfaces, curves, as well as the more advanced notions of manifolds and topology or metric. Contemporary geometry has many subfields: Euclidean geometry is geometry in its classical sense. The educational curriculum of the majority of nations includes the study of points, lines, planes, angles, triangles, congruence, similarity, solid figures, circles, analytic geometry. Euclidean geometry also has applications in computer science, various branches of modern mathematics. Differential geometry uses techniques of linear algebra to study problems in geometry.
Geometry
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Visual checking of the Pythagorean theorem for the (3, 4, 5) triangle as in the Chou Pei Suan Ching 500–200 BC.
Geometry
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An illustration of Desargues' theorem, an important result in Euclidean and projective geometry
Geometry
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Geometry lessons in the 20th century
Geometry
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A European and an Arab practicing geometry in the 15th century.
16.
Ancient Greek
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Ancient Greek includes the forms of Greek used in ancient Greece and the ancient world from around the 9th century BC to the 6th century AD. It is often roughly divided into the Archaic period, Hellenistic period. It is antedated by Mycenaean Greek. The language of the Hellenistic phase is known as Koine. Prior to the Koine period, Greek of earlier periods included several regional dialects. Ancient Greek was the language of Homer and of classical Athenian historians, philosophers. It has been a standard subject of study in educational institutions of the West since the Renaissance. This article primarily contains information of the language. Ancient Greek was a pluricentric language, divided into many dialects. The main dialect groups are Doric, many of them with several subdivisions. Some dialects are found in literary forms used in literature, while others are attested only in inscriptions. There are also historical forms. Homeric Greek is a literary form of Archaic Greek used by other authors. Homeric Greek had significant differences in pronunciation from Classical Attic and other Classical-era dialects. The early form and development of the Hellenic language family are not well understood because of a lack of contemporaneous evidence.
Ancient Greek
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Inscription about the construction of the statue of Athena Parthenos in the Parthenon, 440/439 BC
Ancient Greek
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Ostracon bearing the name of Cimon, Stoa of Attalos
Ancient Greek
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The words ΜΟΛΩΝ ΛΑΒΕ as they are inscribed on the marble of the 1955 Leonidas Monument at Thermopylae
17.
Polygon
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The points where two edges meet are the polygon's vertices or corners. The interior of the polygon is sometimes called its body. An n-gon is a polygon with n sides; for example, a triangle is a 3-gon. A polygon is a 2-dimensional example of the more general polytope in any number of dimensions. The geometrical notion of a polygon has been adapted in various ways to suit particular purposes. They often define a polygon accordingly. A polygonal boundary may be allowed creating star polygons and other self-intersecting polygons. Other generalizations of polygons are described below. The word "polygon" derives from the Greek adjective πολύς "many" and γωνία "corner" or "angle". It has been suggested that γόνυ "knee" may be the origin of “gon”. Polygons are primarily classified by the number of sides. See table below. Polygons may be characterized by their type of non-convexity: Convex: any line drawn through the polygon meets its boundary exactly twice. As a consequence, all its interior angles are less than 180°. Equivalently, any segment with endpoints on the boundary passes through only interior points between its endpoints.
Polygon
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Historical image of polygons (1699)
Polygon
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Some different types of polygon
Polygon
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The Giant's Causeway, in Northern Ireland
18.
Truncation (geometry)
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In geometry, a truncation is an operation in any dimension that cuts polytope vertices, creating a new facet in place of each vertex. The term originates for the Archimedean solids. It represents a fixed geometric, just like the regular polyhedra. In general all ringed uniform polytopes have a uniform truncation. For example, the icosidodecahedron, has a uniform truncation, the truncated icosidodecahedron, represented as tr or t. In the Coxeter-Dynkin diagram, the effect of a truncation is to ring all the nodes adjacent to the ringed node. A n-sided polygon will have 2n sides. A regular polygon truncated will become another regular polygon: t is. R, is another regular polygon in its dual position. A regular polygon can also be represented by its Coxeter-Dynkin diagram, its complete truncation. Star polygons can also be truncated. A truncated pentagram is actually a double-covered decagon with two sets of overlapping vertices and edges. A great heptagram gives a tetradecagram. The final polyhedron is a cuboctahedron. The middle image is the truncated cube.
Truncation (geometry)
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Truncated cubic honeycomb t{4,3,4} or
Truncation (geometry)
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Truncated square is a regular octagon: t{4} = {8} =
19.
Equilateral triangle
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In geometry, an equilateral triangle is a triangle in which all three sides are equal. In the familiar Euclidean geometry, equilateral triangles are also equiangular;, all three internal angles are also congruent to each other and are each 60°. They are regular polygons, can therefore also be referred to as regular triangles. Thus these are properties that are unique to equilateral triangles. The three medians have equal lengths. The three angle bisectors have equal lengths. For some pairs of triangle centers, the fact that they coincide is enough to ensure that the triangle is equilateral. In particular: A triangle is equilateral if any two of incenter, orthocenter coincide. It is also equilateral if its circumcenter coincides with the Nagel point, or if its incenter coincides with its nine-point center. For any triangle, the three medians partition the triangle into six smaller triangles. A triangle is only if any three of the smaller triangles have either the same inradius. A triangle is equilateral if and only if the circumcenters of any three of the smaller triangles have the same distance from the centroid. Morley's trisector theorem states that, in any triangle, the three points of intersection of the adjacent angle trisectors form an equilateral triangle. A version of the isoperimetric inequality for triangles states that the triangle of greatest area among all those with a given perimeter is equilateral. That is, PA, PB, PC satisfy the triangle inequality that any two of them sum to at least as great as the third.
Equilateral triangle
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A regular tetrahedron is made of four equilateral triangles.
Equilateral triangle
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Equilateral triangle
20.
Compass and straightedge
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The idealized ruler, known as a straightedge, is assumed to be infinite in length, has no markings on it and only one edge. The compass is assumed to collapse when lifted from the page, so may not be directly used to transfer distances. More formally, the only permissible constructions are those granted by Euclid's first three postulates. It turns out to be the case that every point constructible using straightedge and compass may also be constructed using compass alone. The ancient Greek mathematicians first conceived compass-and-straightedge constructions, a number of ancient problems in plane geometry impose this restriction. The ancient Greeks developed many constructions, but in some cases were unable to do so. Gauss showed that some polygons are constructible but that most are not. Some of the most famous straightedge-and-compass problems were proven impossible by Pierre Wantzel in 1837, using the mathematical theory of fields. In spite of existing proofs of impossibility, some persist in trying to solve these problems. Circles can only be drawn starting from two given points: the centre and a point on the circle. The compass may or may not collapse when it's not drawing a circle. The straightedge is infinitely long, but it has no markings on it and has only one straight edge, unlike ordinary rulers. It can only be used to draw a line segment between two points or to extend an existing segment. The modern compass generally does not collapse and several modern constructions use this feature. It would appear that the modern compass is a "more powerful" instrument than the ancient collapsing compass.
Compass and straightedge
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A compass
Compass and straightedge
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Creating a regular hexagon with a ruler and compass
21.
Euclid
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Euclid, sometimes called Euclid of Alexandria to distinguish him from Euclides of Megara, was a Greek mathematician, often referred to as the "father of geometry". He was active in Alexandria during the reign of Ptolemy I. In the Elements, Euclid deduced the principles of what is now called Euclidean geometry from a small set of axioms. Euclid also wrote works on perspective, conic sections, spherical geometry, rigor. Euclid is the anglicized version of the Greek Εὐκλείδης, which means "renowned, glorious". Very original references to Euclid survive, so little is known about his life. The place and circumstances of both his birth and death are unknown and may only be estimated roughly relative to other people mentioned with him. He is usually referred to as" ὁ στοιχειώτης". The historical references to Euclid were written centuries after he lived by Proclus c. 450 AD and Pappus of Alexandria c. 320 AD. Proclus introduces Euclid only briefly in his Commentary on the Elements. This anecdote is questionable since it is similar to a story told about Alexander the Great. 247–222 BC. A detailed biography of Euclid is given by Arabian authors, mentioning, for example, a town of Tyre. This biography is generally believed to be completely fictitious. However, there is little evidence in its favor.
Euclid
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Euclid by Justus van Gent, 15th century
Euclid
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One of the oldest surviving fragments of Euclid's Elements, found at Oxyrhynchus and dated to circa AD 100 (P. Oxy. 29). The diagram accompanies Book II, Proposition 5.
Euclid
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Statue in honor of Euclid in the Oxford University Museum of Natural History
22.
Euclid's Elements
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Euclid's Elements is a mathematical and geometric treatise consisting of 13 books attributed to the ancient Greek mathematician Euclid in Alexandria, Ptolemaic Egypt circa 300 BC. It is a collection of definitions, postulates, mathematical proofs of the propositions. The books cover the ancient Greek version of elementary number theory. It is the oldest extant axiomatic deductive treatment of mathematics. It has proven instrumental in the development of logic and modern science. According to Proclus, the term "element" was used to describe a theorem that helps furnishing proofs of many other theorems. The element in the Greek language is the same as letter. This suggests that theorems in the Elements should be seen as standing as letters to language. Euclid's Elements has been referred to as the most influential textbook ever written. Scholars believe that the Elements is largely a collection of theorems proven by other mathematicians, supplemented by some original work. The Heiberg manuscript, is from a Byzantine workshop around 900 and is the basis of modern editions. Papyrus Oxyrhynchus 29 only contains the statement of one proposition. Although known to, for instance, Cicero, no record exists of the text having been translated prior to Boethius in the fifth or sixth century. The Arabs received the Elements around 760; this version was translated into Arabic under Harun al Rashid circa 800. The Byzantine scholar Arethas commissioned the copying of the extant Greek manuscripts of Euclid in the late ninth century.
Euclid's Elements
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The frontispiece of Sir Henry Billingsley's first English version of Euclid's Elements, 1570
Euclid's Elements
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A fragment of Euclid's "Elements" on part of the Oxyrhynchus papyri
Euclid's Elements
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An illumination from a manuscript based on Adelard of Bath 's translation of the Elements, c. 1309–1316; Adelard's is the oldest surviving translation of the Elements into Latin, done in the 12th-century work and translated from Arabic.
Euclid's Elements
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Euclidis – Elementorum libri XV Paris, Hieronymum de Marnef & Guillaume Cavelat, 1573 (second edition after the 1557 ed.); in-8, 350, (2)pp. THOMAS-STANFORD, Early Editions of Euclid's Elements, n°32. Mentioned in T.L. Heath's translation. Private collection Hector Zenil.
23.
Intersection
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In mathematics, the intersection of two or more objects is another, usually "smaller" object. All objects are presumed to lie in a certain common space except in set theory, where the intersection of arbitrary sets is defined. The intersection is one of basic concepts of geometry. Intuitively, the intersection of two or more objects is a new object that lies in each of original objects. A point is the most common in a geometry. It is always defined, but may be empty. Incidence geometry defines an intersection as an object of lower dimension, incident to each of original objects. In this approach an intersection can be sometimes undefined, such as for parallel lines. In both cases the concept of intersection relies on logical conjunction. Algebraic geometry defines intersections in its own way with the intersection theory. There can be more than one primitive object, such as points that form an intersection. It can be understood ambiguously: either the intersection is all of them, or there are several intersection objects. Constructive solid geometry, Boolean Intersection is one of the ways of combining 2D/3D shapes Meet Weisstein, Eric W. "Intersection". MathWorld.
Intersection
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This circle (black) intersects this line (purple) in two points
24.
Circumscribed circle
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In geometry, the circumscribed circle or circumcircle of a polygon is a circle which passes through all the vertices of the polygon. The center of this circle is called the circumcenter and its radius is called the circumradius. A polygon which has a circumscribed circle is called a cyclic polygon. All regular simple polygons, all isosceles trapezoids, all triangles and all rectangles are cyclic. A related notion is the one of a minimum bounding circle, the smallest circle that completely contains the polygon within it. All triangles are cyclic; i.e. every triangle has a circumscribed circle. Since this equation has three parameters only three points' coordinate pairs are required to determine the equation of a circle. Since a triangle is defined by its three vertices, exactly three points are required to determine a circle, every triangle can be circumscribed. The circumcenter of a triangle can be constructed by drawing any two of the three perpendicular bisectors. The center is the point where the perpendicular bisectors intersect, the radius is the length to any of the three vertices. In coastal navigation, a triangle's circumcircle is sometimes used as a way of obtaining a position line using a sextant when no compass is available. The horizontal angle between two landmarks defines the circumcircle upon which the observer lies. Suppose that A = B = C = are the coordinates of points A, B, C. Using the polarization identity, these equations reduce to the condition that the matrix has a nonzero kernel. Thus the circumcircle may alternatively be described as the locus of zeros of the determinant of this matrix: det = 0.
Circumscribed circle
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Circumscribed circle, C, and circumcenter, O, of a cyclic polygon, P
25.
Line segment
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A closed segment includes both endpoints, while an open line segment excludes both endpoints; a half-open line segment includes exactly one of the endpoints. Examples of line segments include the sides of a square. When the end points both lie on a curve such as a circle, a segment is called a chord. Sometimes one needs to distinguish between "open" and "closed" line segments. Equivalently, a segment is the convex hull of two points. Thus, the segment can be expressed as a convex combination of the segment's two end points. Thus in R 2 the segment with endpoints A = and C = is the following collection of points:. A segment is a connected, non-empty set. More generally than above, the concept of a segment can be defined in an ordered geometry. A pair of line segments can be any one of the following: intersecting, parallel, none of these. Segments play an important role in other theories. For example, a set is convex if the segment that joins any two points of the set is contained in the set. This is important because it transforms some of the analysis of convex sets to the analysis of a segment. As a degenerate orbit this is a elliptic trajectory. In addition to appearing as the diagonals of polygons and polyhedra, line segments appear in numerous other locations relative to other geometric shapes.
Line segment
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historical image – create a line segment (1699)
26.
Equiangular polygon
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In Euclidean geometry, an equiangular polygon is a polygon whose vertex angles are equal. If the lengths of the sides are also equal then it is a regular polygon. Isogonal polygons are equiangular polygons which alternate two edge lengths. The only triangle is the equilateral triangle. Rectangles, including the square, are the only equiangular quadrilaterals. For a convex n-gon each internal angle is 180 °; this is the equiangular polygon theorem. A cyclic polygon is only if the alternate sides are equal. Thus if n is odd, a cyclic polygon is only if it is regular. For prime p, every integer-sided p-gon is regular. Moreover, every integer-sided pk-gon has p-fold rotational symmetry. Williams, R. The Geometrical Foundation of Natural Structure: A Source Book of Design. New York: Dover Publications, 1979. P. 32 A Property of Equiangular Polygons: What Is It About? A discussion of Viviani's theorem at Cut-the-knot.
Equiangular polygon
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An equiangular quadrilateral
27.
Bicentric polygon
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All regular polygons are bicentric. On the other hand, a rectangle with unequal sides is not bicentric, because no circle can be tangent to all four sides. Every triangle is bicentric. This is one version of Euler's formula. Not all quadrilaterals are bicentric. This condition is known as Fuss' theorem. Every regular polygon is bicentric. The radius of the inscribed circle is the apothem. The fact that it will always do so is implied by Poncelet's theorem, which more generally applies for inscribed and circumscribed conics. Weisstein, Eric W. "Bicentric polygon". MathWorld.
Bicentric polygon
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An equilateral triangle
28.
Tangential polygon
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In Euclidean geometry, a tangential polygon, also known as a circumscribed polygon, is a convex polygon that contains an inscribed circle. This is a circle, tangent to each of the polygon's sides. The dual polygon of a tangential polygon is a cyclic polygon, which has a circumscribed circle passing through each of its vertices. All triangles are tangential, as are all regular polygons with any number of sides. A well-studied group of tangential polygons are the tangential quadrilaterals, which include the kites. A polygon has an incircle if and only if all of its internal angle bisectors are concurrent. This common point is the incenter. If such a solution exists, then x1... xn are the tangent lengths of the polygon. But if n is even there are an infinitude of them. For a tangential polygon with an odd number of sides, all sides are only if all angles are equal. A tangential polygon with an even number of sides has all sides only if the alternate angles are equal. A tangential polygon has a smaller area than the same interior angles in the same sequence. In a tangential ABCDEF, the main diagonals AD, BE, CF are concurrent according to Brianchon's theorem. Circumgon
Tangential polygon
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A tangental trapezoid
29.
Apothem
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The apothem of a regular polygon is a line segment from the center to the midpoint of one of its sides. Equivalently, it is the line drawn from the center of the polygon, perpendicular to one of its sides. The word "apothem" can also refer to the length of that segment. Regular polygons are the only polygons that have apothems. Because of this, all the apothems in a polygon will be congruent. For a regular pyramid, the apothem is the height of a trapezoidal lateral face. A = n s a = p a 2. An apothem of a regular polygon will always be a radius of the inscribed circle. It is also the minimum distance between any side of its center. A = p a 2 = 2 = π r 2 The apothem of a regular polygon can be found multiple ways. The apothem can also be found by a = s 2 tan. These formulae can still be used even if only the perimeter p and the number of sides n are known because s = p n. Circumradius of a regular polygon Sagitta Chord Apothem of a regular polygon With interactive animation Apothem of pyramid or truncated pyramid Pegg, Jr. Ed. "Sagitta, Chord". The Wolfram Demonstrations Project.
Apothem
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Apothem of a hexagon
30.
Inscribed figure
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In geometry, an inscribed planar shape or solid is one, enclosed by and "fits snugly" inside another geometric shape or solid. To say that "F is inscribed in figure G" means precisely the same thing as "figure G is circumscribed about figure F". A ellipse inscribed in a convex polygon is tangent to every side or face of the outer figure. Triangles or regular polygons inscribed in circles. A circle inscribed in any polygon is called its incircle, in which case the polygon is said to be a tangential polygon. The circle is said to be its circumscribed circle or circumcircle. The inradius or filling radius of a given outer figure is the radius of the inscribed sphere, if it exists. Every triangle can be inscribed in some circle. Every triangle has an inscribed circle, called the incircle. Every regular polygon can be inscribed in some circle. Every circle can be inscribed in some regular polygon of n sides, for any n ≥ 3. Not every polygon with more than three sides has an inscribed circle; those polygons that do are called tangential polygons. Not every polygon with more than three sides is an inscribed polygon of a circle; those polygons that are so inscribed are called cyclic polygons. Every triangle can be inscribed in an ellipse, called its Steiner circumellipse or simply its Steiner ellipse, whose center is the triangle's centroid. Every triangle has an infinitude of inscribed ellipses.
Inscribed figure
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Inscribed circles of various polygons
31.
Angle
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This plane does not have to be a Euclidean plane. Angles are also formed by the intersection of two planes in other spaces. These are called dihedral angles. Angles formed by the intersection of two curves in a plane are defined as the angle determined at the point of intersection. Angle is also used to designate the measure of an angle or of a rotation. This measure is the ratio of the length of a circular arc to its radius. In the case of a geometric angle, the arc is delimited by the sides. The angle comes from the Latin word angulus, meaning "corner"; cognate words are the Greek ἀγκύλος, meaning "crooked, curved," and the English word "ankle". Both are connected with * ank -, meaning "to bend" or "bow". According to Proclus an angle must be a relationship. In mathematical expressions, it is common to use Greek letters to serve as variables standing for the size of some angle. Lower Roman letters are also used, as are upper case Roman letters in the context of polygons. See the figures in this article for examples. In geometric figures, angles may also be identified by the labels attached to the three points that define them. For example, the angle at vertex A enclosed by the rays AB and AC is denoted B A C ^.
Angle
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An angle enclosed by rays emanating from a vertex.
32.
Rotational symmetries
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An object's degree of rotational symmetry is the number of distinct orientations in which it looks the same. Formally the rotational symmetry is symmetry in Euclidean space. Rotations are direct isometries, i.e. isometries preserving orientation. Therefore a symmetry group of rotational symmetry is a subgroup of E+. With the modified notion of symmetry for vector fields the symmetry group can also be E+. For symmetry with respect to rotations about a point we can take that point as origin. These rotations form the special orthogonal group SO, the group of m×m orthogonal matrices with determinant 1. For m = 3 this is the rotation group SO. For chiral objects it is the same as the full symmetry group. Laws of physics are SO-invariant if they do not distinguish different directions in space. Because of Noether's theorem, rotational symmetry of a physical system is equivalent to the law. Note that "1-fold" symmetry is no symmetry. The notation for n-fold symmetry is Cn or simply "n". The actual symmetry group is specified by the point or axis of symmetry, together with the n. For each point or axis of symmetry, the abstract group type is cyclic group of order n, Zn.
Rotational symmetries
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The starting position in shogi
Rotational symmetries
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The triskelion appearing on the Isle of Man flag.
33.
Reflection symmetries
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Reflection symmetry, line symmetry, mirror symmetry, mirror-image symmetry, is symmetry with respect to reflection. That is, a figure which does not change upon undergoing a reflection has reflectional symmetry. In 2D there is a line/axis of symmetry, in 3D a plane of symmetry. An figure, indistinguishable from its transformed image is called mirror symmetric. The set of operations that preserve a given property of the object form a group. Thus a square has four axes of symmetry, because there are four different ways to have the edges all match. A circle has many axes of symmetry. Triangles with symmetry are isosceles. Quadrilaterals with symmetry are kites, deltoids, rhombuses, isosceles trapezoids. All even-sided polygons have two simple reflective forms, one through edges. For an arbitrary shape, the axiality of the shape measures how close it is to being bilaterally symmetric. It equals 1 for between 2/3 and 1 for any convex shape. For each plane of reflection, the symmetry group is isomorphic with Cs, one of the three types of order two, hence algebraically C2. The fundamental domain is a half-space. In certain contexts there is rotational well as reflection symmetry.
Reflection symmetries
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Many animals, such as this spider crab Maja crispata, are bilaterally symmetric.
Reflection symmetries
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Figures with the axes of symmetry drawn in. The figure with no axes is asymmetric.
Reflection symmetries
–
Mirror symmetry is often used in architecture, as in the facade of Santa Maria Novella, Venice, 1470.
34.
Dihedral group
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In mathematics, a dihedral group is the group of symmetries of a regular polygon, which includes rotations and reflections. They play an important role in group theory, geometry, chemistry. The notation for the dihedral group of n differs in geometry and abstract algebra. In geometry, Dn or Dihn refers to the symmetries of a group of order 2n. In abstract algebra, Dn refers to the dihedral group of n. The geometric convention is used in this article. A regular polygon with n sides has 2 different symmetries: n rotational symmetries and n reflection symmetries. Usually, we take 3 here.The associated rotations and reflections make up the dihedral group D n. If n is odd, each axis of symmetry connects the midpoint of one side to the opposite vertex. If n is even, there are n/2 axes of symmetry connecting the midpoints of opposite sides and n / 2 axes of symmetry connecting opposite vertices. In either case, there are 2 n elements in the symmetry group. Reflecting in one axis of symmetry followed by reflecting in another axis of symmetry produces a rotation through twice the angle between the axes. As with any geometric object, the composition of two symmetries of a regular polygon is again a symmetry of this object. With composition of symmetries to produce another as the binary operation, this gives the algebraic structure of a finite group. The following Cayley table shows the effect of composition in the D3.
Dihedral group
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The symmetry group of a snowflake is Dih 6, a dihedral symmetry, the same as for a regular hexagon.
35.
Triangle
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A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices A, B, C is denoted △ A B C. In Euclidean geometry any three points, when non-collinear, determine a unique plane. This article is about triangles in Euclidean geometry except where otherwise noted. Triangles can be classified according to the lengths of their sides: An equilateral triangle has the same length. An equilateral triangle is also a regular polygon with all angles measuring 60°. An isosceles triangle has two sides of equal length. Some mathematicians define an isosceles triangle to have exactly two equal sides, whereas others define an isosceles triangle as one with at least two equal sides. The latter definition would make all equilateral isosceles triangles. The 45 -- 45 -- 90 right triangle, which appears in the square tiling, is isosceles. A scalene triangle has all its sides of different lengths. Equivalently, it has all angles of different measure. Hatch marks, also called tick marks, are used in diagrams of triangles and geometric figures to identify sides of equal lengths. In a triangle, the pattern is usually no more than 3 ticks.
Triangle
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The Flatiron Building in New York is shaped like a triangular prism
Triangle
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A triangle
36.
Square (geometry)
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In geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles. It can also be defined as a rectangle in which two adjacent sides have equal length. A square with vertices ABCD would be denoted ◻ ABCD. Opposite sides of a square are both equal in length. All four angles of a square are equal. All four sides of a square are equal. The diagonals of a square are equal. The square is the n = 2 case of the families of n-orthoplexes. A square has Schläfli symbol. T, is an octagon. H, is a digon. The area A is A = ℓ 2. In classical times, the second power was described in terms of the area of a square, as in the above formula. This led to the use of the square to mean raising to the second power. The area can also be calculated using the diagonal d according to A = d 2.
Square (geometry)
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A regular quadrilateral (tetragon)
37.
Equilateral
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In geometry, an equilateral triangle is a triangle in which all three sides are equal. In the familiar Euclidean geometry, equilateral triangles are also equiangular;, all three internal angles are each 60 °. They can therefore also be referred to as regular triangles. Thus these are properties that are unique to equilateral triangles. The three medians have equal lengths. The three angle bisectors have equal lengths. For some pairs of triangle centers, the fact that they coincide is enough to ensure that the triangle is equilateral. In particular: A triangle is equilateral if any two of incenter, centroid, or orthocenter coincide. It is also equilateral if its incenter coincides with its nine-point center. For any triangle, the three medians partition the triangle into six smaller triangles. A triangle is only if any three of the smaller triangles have either the same perimeter or the same inradius. A triangle is only if the circumcenters of any three of the smaller triangles have the same distance from the centroid. Morley's theorem states that, in any triangle, the three points of intersection of the adjacent angle trisectors form an equilateral triangle. A version of the isoperimetric inequality for triangles states that the triangle of greatest area with a given perimeter is equilateral. That is, PA, PB, PC satisfy the inequality that any two of them sum to at least as great as the third.
Equilateral
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A regular tetrahedron is made of four equilateral triangles.
Equilateral
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Equilateral triangle
38.
Tessellation
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A tessellation of a flat surface is the tiling of a plane using one or more geometric shapes, called tiles, with no overlaps and no gaps. In mathematics, tessellations can be generalized to higher dimensions and a variety of geometries. A periodic tiling has a repeating pattern. The patterns formed by periodic tilings can be categorized into 17 wallpaper groups. A tiling that lacks a repeating pattern is called "non-periodic". An aperiodic tiling uses a small set of tile shapes that cannot form a repeating pattern. In the geometry of higher dimensions, a space-filling or honeycomb is also called a tessellation of space. A real physical tessellation is a tiling made of materials such as cemented ceramic squares or hexagons. Such tilings may be decorative patterns, or may have functions such as providing durable and water-resistant pavement, floor or wall coverings. Historically, tessellations were used in Ancient Rome and in Islamic art such as in the decorative geometric tiling of the Alhambra palace. In the twentieth century, the work of M. C. Escher often made use of tessellations, both in ordinary Euclidean geometry and in hyperbolic geometry, for artistic effect. Tessellations are sometimes employed for decorative effect in quilting. Tessellations form a class of patterns in nature, for example in the arrays of hexagonal cells found in honeycombs. Tessellations were used by the Sumerians in building wall decorations formed by patterns of clay tiles.
Tessellation
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Zellige terracotta tiles in Marrakech, forming edge-to-edge, regular and other tessellations
Tessellation
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A wall sculpture at Leeuwarden celebrating the artistic tessellations of M. C. Escher
Tessellation
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A temple mosaic from the ancient Sumerian city of Uruk IV (3400–3100 BC), showing a tessellation pattern in coloured tiles
Tessellation
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Roman geometric mosaic
39.
Beehive (beekeeping)
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A beehive is an enclosed structure in which some honey bee species of the subgenus Apis live and raise their young. Natural beehives are naturally occurring structures occupied by honeybee colonies, such as hollowed-out trees, while domesticated honeybees live in man-made beehives, often in an apiary. These man-made structures are typically referred to as "beehives". Only the eastern bee are domesticated by humans. A natural beehive is comparable to a bird's nest built with a purpose to protect the dweller. The beehive's internal structure is a densely packed group of hexagonal cells made of beeswax, called a honeycomb. The bees use the cells to store food and to house the "brood". Artificial hives are commonly transported so that bees can pollinate crops in other areas. A number of patents have been issued for beehive designs. Honey bees use hollow trees as nesting sites. Members of other subgenera have exposed aerial combs. The nest is composed of multiple honeycombs, parallel to each other, with a relatively uniform bee space. It usually has a single entrance. Western honey bees prefer nest cavities approximately 45 litres in volume and avoid those smaller than 10 or larger than 100 litres. Bees usually occupy nests for several years.
Beehive (beekeeping)
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Painted wooden beehives with active honey bees
Beehive (beekeeping)
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Natural beehive in the hollow of a tree
Beehive (beekeeping)
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Hives from the collection of Radomysl Castle, Ukraine, 19th century
Beehive (beekeeping)
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Beehives – watercolour painted by Stanisław Masłowski in Wola Rafałowska village, Poland in 1924, Silesian Museum in Katowice, Poland
40.
Honeycomb
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A honeycomb is a mass of hexagonal wax cells built by honey bees in their nests to contain their larvae and stores of honey and pollen. Beekeepers may remove the entire honeycomb to honey. The structure of the comb may be left basically intact when honey is extracted by uncapping and spinning in a centrifugal machine -- the honey extractor. If the honeycomb is too worn out, the wax can be reused including making sheets of comb foundation with hexagonal pattern. Honeycomb in the "supers" that are not allowed to be used for brood stays light-coloured. However, the term "honeycomb" is not often used for such structures. The nonangled rows of honeycomb cells are always horizontally aligned. Thus, each cell has two vertical walls, with "floors" and "ceilings" composed of two angled walls. The cells slope slightly upwards, towards the open ends. Two possible explanations exist as to why honeycomb is composed of hexagons, rather than any other shape. First, the hexagonal tiling creates a partition with equal-sized cells, while minimizing the total perimeter of the cells. Known as the honeycomb conjecture, this was given by Jan Brożek and proved much later by Thomas Hales. Thus, a hexagonal structure uses the least material to create a lattice of cells within a given volume. In support of this, he notes that queen cells, which are constructed singly, are lumpy with no apparent attempt at efficiency. The closed ends of the honeycomb cells are also an example of geometric efficiency, albeit little-noticed.
Honeycomb
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Honeycomb
Honeycomb
–
A natural comb of Apis dorsata. The lower part of the comb has a number of unoccupied cells.
Honeycomb
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"Artificial honeycomb" plate where bees have already completed some cells
Honeycomb
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Natural honeycombs on a building
41.
Voronoi diagram
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In mathematics, a Voronoi diagram is a partitioning of a plane into regions based on distance to points in a specific subset of the plane. These regions are called Voronoi cells. The Voronoi diagram of a set of points is dual to its Delaunay triangulation. It is also called a Voronoi tessellation, a Voronoi decomposition, a Voronoi partition, or a Dirichlet tessellation. Voronoi diagrams have theoretical applications to a large number of fields, mainly in science and technology but also including visual art. They are also known as Thiessen polygons. In the simplest case, shown in the first picture, we are given a finite set of points in the Euclidean plane. Hence it is a convex polygon. The line segments of the Voronoi diagram are all the points in the plane that are equidistant to the two nearest sites. The Voronoi vertices are the points equidistant to three sites. Let X be a metric space with distance function d. Let K be a set of indices and let k ∈ K be a tuple of nonempty subsets in the space X. Usually they are assumed to be disjoint. In addition, many sites are allowed in the definition, but again, in many cases only finitely many sites are considered. Sometimes the combinatorial structure is referred to as the Voronoi diagram.
Voronoi diagram
–
John Snow's original diagram
Voronoi diagram
–
20 points and their Voronoi cells (larger version below).
Voronoi diagram
–
Periodic
Voronoi diagram
42.
Diagonal
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In geometry, a diagonal is a line segment joining two vertices of a polygon or polyhedron, when those vertices are not on the same edge. Informally, any sloping line is called diagonal. In algebra, a diagonal of a square matrix is a set of entries extending from one corner to the farthest corner. There are also non-mathematical uses. As applied to a polygon, a diagonal is a segment joining any two non-consecutive vertices. Therefore, a quadrilateral has two diagonals, joining opposite pairs of vertices. For re-entrant polygons, some diagonals are outside of the polygon. For n-gons with n=3. 4... the number of regions is 1, 4, 11, 25, 50, 91, 154, 246... This is OEIS sequence A006522. The off-diagonal entries are those not on the main diagonal. A diagonal matrix is one whose off-diagonal entries are all zero. A superdiagonal entry is one, directly to the right of the main diagonal. Just as diagonal entries are those A i j with j = i, the superdiagonal entries are those with 1. In geometric studies, the idea of intersecting the diagonal with itself is not directly, but by perturbing it within an equivalence class.
Diagonal
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A stand of basic scaffolding on a house construction site, with diagonal braces to maintain its structure
Diagonal
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The diagonals of a cube with side length 1. AC' (shown in blue) is a space diagonal with length, while AC (shown in red) is a face diagonal and has length.
43.
Circumradius
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In geometry, the circumscribed circle or circumcircle of a polygon is a circle which passes through all the vertices of the polygon. Its radius is called the circumradius. A polygon which has a circumscribed circle is called a cyclic polygon. All regular simple polygons, all isosceles trapezoids, all rectangles are cyclic. A related notion is the one of a bounding circle, the smallest circle that completely contains the polygon within it. All triangles are cyclic; i.e. every triangle has a circumscribed circle. Since this equation has three parameters only three points' coordinate pairs are required to determine the equation of a circle. Since exactly three points are required to determine a circle, every triangle can be circumscribed. The circumcenter of a triangle can be constructed by drawing any two of the three perpendicular bisectors. The radius is the length to any of the three vertices. In coastal navigation, a triangle's circumcircle is sometimes used as a way of obtaining a line using a sextant when no compass is available. The horizontal angle between two landmarks defines the circumcircle upon which the observer lies. Suppose that A = B = C = are the coordinates of points A, B, C. Using the polarization identity, these equations reduce to the condition that the matrix has a nonzero kernel. Thus the circumcircle may alternatively be described as the locus of zeros of the determinant of this matrix: det = 0.
Circumradius
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Circumscribed circle, C, and circumcenter, O, of a cyclic polygon, P
44.
Inscribed
–
In geometry, an inscribed planar shape or solid is one, enclosed by and "fits snugly" inside another geometric shape or solid. To say that "F is inscribed in figure G" means precisely the same thing as "figure G is circumscribed about F". A ellipse inscribed in a polygon is tangent to every side or face of the outer figure. Regular polygons inscribed in circles. A circle inscribed in any polygon is called its incircle, in which case the polygon is said to be a tangential polygon. The circle is said to circumcircle. The inradius or filling radius of a given outer figure is the radius of the inscribed circle or sphere, if it exists. Every circle has an inscribed triangle with any three given angle measures, every triangle can be inscribed in some circle. Every triangle has an inscribed circle, called the incircle. Every circle has an inscribed regular polygon of n sides, for any n≥3, every regular polygon can be inscribed in some circle. Every circle can be inscribed for any n ≥ 3. Not every polygon with more than three sides has an inscribed circle; those polygons that do are called tangential polygons. Not every polygon with more than three sides is an inscribed polygon of a circle; those polygons that are so inscribed are called cyclic polygons. Every triangle can be inscribed in an ellipse, called its Steiner circumellipse or simply its Steiner ellipse, whose center is the triangle's centroid. Every triangle has an infinitude of inscribed ellipses.
Inscribed
–
Inscribed circles of various polygons
45.
Inradius
–
In geometry, the incircle or inscribed circle of a triangle is the largest circle contained in the triangle; it touches the three sides. The center of the incircle is a center called the triangle's incenter. Every triangle has each tangent to one of the triangle's sides. The center of the incircle, called the incenter, can be found as the intersection of the three internal angle bisectors. The center of an excircle is the intersection of the external bisectors of the other two. The center of this excircle is called the excenter relative to the vertex A, or the excenter of A. Polygons with more than three sides do not all have an incircle tangent to all sides; those that do are called tangential polygons. See also Tangent lines to circles. The radii of the excircles are closely related to the area of the triangle. Suppose △ A B C has an incircle with radius r and center I. Let a be the length of BC, b the length of AC, c the length of AB. Now, the incircle is tangent to AB at some point C′, so ∠ A C ′ I is right. Thus the radius C'I is an altitude of △ I A B. Therefore, △ I A B has base length c and height r, so has area 1 2 c r. Similarly, △ I A C has area 1 2 b r and △ I B C has area 1 2 a r.
Inradius
–
A triangle (black) with incircle (blue), incenter (I), excircles (orange), excenters (J A,J B,J C), internal angle bisectors (red) and external angle bisectors (green)
46.
Reflection symmetry
–
Reflection symmetry, line symmetry, mirror symmetry, mirror-image symmetry, is symmetry with respect to reflection. That is, a figure which does not change upon undergoing a reflection has reflectional symmetry. In 2D there is a line/axis of symmetry, in 3D a plane of symmetry. An object or figure, indistinguishable from its transformed image is called mirror symmetric. The set of operations that preserve a given property of the object form a group. Thus a square has four axes of symmetry, because there are four different ways to fold it and have the edges all match. A circle has infinitely many axes of symmetry. Triangles with reflection symmetry are isosceles. Quadrilaterals with symmetry are kites, deltoids, isosceles trapezoids. All even-sided polygons have one through edges. For an arbitrary shape, the axiality of the shape measures how close it is to being bilaterally symmetric. It equals 1 for between 1 for any convex shape. For each plane of reflection, the group is isomorphic with Cs, one of the three types of order two, hence algebraically C2. The fundamental domain is a half-plane or half-space. In certain contexts there is rotational well as symmetry.
Reflection symmetry
–
Many animals, such as this spider crab Maja crispata, are bilaterally symmetric.
Reflection symmetry
–
Figures with the axes of symmetry drawn in. The figure with no axes is asymmetric.
Reflection symmetry
–
Mirror symmetry is often used in architecture, as in the facade of Santa Maria Novella, Venice, 1470.
47.
Cyclic group
–
In algebra, a cyclic group or monogenous group is a group, generated by a single element. Each element can be written in additive notation. This g is called a generator of the group. Every cyclic group is isomorphic to the additive group of Z, the integers. Every finite cyclic group of order n is isomorphic to the additive group of Z/nZ, the integers modulo n. Every cyclic group is an abelian group, every finitely generated abelian group is a direct product of cyclic groups. A group G is called cyclic if there exists an g in G such that G = ⟨ g ⟩ =. For example, G is cyclic. In fact, G is essentially the same as the set with modulo 6. One can use the χ defined by χ = i. The name "cyclic" may be misleading: it is possible to generate many elements and not form any literal cycles;, every gn is distinct. A group generated in this way is isomorphic to the additive group of the integers. The set with the operation of addition, forms a group. It is an cyclic group, because all integers can be written as a finite sum or difference of copies of the number 1. In this group, 1 and 1 are the only generators.
Cyclic group
–
Example cyclic groups in n-dimensional symmetry
Cyclic group
–
Algebraic structure → Group theory Group theory
48.
John Horton Conway
–
John Horton Conway FRS is an English mathematician active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory. He has also contributed to many branches of recreational mathematics, notably the invention of the cellular automaton called the Game of Life. Conway is currently Professor Emeritus of Mathematics at Princeton University in New Jersey. Conway was born in the son of Cyril Horton Conway and Agnes Boyce. By the age of eleven his ambition was to become a mathematician. After leaving secondary school, Conway entered Gonville and Caius College, Cambridge to study mathematics. Conway, a "terribly introverted adolescent" in school, interpreted his admission as an opportunity to transform himself into a new person: an "extrovert". He began to undertake research in number theory supervised by Harold Davenport. Having solved the open problem posed on writing numbers as the sums of fifth powers, Conway began to become interested in infinite ordinals. He was appointed as College Fellow and Lecturer in Mathematics at the University of Cambridge. After leaving Cambridge in 1986, he took up the appointment at Princeton University. Among amateur mathematicians, he is perhaps most widely known to combinatorial game theory, a theory of partisan games. This he developed with them also co-authored the book Winning Ways for your Mathematical Plays. He also wrote the book On Numbers and Games which lays out the mathematical foundations of CGT. He is also one of the inventors of sprouts, well as philosopher's football.
John Horton Conway
–
John Conway
49.
Elongation (geometry)
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In geometry, a Johnson solid is a strictly convex polyhedron, not uniform, each face of, a regular polygon. There is no requirement that the same polygons join around each vertex. An example of a Johnson solid is the square-based pyramid with equilateral sides; it has 1 square face and 4 triangular faces. As in any strictly convex solid, the total of their angles is less than 360 degrees. Since a regular polygon has angles at least 60 degrees, it follows that at most five faces meet at any vertex. The pentagonal pyramid is an example that actually has a degree-5 vertex. In 1966, Norman Johnson published a list which gave them their names and numbers. He did conjecture that there were no others. Victor Zalgaller in 1969 proved that Johnson's list was complete. However, it is not vertex-transitive, as it has different isometry at different vertices, making a Johnson solid rather than an Archimedean solid. The names are more descriptive than they sound. Most of the Johnson solids can be constructed from the first few, together with the Platonic and Archimedean solids, antiprisms. Bi- means that two copies of the solid in question are joined base-to-base. For rotundae, they can be joined so that like faces or unlike faces meet. In this nomenclature, an octahedron would be a square bipyramid, an icosidodecahedron would be a pentagonal gyrobirotunda.
Elongation (geometry)
–
The elongated square gyrobicupola (J 37), a Johnson solid
50.
Rhombus
–
In Euclidean geometry, a rhombus is a simple quadrilateral whose four sides all have the same length. Another name is equilateral quadrilateral, since equilateral means that all of its sides are equal in length. Every rhombus is a kite. A rhombus with right angles is a square. The word "rhombus" comes from Greek ῥόμβος, meaning "to turn round and round". The word was used both by Euclid and Archimedes, who used the term "solid rhombus" for two circular cones sharing a common base. The surface we refer to as rhombus today is a cross section of this solid rhombus through the apex of each of the two cones. The vertices are at and. This is a special case of the superellipse, with exponent 1. Every rhombus has two diagonals connecting two pairs of parallel sides. Using congruent triangles, one can prove that the rhombus is symmetric across each of these diagonals. It follows that any rhombus has the following properties: Opposite angles of a rhombus have equal measure. The two diagonals of a rhombus are perpendicular;, a rhombus is an orthodiagonal quadrilateral. Its diagonals bisect opposite angles. The first property implies that every rhombus is a parallelogram.
Rhombus
–
Some polyhedra with all rhombic faces
Rhombus
–
Two rhombi.
Rhombus
51.
Kite (geometry)
–
In Euclidean geometry, a kite is a quadrilateral whose four sides can be grouped into two pairs of equal-length sides that are adjacent to each other. In contrast, they are opposite to each other rather than adjacent. Kite quadrilaterals are named for flying kites, which often have this shape and which are in turn named for a bird. The word "deltoid" may also refer to a deltoid curve, an unrelated geometric object. The word "kite" is often restricted to the convex variety. A concave kite is a type of pseudotriangle. If all four sides of a kite have the same length, it must be a rhombus. If a kite is equiangular, meaning that all four of its angles are equal, then it must also thus a square. A kite with three equal 108 ° angles and the convex hull of the lute of Pythagoras. The kites that are also cyclic quadrilaterals are exactly the ones formed from two congruent right triangles. That is, for these kites the two equal angles on opposite sides of the symmetry axis are each 90 degrees. They are in fact bicentric quadrilaterals. Among all the bicentric quadrilaterals with a given two radii, the one with maximum area is a right kite. The tiling that it produces by its reflections is the trihexagonal tiling. Among all quadrilaterals, the shape that has the greatest ratio of its perimeter to its diameter is an equidiagonal kite with 12, 5π / 6, 5π / 12.
Kite (geometry)
–
V4.3.4.3
Kite (geometry)
–
A kite showing its sides equal in length and its inscribed circle.
Kite (geometry)
–
V4.3.4.4
Kite (geometry)
–
V4.3.4.5
52.
Parallelogon
–
A parallelogon is a polygon such that images of the polygon under translations only tile the plane when fitted together along entire sides. Opposite sides must be equal in length and parallel. A less obvious restriction is that a parallelogon can only have six sides; a four-sided parallelogon is a parallelogram. In general a parallelogon has rotational symmetry around its center. Hexagonal parallelogons each have varied geometric symmetric forms. In general they all have order 2. Hexagonal parallelogons enable the possibility of nonconvex polygons. Parallelogram can tile the plane as a distorted square tiling while hexagonal parallelogon can tiling the plane as a regular hexagonal tiling. Tilings and Patterns. New York: W. H. Freeman. ISBN 0-7167-1193-1. CS1 maint: Multiple names: authors list list of 107 isohedral tilings, p.473-481 Fedorov's Five Parallelohedra
Parallelogon
–
A parallelogon is constructed by two or three pairs of parallel line segments. The vertices and edges on the interior of the hexagon are suppressed.
53.
Directed edge
–
In mathematics, more specifically in graph theory, a directed graph is a graph, where the edges have a direction associated with them. More specifically, these entities are addressed as directed multigraphs. More specifically, directed graphs without loops are addressed as simple directed graphs, while directed graphs with loops are addressed as loop-digraphs. Symmetric directed graphs are directed graphs where all edges are bidirected. Simple directed graphs are directed graphs that have no multiple arrows with same target nodes. As already introduced, in case of multiple arrows the entity is usually addressed as directed multigraph. It follows that a complete digraph is symmetric. Oriented graphs are directed graphs having no bidirected edges. Directed acyclic graphs are directed graphs with no directed cycles. Multitrees are DAGs in which no two directed paths from a starting vertex meet back at the same vertex. Oriented polytrees are DAGs formed by orienting the edges of acyclic graphs. Rooted trees are oriented trees in which all edges of the underlying undirected tree are directed away from the roots. Rooted directed graphs are digraphs in which a vertex has been distinguished as the root. Control flow graphs are rooted digraphs used as a representation of the paths that might be traversed during its execution. Signal-flow graphs are directed graphs in which nodes represent system variables and branches represent functional connections between pairs of nodes.
Directed edge
–
A directed graph.
54.
Hexagonal tiling
–
In geometry, the hexagonal tiling or hexagonal tessellation is a regular tiling of the Euclidean plane, in which three hexagons meet at each vertex. It has Schläfli symbol of or t. Conway calls it a hextille. The internal angle of the hexagon is 120 degrees so three hexagons at a point make a full 360 degrees. It is one of three regular tilings of the plane. The other two are the triangular tiling and the square tiling. The hexagonal tiling is the densest way to arrange circles in two dimensions. The Honeycomb conjecture states that the hexagonal tiling is the best way to divide a surface into regions of equal area with the least total perimeter. The optimal three-dimensional structure for making honeycomb was investigated by Lord Kelvin, who believed that the Kelvin structure is optimal. However, the less regular Weaire–Phelan structure is slightly better. This structure exists naturally in the form of graphite, where each sheet of graphene resembles wire, with strong carbon bonds. Tubular graphene sheets have been synthesised; these are known as carbon nanotubes. They have potential applications, due to electrical properties. Silicene is similar. Chicken wire consists of a hexagonal lattice of wires.
Hexagonal tiling
–
Chicken wire fencing
Hexagonal tiling
–
Graphene
Hexagonal tiling
–
Periodic
Hexagonal tiling
55.
Dodecagon
–
In geometry, a dodecagon or 12-gon is any twelve-sided polygon. A regular dodecagon is a closed figure with internal angles of the same size. It has twelve lines of reflective symmetry and symmetry of order 12. A regular dodecagon can be constructed as a truncated hexagon, t, or a twice-truncated triangle, tt. The internal angle at each vertex of a regular dodecagon is °. As 12 = 22 × 3, regular dodecagon is constructible using compass and straightedge: Coxeter states that every parallel-sided 2m-gon can be divided into m/2 rhombs. It can be divided into 15 rhombs, with one example shown below. This decomposition is based with 15 of 240 faces. One of the ways the manipulative pattern blocks are used is in creating a number of different dodecagons. The regular dodecagon has order 24. There are 15 distinct subgroup cyclic symmetries. Each symmetry allows one or more degrees of freedom for irregular forms. Only the g12 subgroup has no degrees of freedom but can seen as directed edges. The interior of such an dodecagon is not generally defined. A skew zig-zag dodecagon has vertices alternating between two parallel planes.
Dodecagon
–
pattern blocks
Dodecagon
–
A regular dodecagon
Dodecagon
–
The Vera Cruz church in Segovia
Dodecagon
–
A 1942 British threepence, reverse
56.
Alternation (geometry)
–
In geometry, an alternation or partial truncation, is an operation on a polygon, polyhedron, tiling, or higher dimensional polytope that removes alternate vertices. Coxeter labels an alternation by a prefixed by an h, standing for half. Because alternation reduce all polygon faces to half as many sides, it can only be applied for polytopes with all even-sided faces. An square face becomes a digon, being degenerate, is usually reduced to a single edge. Tiling with a vertex configuration consisting of all even-numbered elements can be alternated. For example, the alternation a vertex figure with.2 b. 2c is a.3.b.3.c.3 where the three is the number of elements in this vertex figure. A special case is square faces whose order divide into degenerate digons. A snub can be seen as an alternation of a truncated truncated quasiregular polyhedron. In general a polyhedron can be snubbed if its truncation has only even-sided faces. All rectified polyhedra can be snubbed, not just from regular polyhedra. The square antiprism is an example of a general snub, can be represented by ss, with the square antiprism, s. In general most of the results of this operation will not be uniform. Examples: Honeycombs An alternated cubic honeycomb is the tetrahedral-octahedral honeycomb. An hexagonal prismatic honeycomb is the gyrated alternated cubic honeycomb. 4-polytope An truncated 24-cell is the snub 24-cell.
Alternation (geometry)
–
Two snub cubes from truncated cuboctahedron See that red and green dots are placed at alternate vertices. A snub cube is generated from deleting either set of vertices, one resulting in clockwise gyrated squares, and other counterclockwise.
57.
Stellation
–
The new figure is a stellation of the original. The stellation comes from the Latin stellātus, "starred", which in turn comes from Latin stella, "star". He stellated the regular dodecahedron to obtain two regular star polyhedra, great stellated dodecahedron. He also stellated the regular octahedron to obtain a regular compound of two tetrahedra. Stellating a regular symmetrically creates a regular star polygon or polygonal compound. These polygons are characterised by the number of times m that the polygonal boundary winds around the centre of the figure. Like all regular polygons, their vertices lie on a circle. M also corresponds to the number of vertices around the circle to get from one end of a given edge to the other, starting at 1. Making m = 1 gives the convex. If m do have a common divisor, then the figure is a regular compound. For example is the regular compound of hexagram, while is a compound of two pentagrams. Some authors use the Schläfli symbol for regular compounds. / 2 stellations if n is odd. Like the heptagon, the octagon also has one, being a star polygon, the other, being the compound of two squares. A polyhedron is stellated by extending the face planes of a polyhedron until they meet again to form a new polyhedron or compound.
Stellation
–
Magnus Wenninger with some of his models of stellated polyhedra in 2009
Stellation
–
Construction of a stellated dodecagon: a regular polygon with Schläfli symbol {12/5}.
Stellation
–
Marble floor mosaic by Paolo Uccello, Basilica of St Mark, Venice, c. 1430
58.
Hexagram
–
A hexagram or sexagram is a six-pointed geometric star figure with the Schläfli symbol, 2, or. It is the compound of two equilateral triangles. The intersection is a regular hexagon. It is used for example in Hanafism, Jewish identity, Hinduism and occultism. In mathematics, the system for the simple Lie group G2 is in the form of a hexagram, with 6 long roots and 6 short roots. A six-pointed star, like a regular hexagon, can be created using a straight edge: Make a circle of any size with the compass. With the pivot on the last point found, similarly repeat until six such points have been marked. With a straight edge, join alternate points on the circumference to form two equilateral triangles. The hexagram is a symbol called satkona yantra or sadkona yantra found on ancient South Indian Hindu temples. It symbolizes if maintained, results in "moksha," or "nirvana". Some researchers have theorized that the hexagram represents the astrological chart at the time of David's anointment as king. The hexagram is also known as the "King's Star" in astrological circles. Curiously the hexagram is not found among these signs. Six pointed stars have also been found in cosmological diagrams in Hinduism, Buddhism, Jainism. The reasons behind this symbol's common appearance in the West are unknown.
Hexagram
–
Diagram showing the two mystic syllables Om and Hrim
Hexagram
–
A regular hexagram
Hexagram
–
The Star of David in the oldest surviving complete copy of the Masoretic text, the Leningrad Codex, dated 1008.
Hexagram
–
Star of David on the Salt Lake Assembly Hall
59.
Triangular tiling
–
In geometry, the triangular tiling or triangular tessellation is one of the three regular tilings of the Euclidean plane. Because the internal angle of the equilateral triangle is 60 degrees, six triangles at a point occupy a full 360 degrees. The triangular tiling has Schläfli symbol of. Conway calls a deltille, named from the shape of the Greek letter delta. It is one of three regular tilings of the plane. The other two are the square tiling and the hexagonal tiling. There are 9 distinct uniform colorings of a triangular tiling. There is one class of Archimedean colorings, 111112, not 1-uniform, containing alternate rows of triangles where every third is colored. There are infinitely such Archimedean colorings that can be created by arbitrary horizontal shifts of the rows. The arrangement of the tiling is called an A2 lattice. It is the 2-dimensional case of a simplectic honeycomb. The A* 2 lattice can be constructed by the union of all three A2 lattices, equivalent to the A2 lattice. + + = dual of the tiling are the centers of the densest possible circle packing. Every circle is in contact with 6 other circles in the packing. The density is π ⁄ √ 90.69 %.
Triangular tiling
–
Periodic
Triangular tiling
–
Triangular tiling
Triangular tiling
60.
Square
–
In geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles. It can also be defined as a rectangle in which two adjacent sides have equal length. A square with vertices ABCD would be denoted ◻ ABCD. Opposite sides of a square are both equal in length. All four angles of a square are equal. All four sides of a square are equal. The diagonals of a square are equal. The square is the n = 2 case of the families of n-orthoplexes. A square has Schläfli symbol. T, is an octagon. H, is a digon. The area A is A = ℓ 2. In classical times, the second power was described in terms of the area of a square, as in the above formula. This led to the use of the square to mean raising to the second power. The area can also be calculated using the diagonal d according to A = d 2.
Square
–
A regular quadrilateral (tetragon)
61.
Rhombitrihexagonal tiling
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In geometry, the rhombitrihexagonal tiling is a semiregular tiling of the Euclidean plane. There are one triangle, one hexagon on each vertex. It has Schläfli symbol of rr. John Conway calls a rhombihexadeltille. It can be considered a cantellated by an expanded hexagonal tiling by Alicia Boole Stott's operational language. There are 3 regular and 8 semiregular tilings in the plane. There is only one uniform coloring in a rhombitrihexagonal tiling. With edge-colorings there is a half symmetry form orbifold notation. The hexagons can be considered with two types of edges. It has Coxeter diagram, Schläfli symbol s2. The bicolored square can be distorted into isosceles trapezoids. In the limit, where the rectangles degenerate into a triangular tiling results, constructed as a snub triangular tiling. There is one 2-uniform tilings, having hexagons dissected into 6 triangles. Every circle is in contact with 4 other circles in the packing. The translational domain contains 6 distinct circles.
Rhombitrihexagonal tiling
–
An ornamental version
Rhombitrihexagonal tiling
–
Rhombitrihexagonal tiling
Rhombitrihexagonal tiling
–
Church floor tiling, Sevilla, Spain
Rhombitrihexagonal tiling
–
Periodic
62.
Star figure
–
A regular polygram can either be in a set of regular polygon compounds. The polygram names combine a prefix, such as penta -, with the Greek suffix - gram. Synonyms using other prefixes exist. The - suffix derives from γραμμῆς meaning a line. A regular polygram, as a regular polygon, is denoted by its Schläfli symbol, where p and q are relatively prime and q ≥ 2. These figures are called regular compound polygons. List of regular compounds #Stars Cromwell, P.; Polyhedra, CUP, Hbk. 1997, ISBN 0-521-66432-2. Pbk. ISBN 0-521-66405-5. P. 175 Grünbaum, B. and G.C. Shephard; Tilings and Patterns, New York: W. H. Freeman & Co. ISBN 0-7167-1193-1. Grünbaum, B.; Polyhedra with Hollow Faces, Proc of NATO-ASI Conference on Polytopes... etc. ed T. Bisztriczky et al. Kluwer Academic pp. 43–70. John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 Robert Lachlan, An Elementary Treatise on Modern Pure Geometry. London: Macmillan, 1893, p. 83 polygrams. Branko Grünbaum, Metamorphoses of polygons, published in The Lighter Side of Mathematics: Proceedings of the Eugène Strens Memorial Conference on Recreational Mathematics and its History,
Star figure
–
Regular polygrams {n/d}, with red lines showing constant d, and blue lines showing compound sequences k{n/d}
63.
Star polygon
–
In geometry, a star polygon is a type of non-convex polygon. Only the regular star polygons have been studied in any depth; star polygons in general appear not to have been formally defined. The first usage is included in polygrams which includes polygons like the pentagram but also compound figures like the hexagram. Star polygon names combine a prefix, such as penta -, with the Greek suffix - gram. Synonyms using other prefixes exist. For example, enneagram is also known as a nonagram, using the ordinal nona from Latin. The - suffix derives from γραμμή meaning a line. A regular polygon is denoted by its Schläfli symbol, where p and q are relatively prime and q ≥ 2. The group of is dihedral group Dn of order 2n, independent of k. A regular polygon can also be obtained as a sequence of stellations of a convex regular core polygon. Regular star polygons were first studied systematically by Thomas Bradwardine, later Kepler. If q are not coprime, a degenerate polygon will result with coinciding vertices and edges. For example can be labeled with two sets of vertices 1-6. This should be seen not a double-winding of a single unicursal hexagon. For |n/d|, the inner vertices have an exterior angle, β, as 360°/n.
Star polygon
–
{5/2}
64.
Skew regular polygon
–
In Euclidean geometry, a regular polygon is a polygon, equiangular and equilateral. Regular polygons may be star. These properties apply to all regular polygons, whether star. A regular n-sided polygon has rotational symmetry of order n. All vertices of a regular polygon lie on a common circle, i.e. they are concyclic points. That is, a regular polygon is a cyclic polygon. Thus a regular polygon is a tangential polygon. A n-sided polygon can be constructed with compass and straightedge if and only if the odd prime factors of n are distinct Fermat primes. See constructible polygon. The symmetry group of an n-sided regular polygon is dihedral group Dn: D2, D3, D4... It consists of the rotations in Cn, together with reflection symmetry in n axes that pass through the center. If n is then half of these axes pass through two opposite vertices, the other half through the midpoint of opposite sides. If n is odd then all axes pass through the midpoint of the opposite side. All simple polygons are convex. Those having the same number of sides are also similar.
Skew regular polygon
–
The zig-zagging side edges of a n - antiprism represent a regular skew 2 n -gon, as shown in this 17-gonal antiprism.
Skew regular polygon
–
Regular convex and star polygons with 3 to 12 vertices labelled with their Schläfli symbols
65.
Cube
–
Beryllium copper, also known as copper beryllium, beryllium bronze and spring copper, is a copper alloy with 0.5—3% beryllium and sometimes other elements. Beryllium copper combines high strength with non-sparking qualities. Beryllium has excellent metalworking, machining properties. Beryllium has specialized applications in tools for hazardous environments, musical instruments, precision measurement devices, bullets, aerospace. Beryllium alloys present a toxic hazard during manufacture. Beryllium copper is a ductile, machinable alloy. Beryllium is resistant to non-oxidizing acids, to plastic decomposition products, to galling. Beryllium can be heat-treated for increased strength, electrical conductivity. Beryllium copper attains the greatest strength of any copper-based alloy. In as finished objects, beryllium copper presents no known health hazard. However, inhalation of dust, fume containing beryllium can cause the serious lung condition, chronic beryllium disease. That disease affects primarily the lungs, restricting the exchange of oxygen between the bloodstream. The International Agency for Research on Cancer lists beryllium as a Group 1 Human Carcinogen. The National Toxicology Program also lists beryllium as a carcinogen. Beryllium copper is a non-ferrous alloy used in springs, spring wire, other parts that must retain their shape under repeated stress and strain.
Cube
–
Example of a non-sparking tool made of beryllium copper
Cube
66.
Giant's Causeway
–
The Giant's Causeway is an area of about 40,000 interlocking basalt columns, the result of an ancient volcanic eruption. It is also known as Clochán an Aifir or Clochán na tha Giant's Causey in Ulster-Scots. It is located of the town of Bushmills. In a 2005 poll of Radio Times readers, the Giant's Causeway was named as the fourth greatest natural wonder in the United Kingdom. The tops of the columns form stepping stones that disappear under the sea. Most of the columns are hexagonal, although there are also some with four, five, eight sides. The solidified lava in the cliffs is 28 metres thick in places. The remainder of the site is owned by a number of private landowners. As the lava cooled, contraction occurred. The size of the columns is primarily determined by the speed at which lava from a volcanic eruption cools. The extensive network produced the distinctive columns seen today. The basalts were originally part of a volcanic plateau called the Thulean Plateau which formed during the Paleocene. According to legend, the columns are the remains of a causeway built by a giant. Fionn built the causeway across the North Channel so that the two giants could meet. In one version of the story, Fionn defeats Benandonner.
Giant's Causeway
–
The Giant's Causeway
Giant's Causeway
–
Engraving of Susanna Drury's A View of the Giant's Causeway: East Prospect, 1768
Giant's Causeway
–
Red basaltic prisms
Giant's Causeway
–
Giant's Causeway at sunset
67.
Hexagonal grid
–
In geometry, the hexagonal tiling or hexagonal tessellation is a regular tiling of the Euclidean plane, in which three hexagons meet at each vertex. It has Schläfli symbol of or t. Conway calls it a hextille. The internal angle of the hexagon is 120 degrees so three hexagons at a point make a full 360 degrees. It is one of three regular tilings of the plane. The other two are the triangular tiling and the square tiling. The hexagonal tiling is the densest way to arrange circles in two dimensions. The Honeycomb conjecture states that the hexagonal tiling is the best way to divide a surface into regions of equal area with the least total perimeter. The optimal three-dimensional structure for making honeycomb was investigated by Lord Kelvin, who believed that the Kelvin structure is optimal. However, the less regular Weaire–Phelan structure is slightly better. This structure exists naturally in the form of graphite, where each sheet of graphene resembles wire, with strong carbon bonds. Tubular graphene sheets have been synthesised; these are known as carbon nanotubes. They have potential applications, due to electrical properties. Silicene is similar. Chicken wire consists of a hexagonal lattice of wires.
Hexagonal grid
–
Chicken wire fencing
Hexagonal grid
–
Hexagonal tiling
Hexagonal grid
–
Graphene
Hexagonal grid
–
Periodic
68.
Wax
–
Waxes are a diverse class of organic compounds that are hydrophobic, malleable solids near ambient temperatures. They include higher lipids, typically with melting points above about 40 ° C, melting to give low viscosity liquids. Waxes are soluble in organic, nonpolar solvents. Natural waxes of different types occur in petroleum. Waxes are organic compounds that characteristically consist of alkyl chains. They may also include functional groups such as fatty acids, primary and secondary long chain alcohols, unsaturated bonds, aromatics, amides, ketones, aldehydes. They frequently contain fatty acid esters well. Synthetic waxes are often long-chain hydrocarbons that lack functional groups. Waxes are synthesized by many animals. Those of origin typically consist of wax esters derived from a variety of carboxylic acids and fatty alcohols. In waxes of origin characteristic mixtures of unesterified hydrocarbons may predominate over esters. The composition depends not only on species, but also on geographic location of the organism. Other insects secrete waxes. A major component of the beeswax used in constructing honeycombs is the ester palmitate, an ester of triacontanol and palmitic acid. Its melting point is 62-65 °C.
Wax
–
Commercial honeycomb foundation, made by pressing beeswax between patterned metal rollers.
Wax
–
Ceroline brand wax for floors and furniture, first half of 20th century. From the Museo del Objeto del Objeto collection
Wax
–
Wax candle.
Wax
–
A typical modern wax sculpture of Cecilia Cheung at Madame Tussauds Hong Kong.
69.
Compression (physical)
–
The compressive strength of structures is an important consideration. In uniaxial compression the forces are directed along one direction only, so that they act towards decreasing the object's length along that direction. If the stress vector itself is opposite to x, the material is said to be under normal compression or pure compressive stress along x. This is the only type of static compression that liquids and gases can bear. In a longitudinal wave, or wave, the medium is displaced in the wave's direction, resulting in areas of compression and rarefaction. When put under compression, every material will suffer some deformation, that causes the relative positions of its atoms and molecules to change. The deformation may be permanent, or may be reversed when the compression forces disappear. In the latter case, the deformation gives rise to reaction forces that oppose the compression forces, may eventually balance them. Gases can not bear steady biaxial compression, they will deform promptly and permanently and will not offer any permanent reaction force. However they can bear isotropic compression, may be compressed in other ways momentarily, for instance in a sound wave. The deformation may not be uniform and may not be aligned with the compression forces. What happens in the directions where there is no compression depends on the material. Most materials will expand in those directions, but some special materials will remain unchanged or even contract. By inducing compression, mechanical properties such as compressive strength or modulus of elasticity, can be measured. Compression machines range from very small table top systems to ones with over 53 MN capacity.
Compression (physical)
–
Tightening a corset applies biaxial compression to the waist.
Compression (physical)
–
Compression test on a universal testing machine
70.
Hexagonal prism
–
In geometry, the hexagonal prism is a prism with hexagonal base. This polyhedron has 8 faces, 12 vertices. Since it has eight faces, it is an octahedron. However, the octahedron is primarily used to refer to the regular octahedron, which has eight triangular faces. Because of the dissimilarity of the various eight-sided figures, the term is rarely used without clarification. Before sharpening, many pencils take the shape of a hexagonal prism. It can be seen as a truncated hexagonal hosohedron, represented by Schläfli symbol t. Alternately it can be seen as the Cartesian product of a regular hexagon and a line segment, represented by the product ×. The dual of a hexagonal prism is a hexagonal bipyramid. The group of a right hexagonal prism is D6h of order 24. The group is D6 of order 12. For p < 6, the members of the sequence are omnitruncated polyhedra, shown below as spherical tilings. For p > 6, they are tilings of the hyperbolic plane, starting with the triheptagonal tiling. Uniform Honeycombs in 3-Space VRML models The Uniform Polyhedra Virtual Reality Polyhedra The Encyclopedia of Polyhedra Prisms and antiprisms Weisstein, Eric W. "Hexagonal prism". MathWorld.
Hexagonal prism
–
Uniform Hexagonal prism
71.
Parallelohedron
–
In geometry a parallelohedron is a polyhedron that can tessellate 3-dimensional spaces with face-to-face contacts via translations. This requires all opposite faces be congruent. Parallelohedra can only hexagons with parallel opposite edges. There are 5 types, first identified by Evgraf Fedorov in his studies of crystallographic systems. The vertices of parallelohedra can be computed by linear combinations of 3 to 6 vectors. Each vector can have any length greater with zero length becoming degenerate, or becoming a smaller parallelohedra. A mn means n directional vectors, each containing m coparallel congruent edges. Each has higher symmetry geometries as well. There are 5 types of parallelohedra, although each has forms of varied symmetry. New York: Dover, pp. 29–30, p. 257, 1973. Tutton, A. E. H. Crystallography and Practical Crystal Measurement, 2nd ed. London: Lubrecht & Cramer, 1964. Weisstein, Eric W. "Primary parallelohedron". MathWorld. Weisstein, Eric W. "Space-filling polyhedron". MathWorld.
Parallelohedron
–
Images
72.
Hexagonal prismatic honeycomb
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The triangular prismatic honeycomb or triangular prismatic cellulation is a space-filling tessellation in Euclidean 3-space. It is composed entirely of triangular prisms. It is constructed from a triangular tiling extruded into prisms. It is one of 28 convex uniform honeycombs. The hexagonal hexagonal prismatic cellulation is a space-filling tessellation in Euclidean 3-space made up of hexagonal prisms. It is constructed from a hexagonal tiling extruded into prisms. It is one of 28 convex uniform honeycombs. This honeycomb can be alternated into the gyrated tetrahedral-octahedral honeycomb, with pairs of tetrahedra existing in the alternated gaps. The trihexagonal trihexagonal prismatic cellulation is a space-filling tessellation in Euclidean 3-space. It is composed of hexagonal prisms and triangular prisms in a ratio of 1:2. It is constructed from a trihexagonal tiling extruded into prisms. It is one of 28 convex uniform honeycombs. Prismatic cellulation is a space-filling tessellation in Euclidean 3-space. It is composed of dodecagonal prisms, triangular prisms in a ratio of 1:2. It is constructed from a truncated hexagonal tiling extruded into prisms.
Hexagonal prismatic honeycomb
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Triangular prismatic honeycomb
73.
Conway criterion
Conway criterion
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A golygon which is a Prototile satisfying the Conway criterion. The four midpoints of centrosymmetric symmetry are indicated by black dots.
Conway criterion
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Periodic
74.
Pascal's theorem
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The most natural setting for Pascal's theorem is in a projective plane since no exceptions need be made for parallel lines. However, with the correct interpretation of what happens when some opposite sides of the hexagon are parallel, the theorem remains valid in the Euclidean plane. This theorem is a generalization of Pappus's theorem – Pappus's theorem is the special case of a degenerate conic of two lines. Pascal's theorem is the polar projective dual of Brianchon's theorem. Par B. P.". Pascal's theorem is a special case of the Cayley–Bacharach theorem. This can be proven independently using a property of pole-polar. Six is the minimum number of points on a conic about which special statements can be made, as five points determine a conic. Then if 2n of those points lie on a common line, the last point will be on that line, too. This configuration of 60 lines is called the Hexagrammum Mysticum. The 60 points formed in this way are now known as the Kirkman points. The Pascal lines also pass, three at a time, through 20 Steiner points. There are 20 Cayley lines which consist of three Kirkman points. The Steiner points also lie, four at a time, on 15 Plücker lines. Furthermore, the 20 Cayley lines pass four at a time through 15 points known as the Salmon points.
Pascal's theorem
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Pascal line GHK of self-crossing hexagon ABCDEF inscribed in ellipse. Opposite sides of hexagon have the same color.
75.
Conic section
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In mathematics, a conic section is a curve obtained as the intersection of the surface of a cone with a plane. The three types of conic section are the ellipse. The type of conic is determined by the value of the eccentricity. This equation may be written in matrix form, some geometric properties can be studied as algebraic conditions. In the Euclidean plane, the conic sections appear to be quite different from one another, but share many properties. By extending the geometry to a projective plane this apparent difference vanishes, the commonality becomes evident. Further extension, by expanding the real coordinates to admit complex coordinates, provides the means to see this unification algebraically. The conic sections have provided a rich source of beautiful results in Euclidean geometry. A conic is the curve obtained as the intersection of a plane, called the cutting plane, with the surface of a double cone. Planes that pass through the vertex of the cone will intersect the cone in a point, a line or a pair of intersecting lines. These are called degenerate conics and some authors do not consider them to be conics at all. Unless otherwise stated, we shall assume that "conic" refers to a non-degenerate conic. There are three types of conics, hyperbola. The circle is a special kind of ellipse, although historically it had been considered as a fourth type. The ellipse arise when the intersection of the plane is a closed curve.
Conic section
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Table of conics, Cyclopaedia, 1728
Conic section
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Types of conic sections: 1. Parabola 2. Circle and ellipse 3. Hyperbola
Conic section
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Diagram from Apollonius' Conics, in a 9th century Arabic translation
Conic section
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The paraboloid shape of Archeocyathids produces conic sections on rock faces
76.
Extended side
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In plane geometry, an extended side or sideline of a polygon is the line that contains one side of the polygon. The extension of a side arises in various contexts. In an obtuse triangle, the altitudes from the angled vertices intersect the corresponding extended base sides but not the base sides themselves. Trilinear coordinates locate a point in the plane from the extended sides of a reference triangle. In a triangle, each of an external angle bisector with the opposite extended side, are collinear. An ex-tangential quadrilateral is a quadrilateral for which there exists a circle, tangent to all four extended sides. The excenter lies at the intersection of six angle bisectors.
Extended side
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Each of a triangle's excircles (orange) is tangent to one of the triangle's sides and to the other two extended sides.
77.
Lemoine hexagon
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There are two definitions of the hexagon that differ based on the order in which the vertices are connected. The Lemoine hexagon can be drawn defined in two ways, first as a simple hexagon with vertices at the intersections as defined before. Nevertheless, the Lemoine hexagon is a cyclic polygon, meaning that its vertices all lie on a common circle. The circumcircle of the Lemoine hexagon is known as the first Lemoine circle. Lemoine, É. "Sur quelques propriétés d'un point remarquable d'un triangle", Association francaise pour l’avancement des sciences, Congrès, pp. 90–95. Mackay, J. S. "Symmedians of their concomitant circles", Proceedings of the Edinburgh Mathematical Society, 14: 37 -- 103, doi:10.1017 / S0013091500031758. Weisstein, Eric W. "Lemoine Hexagon". MathWorld.
Lemoine hexagon
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The Lemoine hexagon, shown with self-intersecting connectivity, circumscribed by the first Lemoine circle
78.
Symmedian point
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In geometry, symmedians are three particular geometrical lines associated with every triangle. They are constructed by reflecting the line over the corresponding bisector. The three symmedians meet at a triangle center called the Lemoine point. Ross Honsberger called its existence "one of the crown jewels of modern geometry". The symmedians illustrate this fact. In the diagram, the medians intersect at the centroid G. Because the symmedians are isogonal to the medians, the symmedians also intersect at a single point, L. This point is called alternatively Grebe point.
Symmedian point
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A triangle with medians (blue), angle bisectors (green) and symmedians (red). The symmedians intersect in the symmedian point K, the angle bisectors in the incenter I and the medians in the centroid G.
79.
Concurrent lines
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In geometry, three or more lines in a plane or higher-dimensional space are said to be concurrent if they intersect at a single point. The point where the three altitudes meet is the orthocenter. Angle bisectors are rays bisecting the associated angle. They all meet at the incenter. Medians connect each vertex of a triangle to the midpoint of the opposite side. The three medians meet at the centroid. Perpendicular bisectors are lines running out at 90 degree angles. The three perpendicular bisectors meet at the circumcenter. Other sets of lines associated with a triangle are concurrent well. For example: Any median is concurrent with two other area bisectors each of, parallel to a side. The three cleavers concur at the center of the Spieker circle, the incircle of the medial triangle. A splitter of a triangle is a segment having one endpoint at one of the three vertices of the triangle and bisecting the perimeter. The three splitters concur at the Nagel point of the triangle. Thus if there are three of them, they concur at the incenter. The Napoleon generalizations of them are points of concurrency.
Concurrent lines
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This article needs additional citations for verification. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed. (March 2011)
80.
Circumcircle
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In geometry, the circumscribed circle or circumcircle of a polygon is a circle which passes through all the vertices of the polygon. The center of this circle is called the circumcenter and its radius is called the circumradius. A polygon which has a circumscribed circle is called a cyclic polygon. All rectangles are cyclic. A related notion is the one of a minimum bounding circle, the smallest circle that completely contains the polygon within it. All triangles are cyclic; i.e. every triangle has a circumscribed circle. Since this equation has three parameters only three points' coordinate pairs are required to determine the equation of a circle. Since a triangle is defined by its three vertices, exactly three points are required to determine a circle, every triangle can be circumscribed. The circumcenter of a triangle can be constructed by drawing any two of the three perpendicular bisectors. The center is the point where the perpendicular bisectors intersect, the radius is the length to any of the three vertices. In coastal navigation, a triangle's circumcircle is sometimes used as a way of obtaining a position line using a sextant when no compass is available. The horizontal angle between two landmarks defines the circumcircle upon which the observer lies. Suppose that A = B = C = are the coordinates of points A, B, C. Using the polarization identity, these equations reduce to the condition that the matrix has a nonzero kernel. Thus the circumcircle may alternatively be described as the locus of zeros of the determinant of this matrix: det = 0.
Circumcircle
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Circumscribed circle, C, and circumcenter, O, of a cyclic polygon, P
81.
Acute triangle
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An acute triangle is a triangle with all three angles acute. An obtuse triangle is one with two acute angles. Since a triangle's angles must sum to °, no triangle can have more than one obtuse angle. Obtuse triangles are the two different types of oblique triangles -- triangles that are not right triangles because they have no 90 ° angle. However, while the circumcenter are in an acute triangle's interior, they are exterior to an obtuse triangle. The orthocenter is the point of the triangle's three altitudes, each of which perpendicularly connects a side to the opposite vertex. In the case of an acute triangle, all three of these segments lie entirely in the triangle's interior, so they intersect in the interior. But for an obtuse triangle, the altitudes from the two acute angles intersect only the extensions of the opposite sides. These altitudes fall entirely outside the triangle, resulting with each other occurring in the triangle's exterior. The right triangle is the in-between case: both its orthocenter lie on its boundary. This implies that the longest side in an obtuse triangle is the one opposite the obtuse-angled vertex. However, an obtuse triangle has only one inscribed square, one of whose sides coincides with part of the longest side of the triangle. All triangles in which the Euler line is parallel to one side are acute. This property holds for BC if and only if = 3. Ono's inequality for the A, 27 2 2 2 ≤ 6, holds for all acute triangles but not for all obtuse triangles.
Acute triangle
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Right
82.
Tangent line
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In geometry, the tangent line to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it through a pair of infinitely close points on the curve. A similar definition applies in n-dimensional Euclidean space. Similarly, the tangent plane to a surface at a given point is the plane that the surface at that point. The concept of a tangent has been extensively generalized; see Tangent space. The word "tangent" comes from the Latin tangere,'to touch'. Euclid makes several references to the tangent to a circle in book III of the Elements. In Apollonius work Conics he defines a tangent as being a line such that no straight line could fall between it and the curve. Archimedes found the tangent by considering the path of a point moving along the curve. Independently Descartes used his method of normals based on the observation that the radius of a circle is always normal to the circle itself. These methods led in the 17th century. René-François de Sluse and Johannes Hudde found algebraic algorithms for finding tangents. Further developments included those of Isaac Barrow, leading to the theory of Isaac Newton and Gottfried Leibniz. An 1828 definition of a tangent was "a right line which when produced, does not cut it". This old definition prevents inflection points from having any tangent.
Tangent line
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Tangent to a curve. The red line is tangential to the curve at the point marked by a red dot.
83.
Brianchon's theorem
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In geometry, Brianchon's theorem is a theorem stating that when a hexagon is circumscribed around a conic section, its principal diagonals meet in a single point. It is named after Charles Julien Brianchon. Let ABCDEF be a hexagon formed by six tangent lines of a conic section. Then lines AD, BE, CF intersect at a single point. The polar projective dual of this theorem give Pascal's theorem. Brianchon's theorem is true in both the real projective plane. However, its statement in the affine plane is in a sense more complicated than that in the projective plane. Consider, for example, five tangent lines to a parabola. In two instances, a line from a vertex to the opposite vertex would be a line parallel of the five tangent lines. Brianchon's theorem stated only for the affine plane would therefore have to be stated differently in such a situation. The dual of Brianchon's theorem has exceptions in the affine plane but not in the projective plane. Brianchon's theorem can be proved by the idea of radical axis or reciprocation. Seven circles theorem Pascal theorem
Brianchon's theorem
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Contents
84.
Centroid
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In mathematics and physics, the centroid or geometric center of a plane figure is the arithmetic mean position of all the points in the shape. The definition extends to any object in n-dimensional space: its centroid is the mean position of all the points in all of the coordinate directions. Informally, it is the point at which an infinitesimally thin cutout of the shape could be perfectly balanced on the tip of a pin. In physics, the center of mass is the mean of all points weighted by the local density or specific weight. If a physical object has uniform density, then its center of mass is the same as the centroid of its shape. In geography, the centroid of a radial projection of a region of the Earth's surface to level is known as the region's geographical center. The geometric centroid of a object always lies in the object. A non-convex object might have a centroid, outside the figure itself. The centroid for example, lies in the object's central void. If the centroid is defined, it is a fixed point of all isometries in its group. In particular, the geometric centroid of an object lies in the intersection of symmetry. The centroid of many figures can be determined by this principle alone. In particular, the centroid of a parallelogram is the point of its two diagonals. This is not true for other quadrilaterals. For the same reason, the centroid of an object with translational symmetry is undefined, because a translation has no fixed point.
Centroid
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Centroid of a triangle
85.
Triangular antiprism
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In geometry, an octahedron is a polyhedron with eight faces. A regular octahedron is a solid composed of eight equilateral triangles, four of which meet at each vertex. A regular octahedron is the dual polyhedron of a cube. It is a rectified tetrahedron. It is a square bipyramid in any of three orthogonal orientations. It is also a antiprism in any of four orientations. An octahedron is the three-dimensional case of the more general concept of a cross polytope. A regular octahedron is a 3-ball in the Manhattan metric. The third correspond to the B2 and A2 Coxeter planes. The octahedron can also be projected onto the plane via a stereographic projection. This projection is conformal, preserving angles but not lengths. Straight lines on the sphere are projected as circular arcs on the plane. Additionally the tensor of the stretched octahedron is I =. These reduce to the equations for the regular octahedron when x m = m = z m = a 2 2. This compound, called the stella octangula, is its first and only stellation.
Triangular antiprism
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(Click here for rotating model)
Triangular antiprism
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Fluorite octahedron.
Triangular antiprism
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Two identically formed rubik's snakes can approximate an octahedron.
86.
Skew polygon
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In geometry, a skew polygon is a polygon whose vertices are not all coplanar. Skew polygons must have at least 4 vertices. The interior surface of such a polygon is not uniquely defined. Skew infinite polygons have vertices which are not all collinear. Antiprismatic polygon has vertices which alternate on two parallel planes, thus must be even-sided. Regular polygon in 3 dimensions are always zig-zag. A regular polygon is isogonal with equal edge lengths. In 3 dimensions a regular polygon is a zig-zag skew, with vertices alternating between two parallel planes. The sides of an n-antiprism can define a regular skew 2n-gons. A regular n-gonal can be given a symbol # as a blend of a regular polygon, an orthogonal line segment. The operation between sequential vertices is glide reflection. Examples are shown on pentagon antiprisms. The star antiprisms also generate regular skew polygons with different order of the top and bottom polygons. A regular compound 2n-gon can be similarly constructed by adding a second skew polygon by a rotation. These shares the same vertices as the prismatic compound of antiprisms.
Skew polygon
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The (red) side edges of tetragonal disphenoid represent a regular zig-zag skew quadrilateral.
87.
Vertex-transitive
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In geometry, a polytope is isogonal or vertex-transitive if, loosely speaking, all its vertices are equivalent. We say that for any two vertices there exists a symmetry of the polytope mapping the first isometrically onto the second. All vertices of a n-dimensional isogonal figure exist on an - sphere. The isogonal has long been used for polyhedra. Vertex-transitive is a synonym borrowed from modern ideas such as graph theory. All regular polygons, regular star polygons are isogonal. The dual of an isogonal polygon is an isotoxal polygon. Some even-sided apeirogons which alternate two edge lengths, for example a rectangle, are isogonal. All planar 2n-gons have dihedral symmetry with reflection lines across the mid-edge points. 2D tiling has a single kind of vertex. Geometrically distorted variations of uniform polyhedra and tilings can also be given the configuration. Quasi-regular if it is also isotoxal but not isohedral. Semi-regular if every face is a regular polygon but it is not isotoxal. Uniform if every face is a regular polygon, i.e. it is regular, semi-regular. Noble if it is also isohedral.
Vertex-transitive
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Isogonal apeirogons
88.
Octahedron
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In geometry, an octahedron is a polyhedron with eight faces. A regular octahedron is a solid composed of eight equilateral triangles, four of which meet at each vertex. A regular octahedron is the dual polyhedron of a cube. It is a rectified tetrahedron. It is a square bipyramid in any of three orthogonal orientations. It is also a antiprism in any of four orientations. An octahedron is the three-dimensional case of the more general concept of a cross polytope. A regular octahedron is a 3-ball in the Manhattan metric. The third correspond to the B2 and A2 Coxeter planes. The octahedron can also be projected onto the plane via a stereographic projection. This projection is conformal, preserving angles but not lengths. Straight lines on the sphere are projected as circular arcs on the plane. Additionally the tensor of the stretched octahedron is I =. These reduce to the equations for the regular octahedron when x m = m = z m = a 2 2. This compound, called the stella octangula, is its first and only stellation.
Octahedron
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(Click here for rotating model)
Octahedron
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Fluorite octahedron.
Octahedron
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Two identically formed rubik's snakes can approximate an octahedron.
89.
Petrie polygon
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Projected graphs are useful in visualizing symmetric structure of the regular polytopes. John Flinders Petrie was the only son of Egyptologist Flinders Petrie. He was born in 1907 and as a schoolboy showed remarkable promise of mathematical ability. In periods of intense concentration he could answer questions about complicated four-dimensional objects by visualizing them. He first noted the importance of the regular skew polygons which appear on the surface of regular polyhedra and higher polytopes. In 1938 Petrie collaborated with Coxeter, Patrick du Val, H.T. Flather to produce The Fifty-Nine Icosahedra for publication. Realizing the geometric facility of the skew polygons used by Petrie, Coxeter named them after his friend when he wrote Regular Polytopes. In a few months after his retirement, Petrie was killed by a while attempting to cross a motorway near his home in Surrey. The idea of Petrie polygons was later extended to semiregular polytopes. The Petrie polygon of the regular polyhedron has h sides, where h+2=24/. The regular duals, and, are contained within the same projected Petrie polygon. The Petrie polygon projections are most useful for visualization of polytopes of dimension four and higher. Each edge is shared by 2 petrie polygon faces. π, has 4 vertices, 3 skew square faces.
Petrie polygon
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Net
90.
Regular polytope
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In mathematics, a regular polytope is a polytope whose symmetry group acts transitively on its flags, thus giving it the highest degree of symmetry. Regular polytopes are the generalized analog in any number of dimensions of regular polyhedra. The strong symmetry of the regular polytopes gives an aesthetic quality that interests both non-mathematicians and mathematicians. Classically, a regular polytope in n dimensions may be defined as having regular vertex figures. All vertices are alike. Note, however, that this definition does not work for abstract polytopes. Regular vertex figures as. Regular polytopes are classified primarily according to their dimensionality. They can be further classified according to symmetry. For example, the regular octahedron share the same symmetry, as do the regular dodecahedron and icosahedron. Indeed, symmetry groups are sometimes named after for example the tetrahedral and icosahedral symmetries. Three special classes of regular polytope exist in every dimensionality: Regular Measure polytope Cross polytope In two dimensions there are infinitely many regular polygons. In four dimensions there are several more regular polyhedra and 4-polytopes besides these three. In above, these are the only ones. See also the list of regular polytopes.
Regular polytope
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Platonic solids
Regular polytope
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A regular pentagon is a polygon, a two-dimensional polytope with 5 edges, represented by Schläfli symbol {5}.
Regular polytope
Regular polytope
91.
Orthogonal projection
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In linear algebra and functional analysis, a projection is a linear transformation P from a vector space to itself such that P 2 = P. That is, whenever P is applied twice to any value, it gives the same result as if it were applied once. It leaves its image unchanged. Though abstract, this definition of "projection" generalizes the idea of graphical projection. One can also consider the effect of a projection on a geometrical object by examining the effect of the projection on points in the object. For example, the function which maps the point to the point is an orthogonal projection onto the x -- y plane. This function is represented by the matrix P =. The action of this matrix on an arbitrary vector is P =. To see that P is indeed i.e. P = P2, we compute P 2 = P = = P. A simple example of a non-orthogonal projection is P =. Via matrix multiplication, one sees that P 2 = = = P. Proving that P is indeed a projection. The P is orthogonal if and only if α = 0. Let W be a finite dimensional vector space and P be a projection on W. Suppose the subspaces U and V are the range and kernel of P respectively.
Orthogonal projection
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The transformation P is the orthogonal projection onto the line m.
92.
3-3 duoprism
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In geometry of 4 dimensions, a 3-3 duoprism, the smallest p-q duoprism, is a 4-polytope resulting from the Cartesian product of two triangles. It has 9 vertices, 15 faces, in 6 triangular prism cells. It has symmetry, order 72. There are three constructions for the honeycomb with two lower symmetries. The regular complex polytope 32, in C 2 has a real representation as a 3-3 duoprism in 4-dimensional space. 32 has 9 vertices, 6 3-edges. Its symmetry is order 12. It also has 3 × 3, with symmetry 33, order 9. This is the symmetry if the blue 3-edges are considered distinct. The dual of a 3-3 duoprism is called a 3-3 duopyramid. It has 9 tetragonal disphenoid cells, 18 triangular faces, 6 vertices. It can be seen in orthogonal projection connecting all pairs just like a 5-simplex seen in projection. It has 9 2-edges corresponding to the connecting edges of the 3-3 duopyramid, while the 6 edges connecting the two triangles are not included. It can be seen with 3 sets of colored edges. This arrangement of edges makes a complete bipartite graph with each vertex from one triangle is connected to every vertex on the other.
3-3 duoprism
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3-3 duoprisms
93.
3-3 duopyramid
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In geometry of 4 dimensions, a 3-3 duoprism, the smallest p-q duoprism, is a 4-polytope resulting from the Cartesian product of two triangles. It has 15 faces, in 6 triangular prism cells. It has order 72. There are three constructions for the honeycomb with two lower symmetries. The regular complex polytope 32, in C 2 has a real representation as a 3-3 duoprism in 4-dimensional space. 32 has 9 vertices, 6 3-edges. Its symmetry is 32, order 18. It also has × 3, with symmetry 33, order 9. This is the symmetry if the red and blue 3-edges are considered distinct. The dual of a 3-3 duoprism is called a 3-3 duopyramid. It has 9 tetragonal disphenoid cells, 6 vertices. It can be seen in orthogonal projection as a 6-gon circle of vertices, edges connecting all pairs, just like a 5-simplex seen in projection. It has 9 2-edges corresponding to the connecting edges of the 3-3 duopyramid, while the 6 edges connecting the two triangles are not included. It can be seen in a hexagonal projection with 3 sets of colored edges. This arrangement of edges makes a complete graph with each vertex from one triangle is connected to every vertex on the other.
3-3 duopyramid
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3-3 duoprisms
94.
5-simplex
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In five-dimensional geometry, a 5-simplex is a self-dual regular 5-polytope. It has six vertices, 15 edges, 20 triangle faces, 6 5-cell facets. It has a dihedral angle of approximately 78.46 °. It can also be called hexa-5-tope, as a 6-facetted polytope in 5-dimensions. The hexateron is derived from hexa - for having six facets and teron for having four-dimensional facets. By Jonathan Bowers, a hexateron is given the hix. The hexateron can be constructed from a 5-cell by adding a 6th vertex such that it is equidistant from all the other vertices of the 5-cell. These construction can be seen as facets of the rectified 6-cube respectively. It is first in a dimensional series of uniform honeycombs, expressed by Coxeter as 3k1 series. A 4-dimensional case exists as 3-sphere tiling, a tetrahedral dihedron. It is first in a dimensional series of uniform honeycombs, expressed as 3k1 series. A 4-dimensional case exists as 3-sphere tiling, a tetrahedral hosohedron. The 5-simplex, as 220 polytope is first in dimensional series 22k. The regular 5-simplex is one of 19 uniform polytera based on all shown here in A5 Coxeter plane orthographic projections. The 5-simplex can also be considered an apex point above the hyperplane.
5-simplex
95.
Platonic solid
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In three-dimensional space, a Platonic solid is a regular, convex polyhedron. It is constructed by congruent polygonal faces with the same number of faces meeting at each vertex. Five solids meet those criteria: Geometers have studied the mathematical symmetry of the Platonic solids for thousands of years. They are named for the Greek philosopher Plato who theorized in his dialogue, the Timaeus, that the classical elements were made of these regular solids. The Platonic solids have been known since antiquity. Dice go back with shapes that predated formal charting of Platonic solids. The ancient Greeks studied the Platonic solids extensively. Some sources credit Pythagoras with their discovery. The Platonic solids are prominent in the philosophy of Plato, their namesake. Plato wrote in the dialogue Timaeus c. 360 B.C. in which he associated each of the four classical elements with a regular solid. Earth was associated with the cube, air with the octahedron, fire with the tetrahedron. There was intuitive justification for these associations: the heat of fire feels stabbing. Air is made of the octahedron; its minuscule components are so smooth that one can barely feel it. The icosahedron, flows out of one's hand when picked up, as if it is made of tiny little balls. By a highly nonspherical solid, the hexahedron represents "earth".
Platonic solid
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{4,3} Defect 90°
Platonic solid
Platonic solid
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Circogonia icosahedra, a species of radiolaria, shaped like a regular icosahedron.
Platonic solid
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Polyhedral dice are often used in role-playing games.
96.
Archimedean solid
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In geometry, an Archimedean solid is one of the 13 solids first enumerated by Archimedes. They are the semi-regular convex polyhedrons composed of regular polygons excluding the prisms and antiprisms. They differ from the Johnson solids, whose regular polygonal faces do not meet in identical vertices. "Identical vertices" means that for any two vertices, there is a global isometry of the entire solid that takes one vertex to the other. Excluding these two infinite families, there are 13 Archimedean solids. All the Archimedan solids can be made from the Platonic solids with tetrahedral, icosahedral symmetry. The Archimedean solids take their name from Archimedes, who discussed them in a now-lost work. Pappus refers to it, stating that Archimedes listed 13 polyhedra. Kepler may have also found the elongated square gyrobicupola: at least, he once stated that there were 14 Archimedean solids. However, the clear statement of the pseudorhombicuboctahedron's existence was made in 1905, by Duncan Sommerville. There are 13 Archimedean solids. Here the vertex configuration refers to the type of regular polygons that meet at any given vertex. For example, a vertex configuration of means that octagon meet at a vertex. Some definitions of polyhedron include one more figure, "pseudo-rhombicuboctahedron". The number of vertices is 720 ° divided by the defect.
Archimedean solid
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The truncated icosidodecahedron, is the largest Archimedean solid, by volume with unit edge length, as well as having the most vertices and edges.
97.
Truncated tetrahedron
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In geometry, the truncated tetrahedron is an Archimedean solid. It has 18 edges. It can be constructed by truncating all 4 vertices of a regular tetrahedron at one third of the original edge length. A deeper truncation, removing a tetrahedron of half the original edge length from each vertex, is called rectification. The rectification of a tetrahedron produces an octahedron. A truncated tetrahedron is the Goldberg polyhedron GIII, containing triangular and hexagonal faces. A truncated tetrahedron can be called a cantic cube, with Coxeter diagram, having half of the vertices of the cantellated cube. There are two dual positions of this construction, combining them creates the uniform compound of two truncated tetrahedra. The densest packing of the Archimedean truncated tetrahedron is believed to be Φ = 207/208, as reported by two independent groups using Monte Carlo methods. This projection is conformal, preserving angles but not areas or lengths. Straight lines on the sphere are projected as circular arcs on the plane. A lower symmetry version of the truncated tetrahedron is called a Friauf polyhedron in crystals such as complex metallic alloys. This form fits 5 Friauf polyhedra around an axis, giving a 72-degree dihedral angle on a subset of 6-6 edges. Its named after J. B. Friauf and his 1927 paper "The crystal structure of the intermetallic compound MgCu2".
Truncated tetrahedron
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(Click here for rotating model)
Truncated tetrahedron
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Triakis tetrahedron (dual polyhedron)
98.
Truncated octahedron
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In geometry, the truncated octahedron is an Archimedean solid. It has 14 faces, 36 edges, 24 vertices. Since each of its faces has point symmetry the truncated octahedron is a zonohedron. It is also the Goldberg polyhedron GIV, containing square and hexagonal faces. Like the cube, it can tessellate 3-dimensional space, as a permutohedron. Its dual polyhedron is the tetrakis hexahedron. If the truncated octahedron has length, its dual tetrakis cube has edge lengths 9/8 √ 2 and 3/2 √ 2. A truncated octahedron is constructed from a regular octahedron with side length 3a by the removal of six right square pyramids, one from each point. These pyramids have both base lateral length of a, to form equilateral triangles. The base area is then a2. Note that this shape is exactly similar to half an octahedron or Johnson solid J1. The truncated octahedron has five orthogonal projections, centered, on two types of edges, two types of faces: square. The last two correspond to the B2 and A2 Coxeter planes. The truncated octahedron can also be represented as a spherical tiling, projected onto the plane via a stereographic projection. This projection is conformal, preserving angles but not areas or lengths.
Truncated octahedron
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(Click here for rotating model)
Truncated octahedron
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Tetrakis hexahedron (dual polyhedron)
99.
Truncated icosahedron
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In geometry, the truncated icosahedron is an Archimedean solid, one of 13 convex isogonal nonprismatic solids whose faces are two or more types of regular polygons. It has 12 regular pentagonal faces, 20 regular hexagonal faces, 90 edges. It is the Goldberg GV, containing pentagonal and hexagonal faces. This geometry is associated with footballs typically patterned with black pentagons. Geodesic domes such as those whose architecture Buckminster Fuller pioneered are often based on this structure. It also corresponds to the geometry of the fullerene molecule. It is used in the bitruncated order-5 dodecahedral honeycomb. This creates 12 new pentagon leaves the original 20 triangle faces as regular hexagons. Thus the length of the edges is one third of that of the original edges. The edges have length 2. The last two correspond to the A2 and H2 Coxeter planes. The truncated icosahedron can also be projected onto the plane via a stereographic projection. This projection is conformal, preserving angles but not lengths. Straight lines on the sphere are projected as circular arcs on the plane. The angle between the segments joining the vertices connected by shared edge is approximately 23.281446 °.
Truncated icosahedron
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(Click here for rotating model)
Truncated icosahedron
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Pentakis dodecahedron (dual polyhedron)
Truncated icosahedron
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Fullerene C 60 molecule
Truncated icosahedron
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Truncated icosahedral radome on a weather station
100.
Soccer
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Association football, more commonly known as football or soccer, is a team sport played between two teams of eleven players with a spherical ball. It dependencies, making it the world's most popular sport. The game is played with a goal at each end. The object of the game is by getting the ball into the opposing goal. Other players mainly may also use their head or torso. The team that scores the most goals by the end of the match wins. The Laws of the Game were originally codified by The Football Association in 1863. Association football is governed internationally by the International Federation of Association Football, which organises every four years. The soccer originated in England, first appearing in the 1880s as an Oxford" - er" abbreviation of the word "association". Within the English-speaking world, football is now usually called football in the United Kingdom and mainly soccer in Canada and the United States. According to FIFA, the competitive game cuju is the earliest form of football for which there is scientific evidence. The intent was kicking a ball through an opening into a net. It was remarkably similar to modern football, though similarities to rugby occurred. During the Han Dynasty, rules were established. Episkyros were Greek ball games.
Soccer
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The attacking player (No. 10) attempts to kick the ball beyond the opposing team's goalkeeper and between the goalposts and beneath the crossbar to score a goal
Soccer
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Children playing cuju in Song dynasty China
Soccer
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Ebenezer Cobb Morley, who is regarded as the "father of football"
Soccer
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A women's international match between the United States and Germany
101.
Fullerene
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A fullerene is a molecule of carbon in the form of a hollow sphere, ellipsoid, tube, many other shapes. Spherical fullerenes, also referred to as Buckminsterfullerenes, resemble the balls used in football. Cylindrical fullerenes are also called carbon nanotubes. Fullerenes are similar to graphite, composed of stacked graphene sheets of linked hexagonal rings; they may also contain pentagonal rings. The name was an homage to Buckminster Fuller, whose geodesic domes it resembles. The structure was also identified some from an electron microscope image, where it formed the core of a "bucky onion". Fullerenes have since been found to occur in nature. More recently, fullerenes have been detected in outer space. According to astronomer Letizia Stanghellini, "It's possible that buckyballs from outer space provided seeds on Earth." The icosahedral cage was mentioned in 1965 as a possible topological structure. Eiji Osawa of Toyohashi University of Technology predicted the existence of C60 in 1970. Neither it nor any translations of it reached Europe or the Americas. Also in 1970, R. W. Henson made a model of C60. Unfortunately, the evidence for this new form of carbon was not accepted, even by his colleagues. The results were acknowledged in Carbon in 1999.
Fullerene
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The C 60 fullerene in crystalline form
Fullerene
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Buckminsterfullerene C 60 (left) and carbon nanotubes (right) are two examples of structures in the fullerene family.
Fullerene
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Fullerite (scanning electron microscope image)
Fullerene
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C 60 in solution
102.
Truncated cuboctahedron
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In geometry, the truncated cuboctahedron is an Archimedean solid. It has 12 square faces, 8 regular hexagonal faces, 72 edges. Since each of its faces has point symmetry, the truncated cuboctahedron is a zonohedron. If you truncate a cuboctahedron by cutting the corners off, you do not get this uniform figure: some of the faces will be rectangles. However, the resulting figure is topologically equivalent to a truncated cuboctahedron and can always be deformed until the faces are regular. Compare to small rhombicuboctahedron. One unfortunate point of confusion: There is a nonconvex uniform polyhedron by the same name. See nonconvex great rhombicuboctahedron. Many other lower symmetry toroids can also be constructed by removing a subset of these dissected components. For example, removing half of the triangular cupolas creates a genus 3 torus, which has tetrahedral symmetry. There is only one uniform coloring of the faces of one color for each type. A 2-uniform coloring, with tetrahedral symmetry, exists with alternately colored hexagons. The truncated cuboctahedron can also be represented as a spherical tiling, projected onto the plane via a stereographic projection. This projection is conformal, preserving angles but not areas or lengths. Straight lines on the sphere are projected as circular arcs on the plane.
Truncated cuboctahedron
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(Click here for rotating model)
Truncated cuboctahedron
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Disdyakis dodecahedron (dual polyhedron)
103.
Truncated icosidodecahedron
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In geometry, the truncated icosidodecahedron is an Archimedean solid, one of thirteen convex isogonal nonprismatic solids constructed by two or more types of regular polygon faces. Since each of its faces has point symmetry, the truncated icosidodecahedron is a zonohedron. If one truncates an icosidodecahedron by cutting the corners off, one does not get this uniform figure: instead of squares the truncation has golden rectangles. However, the resulting figure is topologically equivalent to this and can always be deformed until the faces are regular. Compare to small rhombicosidodecahedron. One unfortunate point of confusion is that there is a nonconvex uniform polyhedron of the same name. See nonconvex great rhombicosidodecahedron. V = 399 a 3. If a set of all 13 Archimedean solids were constructed with all edge lengths equal, the truncated icosidodecahedron would be the largest. The truncated icosidodecahedron has seven orthogonal projections, centered on a vertex, on three types of edges,: square, hexagonal and decagonal. The last two correspond to the A2 and H2 Coxeter planes. The truncated icosidodecahedron can also be represented as a spherical tiling, projected onto the plane via a stereographic projection. This projection is conformal, preserving angles but not areas or lengths. Straight lines on the sphere are projected as circular arcs on the plane. Schlegel diagrams are similar, with a perspective projection and straight edges.
Truncated icosidodecahedron
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(Click here for rotating model)
Truncated icosidodecahedron
Truncated icosidodecahedron
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Disdyakis triacontahedron (dual polyhedron)
104.
Tetrahedral symmetry
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A regular tetrahedron has 12 rotational symmetries, a symmetry order of 24 including transformations that combine a reflection and a rotation. The set of orientation-preserving symmetries forms a group referred as the alternating A4 of S4. Chiral and full are discrete point symmetries. They are among the crystallographic point groups of the cubic crystal system. Seen in stereographic projection the edges of the tetrakis hexahedron form 6 circles in the plane. Each of these 6 circles represent a mirror line in tetrahedral symmetry. The intersection of these circles meet at order 2 and 3 gyration points. T, 332, +, or 23, of order 12 – chiral or rotational tetrahedral symmetry. There are three 2-fold rotation axes, like chiral dihedral symmetry 222, with in addition four 3-fold axes, centered between the three orthogonal directions. The three elements of the latter are "anti-clockwise rotation", corresponding to permutations of the three 2-fold axes, preserving orientation. 43m, of order 24 -- achiral or full tetrahedral symmetry, also known as the triangle group. This group has the same rotation axes as T, but with six mirror planes, each through two 3-fold axes. The 2-fold axes are now S4 axes. Td and O are isomorphic as abstract groups: they both correspond to S4, the symmetric group on 4 objects. Td is the union of T and the set obtained by combining each element of O \ T with inversion.
Tetrahedral symmetry
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A regular tetrahedron, an example of a solid with full tetrahedral symmetry
Tetrahedral symmetry
Tetrahedral symmetry
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In the tetrakis hexahedron one full face is a fundamental domain; other solids with the same symmetry can be obtained by adjusting the orientation of the faces, e.g. flattening selected subsets of faces to combine each subset into one face, or replacing each face by multiple faces, or a curved surface.
105.
Octahedral symmetry
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A regular octahedron has 24 rotational symmetries, a symmetry order of 48 including transformations that combine a reflection and a rotation. A cube has the same set of symmetries, since it is the dual of an octahedron. Octahedral symmetry are the discrete point symmetries with the largest symmetry groups compatible with translational symmetry. They are among the crystallographic point groups of the cubic crystal system. So e.g. the identity is represented as 0 and the inversion as 0 ′. is represented as 17 and is as 17 ′. The pairs can be seen in the six files below. The position of each permutation in the file corresponds to the ∈. 7 ′ 22 = 17 ′: The reflection 7 ′ applied on the 90 ° rotation 22 gives 17 ′. + of order 24, is chiral octahedral symmetry or rotational octahedral symmetry. Td and O are isomorphic as abstract groups: they both correspond to S4, the symmetric group on 4 objects. Td is the union of T and the set obtained by combining each element of O \ T with inversion. O is the group of the regular octahedron. Oh, *432, or m3m of order 48 - achiral octahedral symmetry or full octahedral symmetry. This group has the same rotation axes as O, but with mirror planes, comprising both the mirror planes of Td and Th. This group is the full group of the cube and octahedron.
Octahedral symmetry
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Each face of the disdyakis dodecahedron is a fundamental domain
Octahedral symmetry
106.
Icosahedral symmetry
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A regular icosahedron has 60 rotational symmetries, a symmetry order of 120 including transformations that combine a reflection and a rotation. A regular dodecahedron has the same set of symmetries, since it is the dual of the icosahedron. The full symmetry group is the product A5 × Z2. Coxeter diagram. Icosahedral symmetry is not compatible with translational symmetry, so there are space groups. These correspond to the icosahedral groups being the triangle groups. The first presentation was given in his paper on icosian calculus. Note that other presentations are possible, for instance as an alternating group. I is of order 60. The I is isomorphic to A5, the alternating group of even permutations of five objects. The group contains 5 versions of Th with 6 versions of D5. The icosahedral group Ih has order 120. It has I as normal subgroup of index 2. Ih − 1 acts as the identity. It acts on the compound of ten tetrahedra: I acts on − 1 interchanges the two halves.
Icosahedral symmetry
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Faces of disdyakis triacontahedron are the fundamental domain
Icosahedral symmetry
107.
Goldberg polyhedron
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A Goldberg polyhedron is a convex polyhedron made from hexagons and pentagons. They were first described by Michael Goldberg in 1937. They are defined by three properties: they have rotational icosahedral symmetry. They are not necessarily mirror-symmetric; e.g. G and G are enantiomorphs of each other. A consequence of Euler's polyhedron formula is that there will be exactly twelve pentagons. Icosahedral symmetry ensures that the pentagons are always regular, although many of the hexagons may not be. Typically all of the vertices lie on a sphere. It is a dual polyhedron of a geodesic sphere, with all triangle faces and 6 triangles per vertex, except for 12 vertices with 5 triangles. Simple examples of Goldberg polyhedra include the dodecahedron and truncated icosahedron. Such a polyhedron is denoted G. A dodecahedron is G and a truncated icosahedron is G. A similar technique can be applied to construct polyhedra with tetrahedral symmetry. These polyhedra will have triangles or squares rather than pentagons. These variations are given Roman numeral subscripts: GIII, GIV, GV. The operator dk generates G.
Goldberg polyhedron
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G(1,4)
108.
Chamfered tetrahedron
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In geometry, chamfering or edge-truncation is a Conway polyhedron notation operation that modifies one polyhedron into another. It is similar to expansion, moving faces apart and outward, but also maintain the original vertices. For polyhedra, this operation adds a new hexagonal face in place of each original edge. A polyhedron with e edges will have a chamfered form containing 2e new vertices, 3e new edges, e new hexagonal faces. The chamfer operation applied in series creates progressively larger polyhedra with new hexagonal faces replacing edges from the previous one. The chamfer operator transforms G to G. A regular polyhedron, G, create a Goldberg polyhedra sequence: G, G, G, G, G... The truncated octahedron or truncated icosahedron, G creates a Goldberg sequence: G, G, G, G.... A truncated tetrakis hexahedron or pentakis dodecahedron, G, creates a Goldberg sequence: G, G, G... Like the expansion operation, chamfer can be applied to any dimension. For polygons, it triples the number of vertices. The chamfered tetrahedron is a convex polyhedron constructed as an alternately truncated cube or chamfer operation on a tetrahedron, replacing its 6 edges with hexagons. It is the Goldberg polyhedron GIII, containing triangular and hexagonal faces. It can look a little like a truncated tetrahedron, which has 4 hexagonal and 4 triangular faces, the related Goldberg polyhedron: GIII. Net In geometry, the chamfered cube is a convex polyhedron constructed from the rhombic dodecahedron by truncating the 6 vertices.
Chamfered tetrahedron
Chamfered tetrahedron
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Example chamfered pentagon (v,e) --> (v+2e,3e)
109.
Chamfered cube
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In geometry, chamfering or edge-truncation is a Conway polyhedron notation operation that modifies one polyhedron into another. It is similar to expansion, moving faces apart and also maintain the original vertices. For polyhedra, this operation adds a hexagonal face in place of each original edge. E new hexagonal faces. The operation applied in series creates progressively larger polyhedra with new hexagonal faces replacing edges from the previous one. The operator transforms G to G. G, create a Goldberg polyhedra sequence: G, G, G, G, G... The truncated octahedron or truncated icosahedron, G creates a Goldberg sequence: G, G, G, G.... A truncated tetrakis hexahedron or pentakis dodecahedron, G, creates a Goldberg sequence: G, G, G... Like the operation, chamfer can be applied to any dimension. For polygons, it triples the number of vertices. The chamfered tetrahedron is a convex polyhedron constructed on a tetrahedron replacing its 6 edges with hexagons. It is the Goldberg polyhedron GIII, containing hexagonal faces. It can look a little like a truncated tetrahedron, which has 4 hexagonal and 4 triangular faces, the related Goldberg polyhedron: GIII. Net In geometry, the chamfered cube is a convex polyhedron constructed by truncating the 6 vertices.
Chamfered cube
Chamfered cube
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Example chamfered pentagon (v,e) --> (v+2e,3e)
110.
Chamfered dodecahedron
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In geometry, chamfering or edge-truncation is a Conway polyhedron notation operation that modifies one polyhedron into another. It is similar to expansion, moving faces apart and outward, but also maintain the original vertices. For polyhedra, this operation adds a new hexagonal face in place of each original edge. A polyhedron with e edges will have a chamfered form containing 2e new vertices, 3e new edges, e new hexagonal faces. The chamfer operation applied in series creates progressively larger polyhedra with new hexagonal faces replacing edges from the previous one. The chamfer operator transforms G to G. G, create a Goldberg sequence: G, G, G, G, G... The truncated octahedron or truncated icosahedron, G creates a Goldberg sequence: G, G, G, G.... A truncated tetrakis hexahedron or pentakis dodecahedron, G, creates a Goldberg sequence: G, G, G... Like the expansion operation, chamfer can be applied to any dimension. For polygons, it triples the number of vertices. The chamfered tetrahedron is a convex polyhedron constructed as an alternately truncated cube or chamfer operation on a tetrahedron, replacing its 6 edges with hexagons. It is the Goldberg polyhedron GIII, containing triangular and hexagonal faces. It can look a little like a truncated tetrahedron, which has 4 hexagonal and 4 triangular faces, the related Goldberg polyhedron: GIII. Net In geometry, the chamfered cube is a convex polyhedron constructed from the rhombic dodecahedron by truncating the 6 vertices.
Chamfered dodecahedron
Chamfered dodecahedron
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Example chamfered pentagon (v,e) --> (v+2e,3e)
111.
Johnson solid
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In geometry, a Johnson solid is a strictly convex polyhedron, not uniform, each face of, a regular polygon. There is no requirement that the same polygons join around each vertex. An example of a Johnson solid is the square-based pyramid with equilateral sides; it has 1 square face and 4 triangular faces. As in any strictly convex solid, the total of their angles is less than 360 degrees. Since a regular polygon has angles at least 60 degrees, it follows that at most five faces meet at any vertex. The pentagonal pyramid is an example that actually has a degree-5 vertex. In 1966, Norman Johnson published a list which gave them their names and numbers. He did conjecture that there were no others. Victor Zalgaller in 1969 proved that Johnson's list was complete. However, it is not vertex-transitive, as it has different isometry at different vertices, making a Johnson solid rather than an Archimedean solid. The names are more descriptive than they sound. Most of the Johnson solids can be constructed from the first few, together with the Platonic and Archimedean solids, antiprisms. Bi- means that two copies of the solid in question are joined base-to-base. For rotundae, they can be joined so that like faces or unlike faces meet. In this nomenclature, an octahedron would be a square bipyramid, an icosidodecahedron would be a pentagonal gyrobirotunda.
Johnson solid
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The elongated square gyrobicupola (J 37), a Johnson solid
112.
Triangular cupola
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In geometry, the triangular cupola is one of the Johnson solids. It can be seen as half a cuboctahedron. A Johnson solid is one of 92 strictly convex polyhedra that have regular faces but are not uniform. They were named by Norman Johnson, who first listed these polyhedra in 1966. This isn't a Johnson solid because of its coplanar faces. Merging those coplanar triangles into larger ones, topologically this is another cupola with isosceles trapezoidal side faces. If the base hexagon is replaced by 6 triangles, it generates a coplanar deltahedron with 22 faces. The cupola can form a tessellation of space with square pyramids and/or octahedra, the same way octahedra and cuboctahedra can fill space. The family of cupolae with regular polygons exists up to n=5, higher if isosceles triangles are used in the cupolae. Eric W. Weisstein, Triangular cupola at MathWorld.
Triangular cupola
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Triangular cupola
113.
Elongated triangular cupola
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In geometry, the elongated triangular cupola is one of the Johnson solids. As the name suggests, it can be constructed by elongating a cupola by attaching a hexagonal prism to its base. A Johnson solid is one of 92 strictly convex polyhedra that have regular faces but are not uniform. They were named by Norman Johnson, who first listed these polyhedra in 1966. The elongated cupola can form a tessellation of space with tetrahedra and square pyramids. Eric W. Weisstein, Elongated triangular cupola at MathWorld.
Elongated triangular cupola
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Elongated triangular cupola
114.
Gyroelongated triangular cupola
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In geometry, the gyroelongated triangular cupola is one of the Johnson solids. It can be constructed by attaching a hexagonal antiprism to the base of a cupola. This is called "gyroelongation", which means that an antiprism is joined between the bases of more than one solid. The gyroelongated cupola can also be seen as a gyroelongated triangular bicupola with one triangular cupola removed. Like all cupolae, the polygon has twice as many sides as the top. A Johnson solid is one of 92 strictly convex polyhedra that have regular faces but are not uniform. They were named by Norman Johnson, who first listed these polyhedra in 1966. Eric W. Weisstein, Gyroelongated triangular cupola at MathWorld.
Gyroelongated triangular cupola
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Gyroelongated triangular cupola
115.
Augmented hexagonal prism
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In geometry, the augmented hexagonal prism is one of the Johnson solids. As the name suggests, it can be constructed by augmenting a hexagonal prism by attaching a square pyramid to one of its equatorial faces. When three such pyramids are attached, the result may be a parabiaugmented hexagonal prism, a metabiaugmented hexagonal prism or a triaugmented hexagonal prism. A Johnson solid is one of 92 strictly polyhedra that have regular faces but are not uniform. They were named by Norman Johnson, who first listed these polyhedra in 1966. Eric W. Weisstein, Augmented hexagonal prism at MathWorld.
Augmented hexagonal prism
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Augmented hexagonal prism
116.
Parabiaugmented hexagonal prism
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In geometry, the parabiaugmented hexagonal prism is one of the Johnson solids. As the name suggests, it can be constructed by doubly augmenting a hexagonal prism by attaching square pyramids to two of its nonadjacent, parallel equatorial faces. Attaching the pyramids to nonadjacent, equatorial faces yields a metabiaugmented hexagonal prism. A Johnson solid is one of 92 strictly polyhedra that are not uniform. They were named by Norman Johnson, who first listed these polyhedra in 1966. Eric W. Weisstein, Parabiaugmented hexagonal prism at MathWorld.
Parabiaugmented hexagonal prism
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Parabiaugmented hexagonal prism
117.
Metabiaugmented hexagonal prism
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In geometry, the metabiaugmented hexagonal prism is one of the Johnson solids. As the name suggests, it can be constructed by doubly augmenting a hexagonal prism by attaching square pyramids to two of its nonadjacent, nonparallel equatorial faces. Attaching the pyramids to opposite equatorial faces yields a parabiaugmented hexagonal prism. A Johnson solid is one of 92 strictly polyhedra that are not uniform. They were named by Norman Johnson, who first listed these polyhedra in 1966. Hexagonal prism Eric W. Weisstein, Metabiaugmented hexagonal prism at MathWorld.
Metabiaugmented hexagonal prism
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Metabiaugmented hexagonal prism
118.
Triaugmented hexagonal prism
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In geometry, the triaugmented hexagonal prism is one of the Johnson solids. As the name suggests, it can be constructed by triply augmenting a hexagonal prism by attaching square pyramids to three of its equatorial faces. A Johnson solid is one of 92 strictly polyhedra that are not uniform. They were named by Norman Johnson, who first listed these polyhedra in 1966. Hexagonal prism Eric W. Weisstein, Triaugmented hexagonal prism at MathWorld.
Triaugmented hexagonal prism
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Triaugmented hexagonal prism
119.
Augmented truncated tetrahedron
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In geometry, the augmented truncated tetrahedron is one of the Johnson solids. It is created by attaching a cupola to one hexagonal face of an truncated tetrahedron. A Johnson solid is one of 92 strictly convex polyhedra that have regular faces but are not uniform. They were named by Norman Johnson, who first listed these polyhedra in 1966. Eric W. Weisstein, Augmented truncated tetrahedron at MathWorld.
Augmented truncated tetrahedron
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Augmented truncated tetrahedron
120.
Triangular hebesphenorotunda
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In geometry, the triangular hebesphenorotunda is one of the Johnson solids. A Johnson solid is one of 92 strictly polyhedra that have regular faces but are not uniform. They were named by Norman Johnson, who first listed these polyhedra in 1966. It is one of the Platonic and Archimedean solids. However, it does have a strong relationship to an Archimedean solid. Most evident is the cluster of three pentagons and four triangles on one side of the solid. The hebesphenorotunda is the only Johnson solid with faces of 3, 4, 5 and 6 sides. These coordinates produce a hebesphenorotunda with edge length 2, resting on the XY plane and having its 3-fold axis of symmetry aligned to the Z-axis. A second, triangular hebesphenorotunda can be obtained by negating the second and third coordinates of each point. The pair will inscribe an icosidodecahedron. If the hexagonal face is scaled by the Golden Ratio, then the convex hull of the result will be the entire icosidodecahedron. Eric W. Weisstein, Triangular hebesphenorotunda at MathWorld.
Triangular hebesphenorotunda
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Triangular hebesphenorotunda
121.
Truncated triakis tetrahedron
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It is constructed from taking a triakis tetrahedron by truncating the order-6 vertices. This creates 4 regular hexagon leaves 12 irregular pentagons. If all of both kinds, are truncated, the resulting solid is an irregular icosahedron, whose dual is a trihexakis truncated tetrahedron. Truncation of only the simpler vertices yields what looks like a tetrahedron with each face raised by a low frustum. The dual to that truncation will be the truncated tetrahedron. Near-miss Johnson solid Johnson Solid Near Misses: Number 22 George Hart's Polyhedron generator - "t6kT"
Truncated triakis tetrahedron
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Truncated triakis tetrahedron
122.
Prismoid
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In geometry, a prismatoid is a polyhedron where all vertices lie in two parallel planes. Its lateral faces can be triangles. If the lateral faces are either parallelograms or trapezoids, it is called a prismoid. For example in 4-dimensions, two polyhedra can be connected with polyhedral sides. A tetrahedral-cuboctahedral cupola. Weisstein, Eric W. "Prismatoid". MathWorld.
Prismoid
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Pyramids
123.
Hexagonal antiprism
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Antiprisms are similar to prisms that the side faces are triangles, rather than quadrilaterals. In the case of a regular 6-sided base, one usually considers the case where its copy is twisted by an angle 180°/n. Extra regularity is obtained by the line connecting the base centers being perpendicular to the base planes, making it a right antiprism. As faces, it has the two n-gonal bases and, connecting 2n isosceles triangles. If faces are all regular, it is a polyhedron. The hexagonal faces can be replaced by coplanar triangles, leading with 24 equilateral triangles. Weisstein, Eric W. "Antiprism". MathWorld. Hexagonal Antiprism: Interactive Polyhedron model Virtual Reality Polyhedra www.georgehart.com: The Encyclopedia of Polyhedra VRML model Conway Notation for Polyhedra Try: "A6"
Hexagonal antiprism
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Uniform Hexagonal antiprism
124.
Hexagonal pyramid
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In geometry, a hexagonal pyramid is a pyramid with a hexagonal base upon which are erected six triangular faces that meet at a point. Like any pyramid, it is self-dual. A hexagonal pyramid with a regular hexagon base has C6v symmetry. Bipyramid, prism and antiprism Weisstein, Eric W. "Hexagonal Pyramid". MathWorld. Virtual Reality Polyhedra www.georgehart.com: The Encyclopedia of Polyhedra Conway Notation for Polyhedra Try: "Y6"
Hexagonal pyramid
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Hexagonal pyramid
125.
Trihexagonal tiling
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In geometry, the trihexagonal tiling is one of 11 uniform tilings of the Euclidean plane by regular polygons. It consists of regular hexagons, arranged so that each hexagon is surrounded by triangles and vice versa. The name derives from the fact that it combines a regular triangular tiling. Its edges form an infinite arrangement of lines. Its dual is the rhombille tiling. Its place in the classification of uniform tilings, was already known to Johannes Kepler in his 1619 book Harmonices Mundi. The pattern has long been used in Japanese basketry, where it is called kagome. The Japanese term for this pattern has been taken up in physics, where it is called a Kagome lattice. It occurs also in the crystal structures of certain minerals. Conway calls combining alternate elements from a hexagonal tiling and triangular tiling. The weaved process gives a chiral wallpaper group symmetry, p6. First appeared in a 1951 paper by his assistant Ichirō Shōji. The kagome lattice in this sense consists of the edges of the trihexagonal tiling. Despite the name, these crossing points do not form a mathematical lattice. It is represented by the edges of the quarter cubic honeycomb, filling space by regular tetrahedra and truncated tetrahedra.
Trihexagonal tiling
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Japanese basket showing the kagome pattern
Trihexagonal tiling
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Trihexagonal tiling
Trihexagonal tiling
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Periodic
Trihexagonal tiling
126.
Truncated trihexagonal tiling
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In geometry, the truncated trihexagonal tiling is one of eight semiregular tilings of the Euclidean plane. There are one square, one dodecagon on each vertex. It has Schläfli symbol of tr. There is only one uniform coloring of a trihexagonal tiling, with faces colored by polygon sides. A 2-uniform coloring has two colors of hexagons. 3-uniform colorings can have 3 colors of squares. The trihexagonal tiling has three related 2-uniform tilings, one being a 2-uniform coloring of the semiregular rhombitrihexagonal tiling. The first dissects the hexagons into 6 triangles. The other two dissect the dodecagons in two different orientations. The Truncated trihexagonal tiling can be used as a packing, placing equal diameter circles at the center of every point. Every circle is in contact with 3 other circles in the packing. Circles can be alternatedly colored in this packing with an even number of sides of all the regular polygons of this tiling. Each dodecagon allows for 7 circles, creating a dense 4-uniform packing. 3-6 kisrhombille tiling is a tiling of the Euclidean plane. It is constructed with 4, 6, 12 triangles meeting at each vertex.
Truncated trihexagonal tiling
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Periodic
Truncated trihexagonal tiling
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Truncated trihexagonal tiling
Truncated trihexagonal tiling
127.
2-uniform tiling
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Euclidean plane tilings by convex regular polygons have been widely used since antiquity. The first mathematical treatment was that of Kepler in his Harmonices Mundi. This means that, for every pair of flags, there is a operation mapping the first flag to the second. This is equivalent to the tiling being an edge-to-edge tiling by regular polygons. There must be six equilateral triangles, three regular hexagons at a vertex, yielding the three regular tessellations. Vertex-transitivity means that for every pair of vertices there is a operation mapping the first vertex to the second. Note that there are two image forms of 34.6 tiling, only one of, shown in the following table. All regular semiregular tilings are achiral. Though these yield the same set of tilings in the plane, in other spaces there are Archimedean tilings which are not uniform. Periodic tilings may be classified by the number of orbits of vertices, edges and tiles. K-uniform tilings with the same vertex figures can be further identified by their wallpaper symmetry. 1-uniform tilings include 8 semiregular ones, with 2 or more types of regular polygon faces. There are 20 2-uniform tilings, 61 3-uniform tilings, 151 4-uniform tilings, 673 6-uniform tilings. Each can be grouped by the m of distinct vertex figures, which are also called m-Archimedean tilings. For edge-to-edge Euclidean tilings, the internal angles of the polygons meeting at a vertex must add to 360 degrees.
2-uniform tiling
–
Periodic
2-uniform tiling
–
A regular tiling has one type of regular face.
2-uniform tiling
128.
Graphene
–
Graphene is an allotrope of carbon in the form of a two-dimensional, atomic-scale, hexagonal lattice in which one atom forms each vertex. It is the basic structural element including graphite, charcoal, carbon nanotubes and fullerenes. It can also be considered as the ultimate case of the family of flat polycyclic aromatic hydrocarbons. Graphene has many properties. It is about 200 times stronger than the strongest steel. It is nearly transparent. Graphene also can be levitated by Golden magnets. Researchers have identified the bipolar transistor effect, large quantum oscillations in the material. Scientists have theorized for years. It has likely been unknowingly produced in small quantities through the use of pencils and other similar applications of graphite. It only studied while supported on metal surfaces. The material was later rediscovered, characterized in 2004 by Andre Geim and Konstantin Novoselov at the University of Manchester. Research was informed by existing theoretical descriptions of its composition, properties. High-quality graphene proved making more research possible. This work resulted in the two winning the Nobel Prize in Physics in 2010 "for groundbreaking experiments regarding the two-dimensional graphene."
Graphene
–
Graphene is an atomic-scale honeycomb lattice made of carbon atoms.
Graphene
–
A lump of graphite, a graphene transistor and a tape dispenser. Donated to the Nobel Museum in Stockholm by Andre Geim and Konstantin Novoselov in 2010.
Graphene
–
Andre Geim and Konstantin Novoselov, 2010
Graphene
–
Scanning probe microscopy image of graphene
129.
E-ELT
–
The European Extremely Large Telescope is an astronomical observatory and the world's largest optical/near-infrared extremely large telescope now under construction. Part of the European Southern Observatory, it is located on top of Cerro Armazones in the Atacama Desert of northern Chile. The facility is expected to take 11 years to construct. Work on the E-ELT site started in June 2014. First light is planned for 2024. On 26 the European Southern Observatory Council selected Cerro Armazones, Chile, as the baseline site for the planned E-ELT. It should allow the telescope to be finished sooner. The ESO Council expected a construction proposal for approval in December 2011. Funding was subsequently included for initial work to begin in early 2012. The project received preliminary approval in June 2012. ESO approved the start of construction with over 90 % funding secured. The phase of the 5-mirror anastigmat was fully funded within the ESO budget. With the recent changes in the design, the construction cost is estimated to be $1.055 billion. The start of operations is planned for 2024. Current technology limits single mirrors to being roughly 8 metres in a single piece.
E-ELT
–
Engineer rendering of the 39-metre European Extremely Large Telescope (E-ELT)
E-ELT
–
ESO Council meets at ESO headquarters in Garching, 2012.
E-ELT
–
A real night-time panorama of Cerro Armazones, chosen in April 2010.
E-ELT
–
Renderings of the 40-metre-class E-ELT at dusk (left) and from above (right)
130.
Carapace
–
In tortoises, the underside is called the plastron. In crustaceans, the carapace functions as a protective cover over the cephalothorax. Where it projects forward beyond the eyes, this projection is called a rostrum. The carapace is calcified to varying degrees in different crustaceans. Zooplankton within the phylum Crustacea also have a carapace. Isopods only have a developed "cephalic shield" carapace covering the head. In arachnids, the carapace is formed into a single plate which carries the eyes, ocularium, ozopores and diverse phaneres. In a few orders, such as Solifugae and Schizomida, the carapace may be subdivided. Alternative terms for the carapace of their relatives, which avoids confusion with crustaceans, are prosomal dorsal shield and peltidium. The carapace is the dorsal part of the shell structure of a turtle, consisting primarily of the animal's rib cage, dermal armor, scutes.
Carapace
–
The molted carapace of a lady crab from Long Beach, New York.
Carapace
–
Diagram of a prawn, with the carapace highlighted in red.
Carapace
–
Diagram of an arachnid, with the carapace highlighted in purple
Carapace
–
A Greek tortoise shell opened to show the skeleton from below
131.
Saturn
–
Saturn is the sixth planet from the Sun and the second-largest in the Solar System, after Jupiter. It is a giant with an average radius about nine times that of Earth. Although it has only one-eighth the average density of Earth, with its larger volume Saturn is just over 95 times more massive. Saturn is named after the Roman god of agriculture; its astronomical symbol represents the god's sickle. Saturn's interior is probably composed of a core of iron -- rock. Saturn has a yellow hue due to ammonia crystals in its upper atmosphere. Saturn's magnetic strength is around one-twentieth of Jupiter's. The outer atmosphere is generally bland and lacking in contrast, although long-lived features can appear. Wind speeds on Saturn can reach 1,800 km/h, not as high as those on Neptune. Sixty-two moons are known to orbit Saturn, of which fifty-three are officially named. This does not include the hundreds of moonlets comprising the rings. Saturn is a giant because it is predominantly composed of hydrogen and helium. It lacks a definite surface, though it may have a solid core. Saturn's rotation causes it to have the shape of an oblate spheroid;, it is flattened at the bulges at its equator. Its polar radii differ by almost 10 %: 60,268 km versus 54,364 km, respectively.
Saturn
–
Saturn in natural color, photographed by Cassini in 2004
Saturn
Saturn
–
Composite image roughly comparing the sizes of Saturn and Earth
Saturn
–
Auroral lights at Saturn’s north pole.
132.
Voyager 1
–
Voyager 1 is a space probe launched by NASA on September 5, 1977. Part of the Voyager program to study the outer Solar System, Voyager 1 launched 16 days after Voyager 2. Having operated for 39 years, 16 days, the spacecraft still communicates with the Deep Space Network to receive routine commands and return data. The probe's primary mission objectives included Saturn's large moon, Titan. Voyager was the first probe to provide detailed images of their moons. On August 2012, Voyager 1 crossed the heliopause to become the first spacecraft to enter interstellar space and study the interstellar medium. In the 1960s, a Grand Tour to study the outer planets was proposed which prompted NASA to begin work on a mission in the early 1970s. Information gathered by the Pioneer 10 spacecraft helped Voyager's engineers design Voyager to cope more effectively with the intense environment around Jupiter. Initially, Voyager 1 was planned as "Mariner 11" of the Mariner program. Due to budget cuts, the mission was renamed the Mariner Jupiter-Saturn probes. As the program progressed, the name was later changed to Voyager, since the probe designs began to differ greatly from previous Mariner missions. Voyager 1 was constructed by the Jet Propulsion Laboratory. Voyager has 16 hydrazine thrusters, referencing instruments to keep the probe's radio antenna pointed toward Earth. Collectively, these instruments are Articulation Control Subsystem, along with redundant units of most instruments and 8 backup thrusters. The spacecraft also included 11 scientific instruments to study celestial objects such as planets as it travels through space.
Voyager 1
–
Voyager 1, artist's impression
Voyager 1
Voyager 1
Voyager 1
–
Voyager 1 lifted off with a Titan IIIE
133.
Cassini-Huygens
–
Cassini–Huygens is an unmanned spacecraft sent to the planet Saturn. It is a Flagship-class NASA -- ESA -- ASI spacecraft. Its mission is ongoing as of 2016. It has studied its many natural satellites since arriving there in 2004. Development started in the 1980s. Its design includes a lander for the moon Titan. The two spacecraft are named after astronomers Giovanni Cassini and Christiaan Huygens. On December 2004, Huygens separated from the orbiter and landed on Saturn's moon Titan on January 14, 2005. It successfully returned data to Earth, using the orbiter as a relay. This was the first landing ever accomplished in the outer Solar System. The probe will dive into the planet to avoid biological contamination of Saturn's moons. As of November 30th 2016, Cassini will enter the final phase of the project. Cassini will dive through the outer ring of 20 times, once every 7 days. The spacecraft will enter areas that have been untouched up until this point, getting the closest look ever at Saturn's outer rings. The first pass of the rings took place on December 2016.
Cassini-Huygens
–
Artist's concept of Cassini 's orbit insertion around Saturn
Cassini-Huygens
–
Huygens' explanation for the aspects of Saturn, Systema Saturnium, 1659.
Cassini-Huygens
–
Saturn's north side (2014)
Cassini-Huygens
134.
Benzene
–
Benzene is an important organic chemical compound with the chemical formula C6H6. The benzene molecule is composed of 6 carbon atoms joined with 1 hydrogen atom attached to each. Because it contains only hydrogen atoms, benzene is classed as a hydrocarbon. Benzene is one of the elementary petrochemicals. Because of the continuous pi bond between the carbon atoms, benzene is classed as an aromatic hydrocarbon, the second - annulene. It is sometimes abbreviated Ph–H. Benzene is a colorless and highly flammable liquid with a sweet smell, is responsible for the aroma around petrol stations. Because benzene has a high number, it is an important component of gasoline. Because benzene is a human carcinogen, most non-industrial applications have been limited. An acidic material was named "flowers of benzoin", or benzoic acid. The hydrocarbon derived from benzoic acid thus acquired the name benzin, benzene. In 1833, Eilhard Mitscherlich produced it by distilling benzoic lime. He gave the name benzin. In 1845, Charles Mansfield, working under August Wilhelm von Hofmann, isolated benzene from tar. Four years later, Mansfield began the industrial-scale production of benzene, based on the coal-tar method.
Benzene
–
A bottle of benzene. The warnings show benzene is a toxic and flammable liquid.
Benzene
–
Geometry of molecule
135.
Aromatic compound
–
Aromatic molecules do not break easily to react with other substances. Organic compounds that are not aromatic are classified as aliphatic compounds—they might be cyclic, but only aromatic rings have special stability. Since the most common aromatic compounds are derivatives of benzene, the word “aromatic” occasionally refers informally to benzene derivatives, so it was first defined. Nevertheless, aromatic compounds exist. In living organisms, for example, the most common aromatic rings are the double-ringed bases in RNA and DNA. An aromatic functional group or other substituent is called an aryl group. The earliest use of the term "aromatic" was in an article by August Wilhelm Hofmann in 1855. Hofmann used the term for a class of benzene compounds, many of which have odors, unlike pure saturated hydrocarbons. In terms of the electronic nature of the molecule, aromaticity describes a conjugated system often made of alternating single and double bonds in a ring. This configuration allows for the electrons in the molecule's system increasing the molecule's stability. The molecule can not be represented by one structure, but rather a hybrid such as with the two resonance structures of benzene. These molecules can not be found with the shorter double bond in another. Rather, the molecule exhibits bond lengths between those of double bonds. The model for benzene consists of two resonance forms, which corresponds to the double and single bonds superimposing to produce six one-and-a-half bonds. Benzene is a more stable molecule than would be expected without accounting for charge delocalization.
Aromatic compound
–
Two different resonance forms of benzene (top) combine to produce an average structure (bottom)
136.
Coronene
–
Coronene is a polycyclic aromatic hydrocarbon comprising six peri-fused benzene rings. Its formula is C 24H 12. It is a yellow material that dissolves as benzene, toluene, dichloromethane. Its solutions emit light fluorescence under UV light. Its spectrum is not symmetrical with its excitation spectrum and varies in the number of bands and their relative intensities depending on the solvent. It has been used as a solvent probe, similar to pyrene. It occurs naturally as the very rare carpathite, characterized by flakes of pure coronene embedded in sedimentary rock. This mineral may be created from ancient vent activity. In earlier times this mineral was also called Karpatite or Pendletonite. Coronene is produced in the petroleum-refining process of hydrocracking, where it can dimerize to a fifteen PAH, trivially named "dicoronylene". The compound is to organic chemists because of its aromaticity. It can be described by a set of three mobile Clar sextets. Hexa-benzopericoronenes investigated in supramolecular electronics. They are known to self-assemble into a phase. One derivative in particular forms carbon nanotubes with electrical properties.
Coronene
–
Coronene
137.
Basalt
–
Flood basalt describes the formation in a series of basalt flows. Basalt commonly features a very glassy matrix interspersed with visible mineral grains. The average density is 3.0 gm/cm3. Physical descriptions without mineralogical context may be unreliable in some circumstances. Although usually characterized as "dark", basaltic rocks exhibit a wide range of shading due to geochemical processes. Due to high concentrations of plagioclase, some basalts can be quite light-coloured, superficially resembling andesite to untrained eyes. These phenocrysts usually are of a calcium-rich plagioclase, which have the highest melting temperatures of the typical minerals that can crystallize from the melt. Gabbro is often marketed commercially as "black granite." These volcanic rocks, with silica contents below 45 % are usually classified as komatiites. Tholeiitic basalt is poor in sodium. Included in this category are most basalts of the ocean floor, most large oceanic islands, continental flood basalts such as the Columbia River Plateau. Alkali basalt is rich in sodium. It may contain feldspathoids, alkali feldspar and phlogopite. Boninite is a high-magnesium form of basalt, erupted generally in back-arc basins, distinguished by trace-element composition. Ocean island basalt Lunar basalt On Earth, most basalt magmas have formed by melting of the mantle.
Basalt
–
Basalt
Basalt
–
Columnar basalt flows in Yellowstone National Park, USA.
Basalt
–
Columnar basalt at Szent György Hill, Hungary
Basalt
–
Vesicular basalt at Sunset Crater, Arizona. US quarter for scale.
138.
Northern Ireland
–
Northern Ireland is a top-level constituent unit of the United Kingdom in the northeast of Ireland. It is variously described amongst other terms. Northern Ireland shares a border to the west with the Republic of Ireland. In 2011, its population was 1,810,863, constituting about 3 % of the UK's population. Northern Ireland was created in 1921, when Ireland was partitioned by an act of the British parliament. Northern Ireland has historically been the most industrialised region of Ireland. After declining as a result of the social turmoil of the Troubles, its economy has grown significantly since the late 1990s. 58.2% of those unemployed had been unemployed for over a year. Prominent artists and persons from Northern Ireland include Van Morrison, Rory McIlroy, Joey Dunlop, Wayne McCullough and George Best. Some people from Northern Ireland prefer to identify as Irish while others prefer to identify as British. In many sports, the island of Ireland fields a single team, a notable exception being football. People from Northern Ireland may compete for either Great Britain or Ireland at the Olympic Games. The region, now Northern Ireland was the bedrock of the Irish war of resistance against English programmes of colonialism in the 16th century. Irish resistance made English control fragmentary. In Northern Ireland, the victories of the Battle of the Boyne in this latter war are still celebrated by some Protestants.
Northern Ireland
–
Scrabo Tower, County Down
Northern Ireland
–
Location of Northern Ireland (dark green) – in Europe (green & dark grey) – in the United Kingdom (green)
Northern Ireland
–
Cannon on Derry 's city walls
Northern Ireland
–
Signing of the Ulster Covenant in 1912 in opposition to Home Rule
139.
Dry Tortugas National Park
–
Dry Tortugas National Park is a national park in the United States about 68 miles west of Key West in the Gulf of Mexico. The park preserves Fort Jefferson and the seven Dry Tortugas islands, most isolated of the Florida Keys. The archipelago's coral reefs are the least disturbed of the Florida Keys reefs. The park is noted for abundant sea life, tropical bird breeding grounds, legends of shipwrecks and sunken treasures. The park's centerpiece is a massive but unfinished coastal fortress. Fort Jefferson is composed of more than 16 million bricks. Among United States forts it is exceeded only by Fort Monroe, Virginia and Fort Adams, Rhode Island. Dry Tortugas is unique in its combination of a largely undisturbed tropical ecosystem with historic artifacts. The park has averaged less than 70,000 annual visitors since the year 2000. Activities include kayaking. Dry Tortugas National Park is Dry Tortugas Biosphere Reserve, established by UNESCO in 1976 under its Man and the Biosphere Programme. The Dry Tortugas is a small archipelago of coral islands about 70 miles west of Florida. They represent the westernmost extent of the Florida Keys, though several reefs and submarine banks continue westward beyond the Tortugas. The area is more than 99 percent water. The park is bordered by the Tortugas Ecological Reserve.
Dry Tortugas National Park
–
Underwater artifact with sea life
140.
James Webb Space Telescope
–
The James Webb Space Telescope, previously known as Next Generation Space Telescope, is a Flagship-class space observatory under construction and scheduled to launch in October 2018. The JWST will offer unprecedented resolution and sensitivity through near-infrared to the mid-infrared. While Hubble has a 2.4-meter mirror, the JWST will be located near the Earth -- Sun L2 point. A large sunshield will keep its mirror and four science instruments below 50 K. JWST's capabilities will enable a broad range of investigations across the fields of astronomy and cosmology. One particular goal involves observing some of the most distant objects in the Universe, such as the formation of the first galaxies. These types of targets are beyond the reach of space-based instruments. Another goal is understanding the formation of planets. This will include direct imaging of exoplanets. It is named after the second administrator of NASA, who played an integral role in the Apollo program. The JWST has a history of major cost delays. The realistic budget estimates were that the observatory would cost $1.6 billion and launch in 2011. NASA has now scheduled the telescope for a 2018 launch. Funding was capped at $8 billion. As of winter 2015 -- 2016, the telescope remained for an October 2018 launch and within the 2011 revised budget.
James Webb Space Telescope
–
Full-scale James Webb Space Telescope model at South by Southwest in Austin
James Webb Space Telescope
–
Two alternate Hubble Space Telescope views of the Carina Nebula, comparing visible (top) and infrared (bottom) astronomy
James Webb Space Telescope
–
Infrared observations can see objects hidden in visible light, HUDF-JD2 shown
James Webb Space Telescope
–
JWST will not be exactly at the L2 point, but circle around it in a halo orbit.
141.
Metropolitan France
–
Metropolitan France is the part of France in Europe. It comprises mainland France and nearby islands in the Atlantic Ocean, the Mediterranean Sea, including Corsica. Overseas France is the collective name for the part of France outside Europe: French overseas regions, territories, the sui generis collectivity of New Caledonia. Metropolitan France and Overseas France together form the French Republic. Metropolitan France accounts for 82.2 % of the land territory, 95.9 % of the population of the French Republic. The five overseas regions -- Mayotte -- have the same political status as metropolitan France's regions. In overseas France, a person from metropolitan France is often called a métro, short for métropolitain. Similar terms existed to describe European colonial powers. By extension "metropolis" and "metropolitan" came to mean "motherland", country as opposed to its colonies overseas. There are some people in overseas France who object to the use of the term la France métropolitaine due to its colonial origins. They prefer to call it "the European territory of France", as the Treaties of the European Union do. Likewise, they oppose treating overseas France and metropolitan France as separate entities. Since the end of the 1990s INSEE has included the five overseas departments in its figures for France. Other branches of the French administration may have different definitions of what la France entière is. The World Bank refers to this as "France" only, not "the whole of France" as INSEE does.
Metropolitan France
–
Metropolitan France
142.
Guadeloupe
–
Guadeloupe is an insular region of France located in the Leeward Islands, part of the Lesser Antilles in the Caribbean. Administratively, it is an overseas region consisting of a single overseas department. It has a area of an estimated population of 400,132 as of January 2015. Guadeloupe's two main islands are Basse-Terre to the west and Grande-Terre to the east, which are separated by a narrow strait, crossed with bridges. They are often referred to as a single island. The department also includes the Dependencies of Guadeloupe which include the smaller islands of Marie-Galante and La Désirade, the Îles des Saintes. Guadeloupe, like the other overseas departments, is an integral part of France. It is thus part of the European Union and the Eurozone, its currency is the euro. As an overseas department, Guadeloupe is not part of the Schengen Area. The prefecture of Guadeloupe is the city of Basse-Terre, which lies on the island of the same name. The official language is French, virtually the entire population except recent arrivals from metropolitan France also speaks Antillean Creole. Christopher Columbus named Santa María de Guadalupe in 1493 after the Virgin Mary, venerated in Extremadura. The island was called "Karukera" by the Arawak people, who settled on there in 300 AD/CE. During the 8th century, the Caribs killed the existing population on the island. In November 1493, Christopher Columbus became the first European to land while seeking fresh water.
Guadeloupe
–
The Battle of the Saintes fought near Guadeloupe between France and Britain, 1782.
Guadeloupe
–
Flag
Guadeloupe
–
A bust of French abolitionist Victor Schoelcher.
Guadeloupe
–
A satellite photo of Guadeloupe.
143.
Martinique
–
Like Guadeloupe, it is an overseas region of France, consisting of a single overseas department. One of the Windward Islands, it is directly north of Saint Lucia, southeast of Puerto Rico, northwest of Barbados, south of Dominica. As with the other overseas departments, Martinique is one of the eighteen regions of France and an integral part of the French Republic. As part of France, Martinique is part of the European Union, its currency is the euro. The official language is French, virtually the entire population also speak Antillean Creole. Martinique owes its name to Christopher Columbus, who sighted the island in 1493, finally landed on 15 June 1502. The island was then called "Jouanacaëra-Matinino", which came from a mythical island described by the Tainos of Hispaniola. According to historian Sydney Daney, the island was called "Jouanacaëra" by the Caribs, which means "the island of iguanas". When Columbus returned to the island in 1502, he rechristened the island as Martinica. The name then evolved into Madinina, Madiana, Matinite. Finally, through the influence of the neighboring island of Dominica, it came to be known as Martinique. The island was occupied first by Arawaks, then by Caribs. The Carib people had migrated from the mainland to the islands about 1201 CE, according to carbon dating of artifacts. They were largely displaced, exterminated and assimilated by the Taino, who were resident on the island in the 1490s. Martinique was charted by Columbus in 1493, but Spain had little interest in the territory.
Martinique
–
Saint-Pierre. Before the total destruction of Saint-Pierre in 1902 by a volcanic eruption, it was the most important city of Martinique culturally and economically, being known as "the Paris of the Caribbean".
Martinique
–
Flag
Martinique
–
The attack on the French ships at Martinique in 1667
Martinique
–
The Battle of Martinique between British and French fleets in 1779
144.
French Guiana
–
French Guiana, officially called Guiana, is an overseas department and region of France, located on the north Atlantic coast of South America in the Guyanas. It borders Suriname to the west. By area, it is the second largest region of France and the largest outermost region within the European Union. The area was originally inhabited by Native Americans. It was colonised by other Europeans, who introduced African slaves and later Asian labourers and Hmong refugees from Laos. The official language is French, while each ethnic community has its own language, of which Guianan Creole is the most widely spoken. Its official currency is the euro. The region is the most prosperous territory in South America with the highest GDP per capita. A large part of Guiana's economy derives from the presence of the Guiana Space Centre, now the European Space Agency's primary site near the equator. Guiana means "land of many waters". French Guiana was originally inhabited by indigenous people: Kalina, Arawak, Emerillon, Galibi, Palikur, Wayampi and Wayana. The French attempted to create a colony there with its settlement of some other Caribbean islands. In this penal colony, the convicts were sometimes used as butterfly catchers. During its existence, France transported approximately 56,000 prisoners to Devil's Island. Fewer than 10% survived their sentence.
French Guiana
–
Forested landscape of Remire-Montjoly.
French Guiana
–
Flag
French Guiana
–
View from the île Royale
French Guiana
–
Liana on a palm branch near a lake in Kourou
145.
Hanksite
–
Hanksite is a sulfate mineral, distinguished as one of only a handful that contain both carbonate and sulfate ion groups. It has the chemical formula: Na22K92Cl. It was first named for American geologist Henry Garber Hanks. Hanksite is normally found as evaporite deposits. Hanksite crystals are large but not complex in structure. It is often found in Death Valley. It is associated in the Searles Lake area. Hanksite is transparent or translucent. The mineral's hardness is approximately 3 to 3.5. The specific gravity is approximately 2.5. It sometimes glows pale yellow in ultra-violet light. Typical growth habits are hexagonal prisms or tabular with pyramidal terminations. The streak of Hanksite is white. It can contain inclusions of clay that the crystal formed around while developing. Halite borax trona nahcolite tincalconite
Hanksite
–
Hanksite crystal from Searles Lake
146.
Hexagonal crystal system
–
In crystallography, the hexagonal crystal family is one of the 6 crystal systems. There are 52 space groups associated with it, which are exactly those whose Bravais lattice is either hexagonal or rhombohedral. The hexagonal crystal family consists of two lattice systems: hexagonal and rhombohedral. Each lattice system consists of one Bravais lattice. Hence, there are 3 lattice points per unit cell in total and the lattice is non-primitive. The Bravais lattices in the hexagonal crystal family can also be described by rhombohedral axes. The unit cell is a rhombohedron. This is a unit cell with parameters a = b = c; α = β = γ ≠ 90°. In practice, the hexagonal description is more commonly used because it is easier to deal with a coordinate system with two 90° angles. However, the rhombohedral axes are often shown in textbooks because this cell reveals 3m symmetry of crystal lattice. However, such a description is rarely used. The hexagonal crystal family consists of two crystal systems: trigonal and hexagonal. A system is a set of point groups in which their corresponding space groups are associated with a system. The trigonal crystal system consists of the 5 point groups that have a single three-fold rotation axis. The crystal structures of alpha-quartz in the previous example are described by two of those 18 space groups associated with the hexagonal lattice system.
Hexagonal crystal system
–
An example of the hexagonal crystals, beryl
Hexagonal crystal system
–
Hexagonal Hanksite crystal
147.
The Hexagon
–
The Hexagon is a multi-purpose theatre and arts venue in Reading, Berkshire, England. The theatre was built by Robert Matthew Johnson Marshall, who also built the adjacent Civic Centre. The original design featured a proscenium but no tower. Upon opening, the venue was comparable to Derby's Assembly Rooms -- which also opened in 1977 -- but the Hexagon was described acoustically superior. As the building was designed operate as a multi-use venue, the arena-style seating was used to avoid limited visibility. This rendered a number of seats unusable during performances that utilised the proscenium. The stalls, which use retractable seats, have less headroom than the balcony above. This results in shallow overhangs. The floor, which usually holds stall seating, is adjustable to allow a contiguous service with the stage, providing a 517 square metres surface. The balconies, which are separated by gaps around the auditorium, are similar to those at Christchurch Town Hall. Similarly, the inclined panels around the room—to introduce reflections—may have been inspired by that venue. The venue allows a number of different seating configurations, which affect the capacity. For performances with a arch, the capacity is 946. This increases for standing with balcony seating. The theatre's diameter is roughly 30 metres.
The Hexagon
–
View of The Hexagon from San Francisco Libre Walk
148.
Theatre
–
The performers may communicate this experience through combinations of gesture, speech, song, music, dance. Elements of art, such as painted stagecraft such as lighting are used to enhance the physicality, presence and immediacy of the experience. The specific place of the performance is also named by the word "theatre" as derived from the Greek θέατρον, itself from θεάομαι. Modern theatre, broadly defined, includes performances of musical theatre. There are the art forms of ballet, opera and various other forms. The city-state of Athens is where western theatre originated. Participation in the city-state's many festivals -- and attendance as an audience member in particular -- was an important part of citizenship. The Greeks also developed the concepts of dramatic criticism and architecture. Actors were either amateur or at best semi-professional. The theatre of ancient Greece consisted of three types of drama: tragedy, comedy, the satyr play. The origins of theatre in ancient Greece, according to the first theoretician of theatre, are to be found in the festivals that honoured Dionysus. The performances were given in semi-circular auditoria cut into capable of seating 10,000 -- 20,000 people. The stage consisted of a floor, dressing room and scene-building area. Since the words were the most important part, clear delivery were paramount. Each might play several parts.
Theatre
–
Sarah Bernhardt as Hamlet, in 1899
Theatre
–
A master (right) and his slave (left) in a Greek phlyax play, circa 350/340 BCE
Theatre
–
Mosaic depicting masked actors in a play: two women consult a "witch"
Theatre
–
Performer playing Sugriva in the Koodiyattam form of Sanskrit theatre.
149.
Reading, Berkshire
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Reading is a large town and unitary authority area in the ceremonial county of Berkshire, England of which it is the county town. It was an important centre in the medieval period, as the site of Reading Abbey, a monastery with strong royal connections. The 19th century saw baking and seed growing businesses. Despite its proximity to London, has a inward commuter flow. The first evidence for Reading as a settlement dates from the 8th century. By 1611, it had a population of over 5000 and had grown rich on its trade in cloth. During the 19th century, the town grew rapidly as a manufacturing centre. It is ranked the UK's economic area according to factors such as employment, health, income and skills. Reading is also a retail centre serving a large area of the Thames Valley, is home to the University of Reading. It hosts one of England's biggest music festivals. The Borough of Reading has a population of 160,825 whereas the town of Reading has a total population of 232,662. This figure currently makes Reading the largest settlement in the UK without city status. Despite this, Reading functions is often mistaken as such by visitors. The town forms the largest part of the Reading/Wokingham Urban Area which had a population of 318,014. This population was originally higher until Bracknell was transferred to the Greater London Urban Area.
Reading, Berkshire
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From top left: the Town Hall and St Laurence's Church, the Maiwand Lion, the Town Centre skyline from the Royal Berkshire Hospital, Reading Abbey and The Oracle
Reading, Berkshire
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The earliest map of Reading, published in 1611 by John Speed
Reading, Berkshire
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View of Reading from Caversham by Joseph Farington in 1793
Reading, Berkshire
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Reading Crown Court
150.
Hexagonal chess
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The term hexagonal chess designates a group of chess variants played on boards composed of hexagons. The best known is Gliński's variant, played on a 91-cell hexagonal board. Since each hexagonal cell not on a edge has six neighbor cells, there is increased mobility for pieces compared to a standard orthogonal chessboard. Different shapes and sizes of hexagon-based boards are used by variants. The nature of the game is also affected by the 30-degree orientation of the boardcells; the board can be horizontally or vertically orientated. The six-sidedness of the symmetric gameboard has also resulted in a number of three-player variants. Two early examples did not include checkmate as the winning objective. More chess-like games for hex-based boards started appearing regularly at the beginning of the 20th century. Hex-celled gameboards have grown in use for strategy games generally; for example they are popularly used in modern wargaming. The game was popular in Eastern Europe, especially in Gliński's native Poland. More than 130,000 board sets were sold. The usual set of chess pieces is increased by one pawn. The board has 11 files, marked by 11 ranks. Ranks each contain 11 cells, rank 7 has 9 cells, rank 8 has 7, so on. Rank 11 contains exactly one cell: f11.
Hexagonal chess
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Gliński's hexagonal chess (1936) is the best known and probably most widely played hexagonal chess.
151.
Taiwan
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Taiwan, officially the Republic of China, is a state in East Asia. Neighbors include the People's Republic of China to the west, the Philippines to the south. Taiwan is the one with the largest economy. The island of Taiwan was mainly inhabited before the 17th century, when Dutch and Spanish colonies opened the island to Han Chinese immigration. After a brief rule by the Kingdom of Tungning, the island was annexed by the last dynasty of China. The Qing ceded Taiwan after the Sino-Japanese War. While Taiwan was under Japanese rule, the Republic of China was established after the fall of the Qing dynasty. Following the Japanese surrender to the Allies in 1945, the ROC took control of Taiwan. In the early 1960s, Taiwan entered a period of industrialization, creating a stable industrial economy. In early 1990s, it changed from a one-party military dictatorship dominated by the Kuomintang to a multi-party democracy with universal suffrage. Its high-tech industry plays a key role in the global economy. It is ranked highly in terms of freedom of the press, health care, public education, human development. The ROC continued to represent China at the United Nations until 1971, when the PRC assumed China's seat, causing the ROC to lose its UN membership. The PRC refused diplomatic relations with any country that recognizes the ROC. Although Taiwan is fully self-governing, most international organizations either allow it to participate only as a non-state actor.
Taiwan
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A young Tsou man
Taiwan
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Flag
Taiwan
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Overview of Fort Zeelandia, painted around 1635
Taiwan
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Japanese colonial soldiers march Taiwanese captured after the Tapani Incident from the Tainan jail to court, 1915.
152.
24-cell
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In geometry, the 24-cell is the convex regular 4-polytope with Schläfli symbol. It is also called C24, icositetrachoron, octaplex, icosatetrahedroid, octacube, polyoctahedron, being constructed of octahedral cells. The boundary of the 24-cell is composed of 24 octahedral cells with three at each edge. Together they have 96 triangular faces, 24 vertices. The figure is a cube. The 24-cell is self-dual. In fact, the 24-cell is the unique convex self-dual Euclidean polytope, neither a polygon nor a simplex. Due to this property, it does not have a good analogue in 3 dimensions. A 24-cell is given as the convex hull of its vertices. The other 16 are the vertices of the dual tesseract. This gives equivalent to cutting a tesseract into 8 cubical pyramids, then attaching them to the facets of a second tesseract. This is equivalent to the dual of a rectified 16-cell. The analogous construction in 3-space gives the rhombic dodecahedron which, however, is not regular. We can further divide the last 16 vertices with an odd number. Each of groups of 8 vertices also define a regular 16-cell.
24-cell
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Schlegel diagram (vertices and edges)
153.
Four-dimensional space
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In mathematics, four-dimensional space is a geometric space with four dimensions. It typically is more specifically Euclidean space, generalizing the rules of Euclidean space. Algebraically, it is generated by applying the rules of vectors and coordinate geometry to a space with four dimensions. In particular, a vector with four components can be used to represent a position in four-dimensional space. Spacetime is not a Euclidean space. Lagrange wrote in his Mécanique analytique that mechanics can be viewed as operating in a four-dimensional space — three dimensions of space, one of time. The possibility of geometry in higher dimensions, including four dimensions in particular, was thus established. An arithmetic of four dimensions called quaternions was defined by William Rowan Hamilton in 1843. This associative algebra was the source of the science of vector analysis in three dimensions as recounted in A History of Vector Analysis. Soon after tessarines and coquaternions were introduced as other four-dimensional algebras over R. Hinton's ideas inspired a fantasy about a "Church of the Fourth Dimension" featured by Martin Gardner in his January 1962 "Mathematical Games column" in Scientific American. In 1886 Victor Schlegel described his method of visualizing four-dimensional objects with Schlegel diagrams. But the geometry of spacetime, being non-Euclidean, is profoundly different from that popularised by Hinton. The study of Minkowski space required new mathematics quite different from that of four-dimensional Euclidean space, so developed along quite different lines. Minkowski's geometry of space-time is not Euclidean, consequently has no connection with the present investigation.
Four-dimensional space
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5-cell
Four-dimensional space
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3D projection of a tesseract undergoing a simple rotation in four dimensional space.
154.
Orthoplex
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In geometry, a cross-polytope, orthoplex, hyperoctahedron, or cocube is a regular, convex polytope that exists in n-dimensions. A 2-orthoplex is a square, a 3-orthoplex is a regular octahedron, a 4-orthoplex is a 16-cell. Its facets are simplexes of the previous dimension, while the cross-polytope's figure is another cross-polytope from the previous dimension. The vertices of a cross-polytope are all the permutations of. The cross-polytope is the convex hull of its vertices. The n-dimensional cross-polytope can also be defined as the closed ball in the ℓ1-norm on Rn:. In 1 dimension the cross-polytope is simply the segment, in 2 dimensions it is a square with vertices. In 3 dimensions it is an octahedron—one of the five convex regular polyhedra known as the Platonic solids. Higher-dimensional cross-polytopes are generalizations of these. The cross-polytope is the dual polytope of the hypercube. The 1-skeleton of a n-dimensional cross-polytope is a Turán graph T. The 4-dimensional cross-polytope also goes by the name 16-cell. It is one of six regular 4-polytopes. These 4-polytopes were first described in the mid-19th century. The infinite tessellations of hypercubes, he labeled as δn.
Orthoplex
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2 dimensions square
155.
Self-dual
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Such involutions sometimes have fixed points, so that the dual of A is A itself. For example, Desargues' theorem is self-dual in this sense under the standard duality in projective geometry. From a category viewpoint, duality can also be seen as a functor, at least in the realm of vector spaces. In the words of Michael Atiyah, Duality in mathematics is not a "principle". A simple, maybe the most simple, duality arises from considering subsets of a fixed set S. To any subset A ⊆ S, the complement Ac consists of all those elements in S which are not contained in A. It is again a subset of S. Taking the complement has the following properties: Applying it twice gives back the original set, i.e. c = A. This is referred by saying that the operation of taking the complement is an involution. An inclusion of sets A ⊆ B is turned into an inclusion in the opposite direction Bc ⊆ Ac. Given B of S, A is contained in Bc if and only if B is contained in Ac. Because of this, many theorems about closed sets are dual to theorems about open sets. For example, any union of open sets is open, so dually, any intersection of closed sets is closed. Because of the duality, the complement of the interior of any set U is equal to the closure of the complement of U. A duality in geometry is provided by the dual construction.
Self-dual
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A set C (blue) and its dual cone C* (red).
156.
Euclidean space
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In geometry, Euclidean space encompasses the two-dimensional Euclidean plane, the three-dimensional space of Euclidean geometry, certain other spaces. It is named after the Greek mathematician Euclid of Alexandria. The term "Euclidean" distinguishes these spaces from other types of spaces considered in modern geometry. Euclidean spaces also generalize to higher dimensions. Greek geometry defined the Euclidean plane and Euclidean three-dimensional space using certain postulates, while the other properties of these spaces were deduced as theorems. Geometric constructions are also used to define rational numbers. Geometric shapes are defined as equations and inequalities. From the modern viewpoint, there is essentially only one Euclidean space of each dimension. With Cartesian coordinates it is modelled by the coordinate space of the same dimension. Euclidean spaces have finite dimension. One way to think of the Euclidean plane is as a set of points satisfying expressible in terms of distance and angle. For example, there are two fundamental operations on the plane. One is translation, which means a shifting of the plane so that every point is shifted by the same distance. Even when used in physical theories, Euclidean space is an abstraction detached from actual physical locations, specific reference frames, so on. The reason for working with arbitrary vector spaces instead of Rn is that it is often preferable to work in a coordinate-free manner.
Euclidean space
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A sphere, the most perfect spatial shape according to Pythagoreans, also is an important concept in modern understanding of Euclidean spaces
157.
Hexagonal number
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A hexagonal number is a figurate number. The formula for the nth hexagonal number h n = 2 n 2 − n = n = 2 n × 2. Only every other triangular number is a hexagonal number. Like a number, the digital root in base 10 of a hexagonal number can only be 1, 3, 6, or 9. The digital pattern, repeating every nine terms, is "1 6 6 1 9 3 1 3 9". No perfect numbers are known, hence all known perfect numbers are hexagonal. The largest number that cannot be written as a sum of at most four hexagonal numbers is 130. Adrien-Marie Legendre proved in 1830 that any integer greater than 1791 can be expressed in this way. Hexagonal numbers can be rearranged by. Hexagonal numbers should not be confused with hexagonal numbers, which model the standard packaging of Vienna sausages. To avoid ambiguity, hexagonal numbers are sometimes called "hexagonal numbers". One can efficiently test whether a positive integer x is an hexagonal number by computing n = 8 x + 1 + 4. If n is an integer, then x is the hexagonal number. If n is not an integer, then x is not hexagonal. Centered hexagonal number Mathworld entry on Hexagonal Number
Hexagonal number
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Contents
158.
Regular tiling
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Euclidean plane tilings by convex regular polygons have been widely used since antiquity. The first mathematical treatment was that of Kepler in his Harmonices Mundi. This means that, for every pair of flags, there is a operation mapping the first flag to the second. This is equivalent to the tiling being an edge-to-edge tiling by regular polygons. There must be six equilateral triangles, three regular hexagons at a vertex, yielding the three regular tessellations. Vertex-transitivity means that for every pair of vertices there is a operation mapping the first vertex to the second. Note that there are two image forms of 34.6 tiling, only one of, shown in the following table. All regular semiregular tilings are achiral. Though these yield the same set of tilings in the plane, in other spaces there are Archimedean tilings which are not uniform. Periodic tilings may be classified by the number of orbits of vertices, edges and tiles. K-uniform tilings with the same vertex figures can be further identified by their wallpaper symmetry. 1-uniform tilings include 8 semiregular ones, with 2 or more types of regular polygon faces. There are 20 2-uniform tilings, 61 3-uniform tilings, 151 4-uniform tilings, 673 6-uniform tilings. Each can be grouped by the m of distinct vertex figures, which are also called m-Archimedean tilings. For edge-to-edge Euclidean tilings, the internal angles of the polygons meeting at a vertex must add to 360 degrees.
Regular tiling
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Periodic
Regular tiling
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A regular tiling has one type of regular face.
Regular tiling
159.
Unicursal hexagram
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The unicursal hexagram is a hexagram or six-pointed star that can be traced or drawn unicursally, in one continuous line rather than by two overlaid triangles. The hexagram can also be depicted inside a circle with the points touching it. It is often depicted with the lines of the hexagram passing over and under one another to form a knot. It is the same shape as depicted in Blaise Pascal's 1639 Hexagrammum Mysticum Theorem. In Aleister Crowley's Thelema, the hexagram is usually depicted with a five-petalled flower in the centre which symbolises a pentacle. The symbol which represents the microcosmic forces interweave with the macro-cosmic forces. The unicursal hexagram was part of the symbol called "The Seal of Orichalcos", prominent in the Waking the Dragons arc of Yu-Gi-Oh! . A unicursal hexagram appears several times in the television Supernatural as a symbol to ward off evil entities. It was also featured prominently in the season 8 episode "As Time Goes By" as the symbol signifying membership of the Men of Letters. It's mentioned that it stands to Atlantis. Serves as a symbol of Inferno fraction in the Heroes of Might and Magic V. 7₄ knot Wigner–Seitz cell As Above, So Below
Unicursal hexagram
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Non-interlaced unicursal hexagram
160.
Honeycomb conjecture
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The conjecture was proven in 1999 by mathematician Thomas C. Hales. Let C be the union of these bounded components. Then lim area ≥ 12 4. Equality is attained for the hexagonal tile. The first record of the conjecture is often attributed to Pappus of Alexandria. The conjecture was proven in 1999 by mathematician Thomas C. Hales, who mentions in his work that there is reason to believe that the conjecture may have been present in the minds of mathematicians before Varro. Weaire–Phelan structure, a counter-example to the Kelvin conjecture on the solution of the similar problem in 3D.
Honeycomb conjecture
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A regular hexagonal grid
161.
International Standard Book Number
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The International Standard Book Number is a unique numeric commercial book identifier. An ISBN is assigned to each variation of a book. For example, an e-book, a hardcover edition of the same book would each have a different ISBN. The ISBN is 13 digits long if assigned after 1 January 2007, 10 digits long if assigned before 2007. The method of assigning an ISBN varies from country to country, often depending on how large the publishing industry is within a country. The initial ISBN configuration of recognition was generated based upon the 9-digit Standard Book Numbering created in 1966. The 10-digit ISBN format was published in 1970 as international standard ISO 2108. The International Standard Serial Number, identifies periodical publications such as magazines; and the International Standard Music Number covers for musical scores. The ISBN configuration of recognition was generated in 1967 in the United Kingdom by Emery Koltay. The 10-digit ISBN format was published as international standard ISO 2108. The United Kingdom continued to use the 9-digit SBN code until 1974. The ISO on-line facility only refers back to 1978. An SBN may be converted by prefixing the digit "0". This can be converted to ISBN 0-340-01381-8; the digit does not need to be re-calculated. Since 1 ISBNs have contained 13 digits, a format, compatible with "Bookland" European Article Number EAN-13s.
International Standard Book Number
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A 13-digit ISBN, 978-3-16-148410-0, as represented by an EAN-13 bar code
162.
Space.com
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Space.com is a space and astronomy news website. Its stories are often syndicated including CNN, MSNBC, Yahoo!, USA Today. Space.com was founded in July 1999. The move was the source of intense media speculation. When the dot-com bubble burst in 2000, many felt that it would collapse. As it expanded, it acquired other web sites such as Starport.com and Explorezone.com. It also acquired Space News. Despite some growth, Space.com was never able to achieve what Dobbs had hoped in 2001, he returned to CNN. He still owns a minority stake. Space.com has enjoyed the participation of several key space-related public figures, Neil Armstrong, Alexei Leonov, Thomas Stafford. For its coverage of the Space Shuttle Columbia disaster, it received the Online Journalism Award for Breaking News by the Online News Association. Media Life Magazine 2 January 2001 Imaginova.com About Us Space.com ouramazingplanet.com
Space.com
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Space.com in October 2006
163.
Monogon
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In geometry a monogon is a polygon with one edge and one vertex. It has Schläfli symbol. Since a monogon has only one vertex, every monogon is regular by definition. In Euclidean geometry a monogon is a degenerate polygon because its endpoints must coincide, unlike any Euclidean segment. Most definitions of a polygon in Euclidean geometry do not admit the monogon. In spherical geometry, a monogon can be constructed as a vertex on a great circle. This forms a dihedron, with two monogonal faces which share one 360 ° edge and one vertex. A hosohedron, has two antipodal vertices at the poles, one 360 degree lune face, one edge between the two vertices. Digon Herbert Busemann, The geometry of geodesics. New York, Academic Press, 1955 Coxeter, H.S.M; Regular Polytopes. Dover Publications Inc. ISBN 0-486-61480-8
Monogon
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On a circle, a monogon is a tessellation with a single vertex, and one 360° arc edge.
164.
Digon
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In geometry, a digon is a polygon with two sides and two vertices. One or both would have to be curved. A regular digon is represented by Schläfli symbol. It may be constructed as a pair of 180 degree arcs connecting antipodal points, when it forms a lune. The digon is the simplest abstract polytope of rank 2. T is a square. H is a monogon. Its two angles are equal. As such, the regular digon is a constructible polygon. Some definitions of a polygon do not consider the digon to be a proper polygon because of its degeneracy in the Euclidean case. A digon as a face of a polyhedron is degenerate because it is a degenerate polygon. But sometimes it can have a topological existence in transforming polyhedra. A spherical lune is a digon whose two vertices are antipodal points on the sphere. A spherical polyhedron constructed from such digons is called a hosohedron. The digon is an important construct in the topological theory of networks such as polyhedral surfaces.
Digon
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On a circle, a digon is a tessellation with two antipodal points, and two 180° arc edges.
165.
Heptagon
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In geometry, a heptagon is a seven-sided polygon or 7-gon. The heptagon is also occasionally referred to as the septagon, using "sept-" together with the Greek suffix "-agon" meaning angle. A regular heptagon, in which all angles are equal, has internal angles of 5π / 7 radians. Its Schläfli symbol is. The area of each of the 14 small triangles is one-fourth of the apothem. This expression cannot be algebraically rewritten without complex components, since the indicated cubic function is casus irreducibilis. This type of construction is called a neusis construction. It is also constructible with compass, angle trisector. Consequently this polynomial is the minimal polynomial of 2cos, whereas the degree of the minimal polynomial for a constructible number must be a power of 2. An approximation for practical use with an error of about 0.2% is shown in the drawing. It is attributed to Albrecht Dürer. Let A lie on the circumference of the circumcircle. Draw BOC. Then B D = 2 B C gives an approximation for the edge of the heptagon. Since 7 is a prime number there is one subgroup with dihedral symmetry: Dih1, 2 cyclic group symmetries: Z7, Z1.
Heptagon
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Cactus
Heptagon
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A regular heptagon
166.
Nonagon
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In geometry, a nonagon /ˈnɒnəɡɒn/ is a nine-sided polygon or 9-gon. The name "enneagon" is arguably more correct, though somewhat less common than "nonagon". A regular nonagon has internal angles of 140 °. Although a regular nonagon is not constructible with compass and straightedge, there are very old methods of construction that produce very close approximations. It can be also constructed by allowing the use of an angle trisector. The following is an approximate construction of a nonagon using compass. Auxiliary line g aims over the point O to the point N, between O and N, therefore no auxiliary line. See also the calculation. The regular enneagon has order 18. There are 2 subgroup dihedral symmetries: Dih3 and Dih1, 3 cyclic group symmetries: Z9, Z1. These 6 symmetries can be seen in 6 distinct symmetries on the enneagon. John Conway labels these by a letter and order. No symmetry is labeled a1. The dihedral symmetries are divided depending on i when reflection lines path through both edges and vertices. Cyclic symmetries in the middle column are labeled as g for their central gyration orders.
Nonagon
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A regular nonagon (enneagon)
167.
Decagon
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In geometry, a decagon is a ten-sided polygon or 10-gon. Each internal angle will always be equal to 144 °. Its Schläfli symbol can also be constructed as a truncated pentagon, t, a quasiregular decagon alternating two types of edges. By simple trigonometry, it can be written algebraically as d = a 5 + 2 5. An alternative method is as follows: a pentagon in a circle by one of the methods shown in constructing a pentagon. Extend a line through the center of the circle to the opposite side of that same circle. Where each line cuts the circle is a vertex of the decagon. The five corners of the pentagon constitute alternate corners of the decagon. Join these points to the new points to form the decagon. In the construction with given circumcircle the circular arc around G with radius GE3 produces the AH, whose division corresponds to the golden ratio. A M ¯ M H ¯ = A H ¯ A M ¯ = 1 + 2 = Φ ≈ 1.618. In the construction with given length the circular arc around D with radius DA produces the segment E10F, whose division corresponds to the golden ratio. The regular decagon has order 20. There are 3 subgroup dihedral symmetries: 4 cyclic group symmetries: Z10, Z5, Z2, Z1. John Conway labels these by a letter and order.
Decagon
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A regular decagon
168.
Hendecagon
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In geometry, a hendecagon or 11-gon is an eleven-sided polygon. A regular hendecagon is represented by Schläfli symbol. A regular hendecagon has internal angles of 147.27 degrees. As 11 is not a prime, the regular hendecagon is not constructible with compass and straightedge. Because 11 is not a prime, construction of a regular hendecagon is still impossible even with the usage of an angle trisector. It can, however, be constructed via neusis construction. Close approximations to the regular hendecagon can be constructed, however. For instance, the Greek mathematicians approximated the side length of a hendecagon inscribed in a unit circle as being 14/25 units long. The regular hendecagon has order 22. Since 11 is a prime number there is one subgroup with dihedral symmetry: Dih1, 2 cyclic group symmetries: Z1. These 4 symmetries can be seen in 4 distinct symmetries on the hendecagon. John Conway labels these by a letter and order. No symmetry is labeled a1. The dihedral symmetries are divided depending on i when reflection lines path through both edges and vertices. Cyclic symmetries in the middle column are labeled as g for their central gyration orders.
Hendecagon
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A regular hendecagon
169.
Tridecagon
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In geometry, a tridecagon or triskaidecagon or 13-gon is a thirteen-sided polygon. A regular tridecagon is represented by Schläfli symbol. As 13 is a prime but not a Fermat prime, the regular tridecagon can not be constructed using a compass and straightedge. However, it is constructible using an angle trisector. An approximate construction of a regular tridecagon using straightedge and compass is shown here. Another possible animation of an approximate construction, also possible with using straightedge and compass. For details, see: Wikibooks: construction description The regular tridecagon has Dih13 symmetry, order 26. Since 13 is a prime number there is one subgroup with dihedral symmetry: Dih1, 2 cyclic group symmetries: Z1. These 4 symmetries can be seen in 4 distinct symmetries on the tridecagon. John Conway labels these by a letter and order. No symmetry is labeled a1. The dihedral symmetries are divided depending on i when reflection lines path through both edges and vertices. Cyclic symmetries in the middle column are labeled as g for their central gyration orders. Each symmetry allows one or more degrees of freedom for irregular forms. Only the g13 subgroup has no degrees of freedom but can seen as directed edges.
Tridecagon
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A regular tridecagon
170.
Tetradecagon
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In geometry, a tetradecagon or tetrakaidecagon or 14-gon is a fourteen-sided polygon. A regular tetradecagon can be constructed as a quasiregular truncated heptagon, t, which alternates two types of edges. However, it is constructible using neusis, for example in the following illustration with use of the angle trisector. Based on the unit circle r = 1 Constructed side lenght of the tetradecagon in GeoGebra a = 0.445041867912629... Side lenght of the tetradecagon a s h o u l d = 2 ⋅ sin = 0.4450418679126288089... 0.2 mm. For details, see: Wikibooks: construction description The regular tetradecagon has Dih14 symmetry, order 28. There are 3 subgroup dihedral symmetries: 4 cyclic group symmetries: Z14, Z7, Z2, Z1. John Conway labels these by a letter and order. No symmetry is labeled a1. The dihedral symmetries are divided depending on i when reflection lines path through both edges and vertices. Cyclic symmetries in the middle column are labeled as g for their central gyration orders. Each symmetry allows one or more degrees of freedom for irregular forms. Only the g14 subgroup has no degrees of freedom but can seen as directed edges. These two forms have half the symmetry order of the regular tetradecagon.
Tetradecagon
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A regular tetradecagon
171.
Pentadecagon
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In geometry, a pentadecagon or pentakaidecagon or 15-gon is a fifteen-sided polygon. A regular pentadecagon is represented by Schläfli symbol. A regular triangle, pentadecagon can completely fill a plane vertex. Compared with the first animation are in the following two images the two circular arcs rotated 90° counterclockwise shown. A compass and straightedge construction for a given side length. Dih15 has 3 dihedral subgroups: Dih5, Dih1. And four more cyclic symmetries: Z15, Z5, Z1, with Zn representing π / n radian rotational symmetry. On the pentadecagon, there are 8 distinct symmetries. John Conway labels these symmetries with a order of the symmetry follows the letter. He gives r30 for Dih15. A1 labels no symmetry. These lower symmetries allows degrees of freedoms in defining irregular pentadecagons. Only the g15 subgroup has no degrees of freedom but can seen as directed edges. Calculation of the circumradius Weisstein, Eric W. "Pentadecagon". MathWorld.
Pentadecagon
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A regular pentadecagon
172.
Hexadecagon
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In mathematics, a hexadecagon or 16-gon is a sixteen-sided polygon. All sides are congruent. Its Schläfli symbol can be constructed as a truncated octagon, t, a twice-truncated square tt. T, is a triacontadigon. As 16 = 24, a regular hexadecagon is constructible using compass and straightedge: this was already known to Greek mathematicians. The total angle measure of any hexadecagon is 2520 degrees. The area of a regular hexadecagon with edge t is A = 4 t 2 cot π 16 = 4 t 2. Since the area of the circumcircle is R 2, the regular hexadecagon fills approximately 97.45 % of its circumcircle. The regular hexadecagon has order 32. There are 4 dihedral subgroups: 5 cyclic subgroups: Z16, Z8, Z4, Z2, Z1, the last implying no symmetry. On the regular hexadecagon, there are 14 distinct symmetries. No symmetry is labeled a1. These two forms have half the symmetry order of the regular hexadecagon. Each symmetry allows one or more degrees of freedom for irregular forms. Only the g16 subgroup has no degrees of freedom but can seen as directed edges.
Hexadecagon
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The hexadecagonal tower from Raphael's The Marriage of the Virgin
Hexadecagon
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A regular hexadecagon
Hexadecagon
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A hexadecagrammic pattern from the Alhambra
173.
Heptadecagon
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In geometry, a heptadecagon or 17-gon is a seventeen-sided polygon. A regular heptadecagon is represented by the Schläfli symbol. This proof represented the first progress in over 2000 years. The explicit construction of a heptadecagon was given by Herbert William Richmond in 1893. The following method of construction uses Carlyle circles, as shown below. Another more recent construction is given by Callagy. The regular heptadecagon has order 34. Since 7 is a prime number there is one subgroup with dihedral symmetry: Dih1, 2 cyclic group symmetries: Z1. These 4 symmetries can be seen in 4 distinct symmetries on the heptadecagon. John Conway labels these by a letter and order. No symmetry is labeled a1. The dihedral symmetries are divided depending on i when reflection lines path through both edges and vertices. Cyclic symmetries in the middle column are labeled as g for their central gyration orders. Each symmetry allows one or more degrees of freedom for irregular forms. Only the g17 subgroup has no degrees of freedom but can seen as directed edges.
Heptadecagon
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A regular heptadecagon
174.
Octadecagon
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An octadecagon or 18-gon is an eighteen-sided polygon. A regular octadecagon can be constructed as a quasiregular truncated enneagon, t, which alternates two types of edges. As 18 = 2 × 32, a regular octadecagon can not straightedge. However, it is constructible using an angle trisector. The approximate construction is very similar to that of the enneagon, as an octadecagon can be constructed as a truncated enneagon. It is also feasible with exclusive use of compass and straightedge. The regular octadecagon has order 36. There are 5 subgroup dihedral symmetries: Dih9, 6 cyclic group symmetries:. These 15 symmetries can be seen in 12 distinct symmetries on the octadecagon. John Conway labels these by a letter and order. No symmetry is labeled a1. The dihedral symmetries are divided depending on i when reflection lines path through both edges and vertices. Cyclic symmetries in the middle column are labeled as g for their central gyration orders. Each symmetry allows one or more degrees of freedom for irregular forms. Only the g18 subgroup has no degrees of freedom but can seen as directed edges.
Octadecagon
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A regular octadecagon
175.
Enneadecagon
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In geometry an enneadecagon or 19-gon is a nineteen-sided polygon. It is also known as a nonadecagon. A regular enneadecagon is represented by Schläfli symbol. The radius of the circumcircle of the regular enneadecagon with side t is R = t 2 csc 180 19. The area, where t is the length, is 19 4 t 2 cot π 19 ≃ 28.4652 t 2. As 19 is a prime but not a Fermat prime, the regular enneadecagon can not be constructed using a compass and straightedge. However, it is constructible using an angle trisector. Another animation of an approximate construction. Based on the unit circle r = 1 Constructed side lenght of the enneadecagon in GeoGebra a = 0.329189180561468... Side lenght of the enneadecagon a s h o u l d = 2 ⋅ sin = 0.329189180561467788... 0.21 mm. The regular enneadecagon has order 38. Since 19 is a prime number there is one subgroup with dihedral symmetry: Dih1, 2 cyclic group symmetries: Z1. These 4 symmetries can be seen in 4 distinct symmetries on the enneadecagon. John Conway labels these by a letter and order.
Enneadecagon
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A regular enneadecagon
176.
Icosagon
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In geometry, an icosagon or 20-gon is a twenty-sided polygon. The sum of any icosagon's interior angles is 3240 degrees. The regular icosagon can also be constructed as a truncated decagon, a twice-truncated pentagon, tt. One interior angle in a regular icosagon is 162 °, meaning that one angle would be °. The area of a regular icosagon with t is A = 5 t 2 ≃ 31.5687 t 2. The Big Wheel on the popular US game show The Price Is Right has an icosagonal cross-section. As a golygonal path, the swastika is considered to be an irregular icosagon. Icosagon can completely fill a plane vertex. There are 5 dihedral symmetries:, 6 group symmetries:. John Conway labels these by a order. Full symmetry of the regular form is r40 and no symmetry is labeled a1. The dihedral symmetries are divided depending on i when reflection lines path through both vertices. Cyclic symmetries in the middle column are labeled as g for their central gyration orders. Each subgroup symmetry allows one or more degrees of freedom for irregular forms. Only the g20 subgroup has no degrees of freedom but can seen as directed edges.
Icosagon
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A regular icosagon
177.
Icositetragon
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In geometry, an icositetragon or 24-gon is a twenty-four-sided polygon. The sum of any icositetragon's interior angles is 3960 degrees. One interior angle in a regular icositetragon is 165 °, meaning that one angle would be °. The area of a regular icositetragon is: A = 6 t 2 cot π 24 = 6 t 2. The icositetragon appeared in Archimedes' approximation of pi, along with dodecagon, tetracontaoctagon, enneacontahexagon. As 24 = 23 × 3, a regular icositetragon is constructible using a compass and straightedge. As a truncated dodecagon, it can be constructed by an edge-bisection of a regular dodecagon. The regular icositetragon has Dih24 symmetry, order 48. There are 7 dihedral symmetries:, 8 group symmetries:. These 16 symmetries can be seen in 22 distinct symmetries on the icositetragon. John Conway labels these by a order. The full symmetry of the regular form is r48 and no symmetry is labeled a1. The dihedral symmetries are divided depending on i when reflection lines path through both vertices. Cyclic symmetries in the middle column are labeled as g for their central gyration orders. Each subgroup symmetry allows one or more degrees of freedom for irregular forms.
Icositetragon
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A regular icositetragon
178.
Triacontagon
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In geometry, a triacontagon or 30-gon is a thirty-sided polygon. The sum of any triacontagon's interior angles is 5040 degrees. The regular triacontagon can also be constructed as a truncated pentadecagon, t. T, is a hexacontagon. One interior angle in a regular triacontagon is 168 °, meaning that one angle would be 12 °. The regular triacontagon has order 60, represented by 30 lines of reflection. Dih30 has 7 dihedral subgroups: and. It also has eight more cyclic symmetries as subgroups:, with Zn representing π / n radian symmetry. John Conway labels these lower symmetries with a order of the symmetry follows the letter. A1 labels no symmetry. These lower symmetries allows degrees of freedoms in defining irregular triacontagons. Only the g30 subgroup has no degrees of freedom but can seen as directed edges. A triacontagram is a 30-sided polygon. , 11 compound star figures with the same vertex configuration. There are also isogonal triacontagrams constructed as deeper truncations of inverted pentadecagrams.
Triacontagon
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A regular triacontagon
179.
Tetracontagon
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In geometry, a tetracontagon or tessaracontagon is a forty-sided polygon or 40-gon. The sum of any tetracontagon's interior angles is 6840 degrees. A regular tetracontagon can also be constructed as a truncated icosagon, t, which alternates two types of edges. Furthermore, it can also be constructed as a twice-truncated decagon, a thrice-truncated pentagon, ttt. One interior angle in a regular tetracontagon is 171 °, meaning that one angle would be 9 °. As a truncated icosagon, it can be constructed by an edge-bisection of a regular icosagon. Transfer this on the circumcircle, there arises the C39. Connect the C39 with the central point M, there arises the angle C39MC1 with 72 °. Halve the C39MC1, there arise the intersection C40 and the angle C40MC1 with 9 °. Connect the C1 with the point C40, there arises the first side length a of the tetracontagon. Finally you transfer the C1C40 repeatedly counterclockwise on the circumcircle until arises a regular tetracontagon. Extend the segment AB by more than two times. Draw each a circular arc about the points A and B, there arise the intersections C and D. Draw a vertical straight line from point C through point D. Draw a parallel line too the segment CD from the point B to the circular arc, there arises the intersection F.
Tetracontagon
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A regular tetracontagon
180.
Pentacontagon
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In geometry, a pentacontagon or pentecontagon or 50-gon is a fifty-sided polygon. The sum of any pentacontagon's interior angles is 8640 degrees. A regular pentacontagon can be constructed as a quasiregular truncated icosipentagon, t, which alternates two types of edges. One interior angle in a regular pentacontagon is 4⁄5 °, meaning that one exterior angle would be 7 1⁄5 °. The regular pentacontagon has order 100, represented by 50 lines of reflection. Dih50 has 5 dihedral subgroups: and. It also has 6 more cyclic symmetries as subgroups:, with Zn representing π / n radian symmetry. John Conway labels these lower symmetries with a order of the symmetry follows the letter. A1 labels no symmetry. These lower symmetries allows degrees of freedom in defining irregular pentacontagons. Only the g50 subgroup has no degrees of freedom but can seen as directed edges. A pentacontagram is a 50-sided polygon. There are 9 regular forms given by and, as well as 16 compound star figures with the same vertex configuration. Weisstein, Eric W. "Pentacontagon". MathWorld.
Pentacontagon
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A regular pentacontagon
181.
Hexacontagon
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In geometry, a hexacontagon or hexecontagon or 60-gon is a sixty-sided polygon. The sum of any hexacontagon's interior angles is 10440 degrees. A regular hexacontagon also can be constructed as a truncated triacontagon, t, or a twice-truncated pentadecagon, tt. T, is a 120-gon. One interior angle in a regular hexacontagon is 174 °, meaning that one angle would be 6 °. Since 60 = 22 × 3 × 5, a regular hexacontagon is constructible using a straightedge. As a truncated triacontagon, it can be constructed by an edge-bisection of a regular triacontagon. The regular hexacontagon has order 120, represented by 60 lines of reflection. Dih60 has 11 dihedral subgroups:, and. And 12 more cyclic symmetries:, and, with Zn representing π/n radian rotational symmetry. These 24 symmetries are related to 32 distinct symmetries on the hexacontagon. John Conway labels these lower symmetries with a order of the symmetry follows the letter. A1 labels no symmetry. These lower symmetries allows degrees of freedom in defining irregular hexacontagons. Only the g60 symmetry has no degrees of freedom but can seen as directed edges.
Hexacontagon
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A regular hexacontagon
182.
Heptacontagon
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In geometry, a heptacontagon or 70-gon is a seventy-sided polygon. The sum of any heptacontagon's interior angles is 12240 degrees. A regular heptacontagon can also be constructed as a truncated triacontapentagon, t, which alternates two types of edges. One interior angle in a regular heptacontagon is 6⁄7 °, meaning that one exterior angle would be 5 1⁄7 °. The regular heptacontagon has order 140, represented by 70 lines of reflection. Dih70 has 7 dihedral subgroups: and. It also has 8 more cyclic symmetries as subgroups:, with Zn representing π / n radian symmetry. John Conway labels these lower symmetries with a order of the symmetry follows the letter. A1 labels no symmetry. These lower symmetries allows degrees of freedoms in defining irregular heptacontagons. Only the g70 subgroup has no degrees of freedom but can seen as directed edges. A heptacontagram is a 70-sided polygon. There are 11 regular forms given by and, as well as 23 regular star figures with the same vertex configuration. Weisstein, Eric W. "Heptacontagon". MathWorld.
Heptacontagon
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A regular heptacontagon
183.
Octacontagon
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In geometry, an octacontagon is an eighty-sided polygon. The sum of any octacontagon's interior angles is 14040 degrees. One interior angle in a regular octacontagon is 1⁄2 °, meaning that one exterior angle would be 4 1⁄2 °. As a truncated tetracontagon, it can be constructed by an edge-bisection of a regular tetracontagon. Dih40 has 9 dihedral subgroups:, and. It also has 10 more cyclic symmetries as subgroups:, with Zn representing π / n radian symmetry. John Conway labels these lower symmetries with a order of the symmetry follows the letter. R160 represents a1 labels no symmetry. These lower symmetries allows degrees of freedoms in defining irregular octacontagons. Only the g80 subgroup has no degrees of freedom but can seen as directed edges. An octacontagram is an 80-sided polygon. There are 15 regular forms given by and, as well as 24 regular star figures with the same vertex configuration. Weisstein, Eric W. "Octacontagon". MathWorld. Naming Polygons and Polyhedra ogdoacontagon
Octacontagon
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A regular octacontagon
184.
Enneacontagon
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In geometry, an enneacontagon or enenecontagon or 90-gon is a ninety-sided polygon. The sum of any enneacontagon's interior angles is 15840 degrees. A regular enneacontagon can be constructed as a truncated tetracontapentagon, t, which alternates two types of edges. One interior angle in a regular enneacontagon is 176 °, meaning that one angle would be 4 °. The regular enneacontagon has order 180, represented by 90 lines of reflection. Dih90 has 11 dihedral subgroups: and. And 12 more cyclic symmetries:, and, with Zn representing π/n radian rotational symmetry. These 24 symmmetries are related to 30 distinct symmetries on the enneacontagon. John Conway labels these lower symmetries with a order of the symmetry follows the letter. A1 labels no symmetry. These lower symmetries allows degrees of freedom in defining irregular enneacontagons. Only the g90 symmetry has no degrees of freedom but can seen as directed edges. An enneacontagram is a 90-sided polygon. There are 11 regular forms given by and, as well as 33 regular star figures with the same vertex configuration. Weisstein, Eric W. "Enneacontagon".
Enneacontagon
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A regular enneacontagon
185.
Hectogon
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In geometry, a hectogon or hecatontagon or 100-gon is a hundred-sided polygon. The sum of any hectogon's interior angles is 17640 degrees. A regular hectogon can be constructed as a truncated pentacontagon, t, or a twice-truncated icosipentagon, tt. One interior angle in a regular hectogon is 2⁄5 °, meaning that one exterior angle would be 3 3⁄5 °. Thus the regular hectogon is not a constructible polygon. The regular hectogon has order 200, represented by 100 lines of reflection. Dih100 has 8 dihedral subgroups:. It also has 9 more cyclic symmetries as subgroups:, with Zn representing π / n radian symmetry. John Conway labels these lower symmetries with a order of the symmetry follows the letter. R200 represents a1 labels no symmetry. These lower symmetries allows degrees of freedom in defining irregular hectogons. Only the g100 subgroup has no degrees of freedom but can seen as directed edges. A hectogram is an 100-sided polygon. There are 19 regular forms given by and, as well as 30 regular star figures with the same vertex configuration. Weisstein, Eric W. "Hectogon".
Hectogon
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A regular hectogon
186.
257-gon
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In geometry, a 257-gon is a polygon with 257 sides. The sum of the interior angles of any non-self-intersecting 257-gon is 91800°. The area of a regular 257-gon is A = 257 4 t 2 π 257 ≈ 5255.751 t 2. The regular 257-gon is for being a constructible polygon:, it can be constructed using a compass and an unmarked straightedge. This is because 257 is a prime, being of the form 22n + 1. Another method involves the use of 24 being Carlyle circles: this method is pictured below. One of these Carlyle circles solves the quadratic equation x2 + x 64 = 0. The regular 257-gon has order 514. Since 257 is a prime number there is one subgroup with dihedral symmetry: Dih1, 2 cyclic group symmetries: Z1. A 257-gram is a 257-sided polygon. Below is a view of, with 257 nearly radial edges, with its star vertex angles 180 ° / 257. Weisstein, Eric W. "257-gon". MathWorld. Robert Dixon Mathographics. New York: Dover, p. 53, 1991.
257-gon
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A regular 257-gon
187.
Chiliagon
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In geometry, a chiliagon or 1000-gon is a polygon with 1000 sides. Philosophers commonly refer to chiliagons to illustrate ideas about the nature and workings of thought, mental representation. The measure of each internal angle in a regular chiliagon is 179.64°. Because 1000 = 23 × 53, the number of sides is neither a power of two. Thus the regular chiliagon is not a constructible polygon. René Descartes uses the chiliagon as an example in his Sixth Meditation to demonstrate the difference between pure imagination. The imagination constructs a "confused representation,", no different from that which it constructs of a myriagon. Therefore, the intellect is not dependent on Descartes claims, as it is able to entertain clear and distinct ideas when imagination is unable to. The example of a chiliagon is also referenced by other philosophers, such as Immanuel Kant. Inspired by Descartes's example, Roderick Chisholm and other 20th-century philosophers have used similar examples to make similar points. Chisholm's speckled hen, which need not have a determinate number of speckles to be successfully imagined, is perhaps the most famous of these. The regular chiliagon has order 2000, represented by 1000 lines of reflection. Dih100 has 15 dihedral subgroups: Dih500, Dih250, Dih125, Dih200, Dih100, Dih50, Dih25, Dih40, Dih20, Dih10, Dih5, Dih8, Dih4, Dih1. John Conway labels these lower symmetries with a order of the symmetry follows the letter. A1 labels no symmetry.
Chiliagon
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A regular chiliagon
188.
Myriagon
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In geometry, a myriagon or 10000-gon is a polygon with 10000 sides. Several philosophers have used the regular myriagon to illustrate issues regarding thought. The measure of each internal angle in a regular myriagon is 179.964°. Because 10000 = 24 × 54, the number of sides is neither a power of two. Thus the regular myriagon is not a constructible polygon. The regular myriagon has order 20000, represented by 10000 lines of reflection. Dih100 has 24 dihedral subgroups:, and. It also has 25 more cyclic symmetries as subgroups:, with Zn representing π / n radian symmetry. John Conway labels these lower symmetries with a order of the symmetry follows the letter. A1 labels no symmetry. These lower symmetries allows degrees of freedom in defining irregular myriagons. Only the g10000 subgroup has no degrees of freedom but can seen as directed edges. A myriagram is an 10000-sided polygon. There are 1999 regular forms given by Schläfli symbols of the form, where n is an integer between 5000, coprime to 10000. There are also 3000 regular star figures in the remaining cases.
Myriagon
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A regular myriagon
189.
65537-gon
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In geometry, a 65537-gon is a polygon with 65537 sides. The sum of the interior angles of any non-self-intersecting 65537-gon is 23592600°. The regular 65537-gon is for being a constructible polygon:, it can be constructed using a compass and an unmarked straightedge. This is because 65537 is a prime, being of the form 22n + 1. The construction is very complex; Hermes spent 10 years completing the 200-page manuscript. The first stages of this method are pictured below. This method faces practical problems, as one of these Carlyle circles solves the quadratic equation x2 + x 16384 = 0. The regular 65537-gon has order 131074. Since 65537 is a prime number there is one subgroup with dihedral symmetry: Dih1, 2 cyclic group symmetries: Z1. A 65537-gram is a 65537-sided polygon. Weisstein, Eric W. "65537-gon". MathWorld. Robert Dixon Mathographics. New York: Dover, p. 53, 1991. Benjamin Bold, Famous Problems of Geometry and How to Solve Them New York: Dover, p. 70, 1982.
65537-gon
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A regular 65537-gon
190.
Megagon
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A megagon or 1000000-gon is a polygon with 1 million sides. Even if drawn at the size of the Earth, a regular megagon would be very difficult to distinguish from a circle. A regular megagon has an interior angle of 179.99964°. The area of a regular megagon with sides of length a is given by A = 250000 a 2 cot π 1000000. The perimeter of a regular megagon inscribed in the circle is: 2000000 sin π 1000000, very close to 2π. The difference between the circumference of this circle comes to less than 1/16 millimeters. Because 1000000 = 26 × 56, the number of sides is not a power of two. Thus the regular megagon is not a constructible polygon. Like René Descartes' example of the chiliagon, the million-sided polygon has been used as an illustration of a well-defined concept that cannot be visualised. The megagon is also used to a circle. The regular megagon has order 2000000, represented by 1000000 lines of reflection. Dih100 has 48 dihedral subgroups:, and. It also has 49 more cyclic symmetries as subgroups:, with Zn representing π / n radian symmetry. John Conway labels these lower symmetries with a order of the symmetry follows the letter. R2000000 represents a1 labels no symmetry.
Megagon
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A regular megagon
191.
Apeirogon
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In geometry, an apeirogon is a generalized polygon with a countably infinite number of sides. It can be considered as the limit of an n-sided polygon as n approaches infinity. Defining half the plane as the interior. This article describes an apeirogon in its linear form as a partition of a line. A regular apeirogon has equal edge lengths, just like any regular polygon. Its Coxeter-Dynkin diagram is. It is the first in the dimensional family of hypercubic honeycombs. This line may be considered by analogy with regular polygons with great number of edges, which resemble a circle. In two dimensions, a regular apeirogon divides the plane into two half-planes as a apeirogonal dihedron. The interior of an apeirogon can be defined by its orientation, filling one plane. Dually the apeirogonal hosohedron has an apeirogonal vertex figure. A apeirogonal hosohedron becomes a apeirogonal prism, with each vertex bounded by two squares and an apeirogon. An apeirogonal prism is a apeirogonal antiprism, with each vertex bounded by three triangles and an apeirogon. An isogonal apeirogon alternates two types of edges. A apeirogon is an isogonal apeirogon with equal edge lengths.
Apeirogon
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Regular apeirogon
192.
Pentagram
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A pentagram is the shape of a five-pointed star drawn with five straight strokes. The pentagram comes from the Greek word πεντάγραμμον, from πέντε, "five" + γραμμή, "line". The word "pentacle" is sometimes used synonymously with "pentagram" The pentalpha is a learned modern revival of a post-classical Greek name of the shape. The pentagram is the simplest regular polygon. The pentagram contains fifteen line segments. It is represented by the Schläfli symbol. The pentagram can be constructed by connecting alternate vertices of a pentagon; see details of the construction. It can also be constructed by extending the edges of a pentagon until the lines intersect. Also, the ratio of the length of the shorter segment to the segment bounded by the two intersecting edges is φ. The pentagram includes ten isosceles triangles: five obtuse isosceles triangles. In all of them, the ratio of the longer side to the shorter side is φ. The acute triangles are golden triangles. The isosceles triangle highlighted via the colored lines in the illustration is a golden gnomon. The pentagram of Venus is the apparent path of the planet Venus as observed from Earth. Groups of five intersections of curves, equidistant from the figure's center, have the geometric relationship.
Pentagram
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Man inscribed in a pentagram, from Heinrich Cornelius Agrippa 's De occulta philosophia libri tres. The five signs at the pentagram's vertices are astrological.
Pentagram
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A regular pentagram
Pentagram
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The occultist and magician Eliphas Levi 's pentagram, which he considered to be a symbol of the microcosm, or human.
Pentagram
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A goat's head inscribed in a downward-pointing pentagram, from La Clef de la Magie Noire by Stanislas de Guaita (1897).
193.
Heptagram
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A heptagram, septagram, or septegram is a seven-point star drawn with seven straight strokes. The name heptagram combines a numeral prefix, hepta-, with the Greek suffix -gram. The -gram suffix derives from γραμμῆς meaning a line. In general, a heptagram is any self-intersecting heptagon. There are two regular heptagrams, labeled as and, with the second number representing the vertex interval step from a regular heptagon. This is the smallest polygon that can be drawn as irreducible fractions. The two heptagrams are sometimes called the heptagram and the great heptagram. The previous one, the regular hexagram, is a compound of two triangles. The smallest star polygon is the pentagram. The next one is the octagram, followed by the regular enneagram, which also has two forms: and, as well as one compound of three triangles. The heptagram was used in Christianity to symbolize the seven days of creation and became a traditional symbol for warding off evil. The heptagram is a symbol of perfection in many Christian sects. The heptagram is used in the symbol for Babalon in Thelema. The heptagram is known among neopagans as the Elven Star or Fairy Star. It is treated as a sacred symbol in modern witchcraft traditions.
Heptagram
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Former Georgian coat of arms, 1918–1921, 1991–2004
Heptagram
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A regular heptagram
Heptagram
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Flag of the Kingdom of Pentortoise
194.
Octagram
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In geometry, an octagram is an eight-angled star polygon. The octagram combine a Greek numeral prefix, octa -, with the Greek suffix - gram. The - suffix derives from γραμμή meaning "line". In general, an octagram is any self-intersecting octagon. The regular octagram is labeled by the Schläfli symbol, which means an 8-sided star, connected by every third point. These variations have Dih4, symmetry: The symbol Rub el Hizb is a Unicode glyph ۞ at U +06 DE. Deeper truncations of the square can produce intermediate star polygon forms with equal spaced vertices and two edge lengths. A truncated square is an t =. A quasitruncated square, inverted as, is t =. The uniform polyhedron stellated truncated hexahedron, t' = t has octagram faces constructed from the cube in this way. There are two octagrammic star figures of the form, the first constructed as two squares = 2, second as four degenerate digons, = 4. There are other isotoxal compounds including rectangular and rhombic forms. An octagonal star can be seen with internal intersecting geometry erased. It can also be dissected by radial lines. Selburose – usage of regular octagram in Norwegian design Stars generally Star Stellated polygons Two-dimensional regular polytopes Grünbaum, B. and G.C.
Octagram
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A regular octagram
195.
Enneagram (geometry)
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In geometry, an enneagram is a nine-pointed plane figure. It is sometimes called a nonagram. The enneagram combines the numeral prefix, ennea -, with the Greek suffix - gram. The - suffix derives from γραμμῆς meaning a line. A regular enneagram is connected in fixed steps. It has two forms, connecting every second and every fourth points respectively. There is also 3, made from the regular enneagon points but connected as a compound of three equilateral triangles. This figure is sometimes known as the star of Goliath, after or 2, the star of David. This geometrical figure should not be confused with the logic puzzles called nonograms. Enneagram can also symbolize the nine gifts or fruits of the Holy Spirit. The heavy metal Slipknot uses the star figure enneagram as a symbol. Nonagon List of regular star polygons Bibliography John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 Nonagram -- from Wolfram MathWorld
Enneagram (geometry)
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Enneagrams shown as sequential stellations
196.
Decagram (geometry)
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In geometry, a decagram is a 10-point star polygon. There is one regular decagram, containing the vertices of a regular decagon, but connected by every third point. Its Schläfli symbol is. The decagram combine a numeral prefix, deca -, with the Greek suffix - gram. The - suffix derives from γραμμῆς meaning a line. For a regular decagram with edge lengths, the proportions of the crossing points on each edge are as shown below. Decagrams have been used as one of the decorative motifs in girih tiles. A regular decagram is a 10-sided polygram, containing the same vertices as regular decagon. Deeper truncations of the regular pentagram can produce intermediate star polygon forms with ten equally spaced vertices and two edge lengths that remain vertex-transitive. List of regular polytopes and compounds#Stars
Decagram (geometry)
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A regular decagram
197.
Hendecagram
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In geometry, a hendecagram is a star polygon that has eleven vertices. The hendecagram combines a Greek numeral prefix, hendeca -, with the Greek suffix - gram. The hendeca- prefix derives from Greek ἕνδεκα meaning "eleven". The - suffix derives from γραμμῆς meaning a line. These same four forms can also be considered as stellations of a regular hendecagon. However, Hilton & Pedersen describe folding patterns for making the hendecagrams, out of strips of paper. May be used to approximate the shape of DNA molecules. Fort Wood, now the base of the Statue of Liberty in New York City, is a fort in the form of an irregular 11-point star. The Topkapı Scroll contains images of an 11-pointed Girih form used in Islamic art. An star-shaped cross-section was used in the Space Shuttle Solid Rocket Booster, for the core of the forward section of the rocket. Hendecagrammic prism Weisstein, Eric W. "Polygram". MathWorld.
Hendecagram
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Fort Wood 's star-shaped walls became the base of the Statue of Liberty.
Hendecagram
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Four forms
198.
Dodecagram
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A dodecagram is a star polygon that has 12 vertices. There is one regular form:. A regular dodecagram has the same arrangement as a regular dodecagon, which may be regarded as. The dodecagram combine a numeral prefix, dodeca -, with the Greek suffix - gram. The - suffix derives from γραμμῆς meaning a line. A regular dodecagram can be seen as t =. Isogonal variations with equal spaced vertices can be constructed with two edge lengths. There are four regular dodecagram star figures, = 2, = 3, = 6. Superimposing all the dodecagrams on each other -- including the degenerate compound of six digons, -- produces the complete graph K12. Dodecagrams can also be incorporated into uniform polyhedra. Below are the three prismatic uniform polyhedra containing regular dodecagrams. Stellation Star polygon List of regular polytopes Weisstein, Eric W. "Dodecagram". MathWorld. Grünbaum, B. and G.C. Shephard; Tilings and Patterns, New York: W. H. Freeman & Co. ISBN 0-7167-1193-1.
Dodecagram
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A regular dodecagram
199.
Uniform polytope
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A uniform polytope of dimension three or higher is a vertex-transitive polytope bounded by uniform facets. The uniform polytopes in two dimensions are the regular polygons. This also includes the regular polytopes. Further, regular faces and vertex figures are allowed, which greatly expand the possible solutions. Nearly every uniform polytope can be represented by a Coxeter diagram. Notable exceptions include the grand antiprism in four dimensions. Equivalently, the Wythoffian polytopes can be generated by applying basic operations to the regular polytopes in that dimension. This approach is the basis of the Conway polyhedron notation. Regular n-polytopes have n orders of rectification. The zeroth rectification is the original form. The rectification is the dual. An extended Schläfli symbol can be used for representing rectified forms, with a single subscript: k-th rectification = tk = kr. Truncation operations that can be applied to regular n-polytopes in any combination. The operation is named for the distance between them. Truncation cuts vertices, runcination cuts faces, sterication cut cells.
Uniform polytope
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Truncated triangle is a uniform hexagon, with Coxeter diagram.
200.
Coxeter group
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In mathematics, a Coxeter group, named after H. S. M. Coxeter, is an abstract group that admits a formal description in terms of reflections. Indeed, the finite Coxeter groups are precisely the finite Euclidean reflection groups; the symmetry groups of regular polyhedra are an example. However, not all Coxeter groups are finite, not all can be described in terms of Euclidean reflections. Finite Coxeter groups were classified in 1935. Coxeter groups find applications in many areas of mathematics. Examples of finite Coxeter groups include the Weyl groups of simple Lie algebras. Standard references include and. M i j = ∞ means no relation of the form m should be imposed. The pair where W is a Coxeter group with generators S= is called Coxeter system. Note that in general S is not uniquely determined by W. For example, the Coxeter groups of type B3 and A1xA3 are isomorphic but the Coxeter systems are not equivalent. A number of conclusions can be drawn immediately from the above definition. The mi i = 1 means that 1 = 2 = 1 for all i; the generators are involutions. If j = 2, then the generators ri and rj commute. This follows by observing that xx = yy = 1, together with xyxy = 1 implies that xy = xy = yx = yx.
Coxeter group
201.
Simple Lie group
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Essentially, simple Lie groups are connected Lie groups which cannot be decomposed as an extension of smaller connected Lie groups, which are not commutative. In theory, a simple group is a connected locally compact non-abelian Lie group G which does not have nontrivial connected normal subgroups. A simple algebra is a non-abelian algebra whose only ideals are 0 and itself. An definition of a simple group follows from the Lie correspondence: a connected Lie group is simple if its Lie algebra is simple. It emerged in the course of classification of simple Lie groups that there exist also several exceptional possibilities not corresponding to any familiar geometry. These exceptional groups account for special configurations in other branches of mathematics, as well as contemporary theoretical physics. All Lie groups are smooth manifolds. Its algebra is simple as a complex Lie algebra. Note that the underlying Lie group may not be simple, although it will still be semisimple. It is often useful to study slightly more general classes of Lie groups than simple groups, generally reductive Lie groups. A connected group is called semisimple if its algebra is a semisimple lie algebra, i.e. a direct sum of simple Lie algebras. It is called reductive if its algebra is a direct sum of trivial Lie algebras. Reductive groups occur naturally in algebra, physics. For example, the G L n of symmetries of an real vector space is reductive. Finite-dimensional representations of simple groups split into direct sums of irreducible representations, which are classified by vectors in the weight lattice satisfying certain properties.
Simple Lie group
202.
E6 (mathematics)
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The E6 comes from the Cartan -- Killing classification of the simple Lie algebras. This classifies Lie algebras into four infinite series labeled five exceptional cases labeled G2. The E6 algebra is thus one of the five exceptional cases. A basis is given on a cubic surface. The dual representation, inequivalent, is also 27-dimensional. In particle physics, E6 plays a role in some grand unified theories. There is a complex algebra of type E6, corresponding to a complex group of complex dimension 78. The adjoint Lie E6 of complex dimension 78 can be considered as a simple real Lie group of real dimension 156. The split form, EI, which has maximal compact subgroup Sp/, fundamental group of order 2 and outer automorphism group of order 2. The quasi-split form EII, which has outer automorphism group of order 2. EIII, which has compact subgroup SO × Spin /, trivial outer automorphism group. EIV, which has compact F4, trivial fundamental group cyclic and outer automorphism group of order 2. The EIV form of E6 is the group of collineations of the projective OP2. It is also the group of determinant-preserving linear transformations of the exceptional Jordan algebra. The exceptional Jordan algebra is 27-dimensional, which explains why the real form of E6 has a complex representation.
E6 (mathematics)
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Algebraic structure → Group theory Group theory
203.
E7 (mathematics)
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The E7 algebra is thus one of the five exceptional cases. The dimension of its fundamental representation is 56. There is a complex Lie algebra of type E7, corresponding to a complex group of complex dimension 133. The complex adjoint Lie E7 of complex dimension 133 can be considered as a simple real Lie group of real dimension 266. The split form, EV, which has maximal compact subgroup SU/, fundamental group cyclic of order 4 and outer automorphism group of order 2. EVI, which has compact subgroup SU · SO /, fundamental group non-cyclic of order 4 and trivial outer automorphism group. EVII, which has maximal subgroup SO · E6 /, infinite cyclic findamental group and outer automorphism group of order 2. For a complete list of real forms of simple Lie algebras, see the list of simple Lie groups. The real form of E7 is the isometry group of the 64-dimensional exceptional compact Riemannian symmetric space EVI. This can be seen systematically using a construction known as the magic square, due to Jacques Tits. The Tits–Koecher construction produces forms of the E7 Lie algebra from Albert algebras, 27-dimensional exceptional Jordan algebras. Over finite fields, the Lang -- Steinberg theorem implies meaning that E7 has no twisted forms: see below. The Dynkin diagram for E7 is given by. There are 126 roots. Given a Dynkin diagram node ordering of: one choice of simple roots is given by the rows of the following matrix:.
E7 (mathematics)
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Algebraic structure → Group theory Group theory
204.
E8 (mathematics)
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The E8 algebra is the largest and most complicated of these exceptional cases. Cartan determined that a simple algebra of type E8 admits three real forms. Each of them gives rise to a simple group of 248, exactly one of, compact. The E8 has dimension 248. Its rank, the dimension of its maximal torus, is 8. Therefore, the vectors of the system are in Euclidean space: they are described explicitly later in this article. There is a Ek for every integer k ≥ 3, infinite dimensional if k is greater than 8. There is a complex algebra of type E8, corresponding to a complex group of complex dimension 248. The complex E8 of complex dimension 248 can be considered as a simple real Lie group of real dimension 496. This has an outer automorphism group of order 2 generated by complex conjugation. The split form, EVIII, which has maximal compact subgroup Spin/, fundamental group of order 2 and has trivial outer automorphism group. EIX, which has maximal compact subgroup E7×SU/, fundamental group of order 2 and has trivial outer automorphism group. For a complete list of real forms of simple Lie algebras, see the list of simple Lie groups. Over finite fields, the Lang–Steinberg theorem implies that H1=0, meaning that E8 has no twisted forms: see below. The characters of dimensional representations of the complex Lie algebras and Lie groups are all given by the Weyl character formula.
E8 (mathematics)
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Zome model of the E 8 root system, projected into three-space, and represented by the vertices of the 421 polytope,
E8 (mathematics)
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Algebraic structure → Group theory Group theory
E8 (mathematics)
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E8 with thread made by hand
205.
E10 (mathematics)
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In papers, E2 and E4 are used as names for F4. The En group is similar to An group, except the node is connected to the 3rd node. The determinant of the Cartan matrix for En is 9-n. E3 is another name for the A1A2 of dimension 11, with Cartan determinant 6. E4 is another name for the A4 of dimension 24, with Cartan determinant 5. E5 is another name for the D5 of dimension 45, with Cartan determinant 4. E6 is the exceptional Lie algebra of dimension 78, with Cartan determinant 3. E7 is the exceptional Lie algebra of dimension 133, with Cartan determinant 2. E8 is the exceptional Lie algebra of dimension 248, with Cartan determinant 1. E9 is another name for the affine Lie E ~ 8 corresponding to the Lie algebra of type E8. E9 has a Cartan matrix with determinant 0. E10 is an infinite-dimensional Kac–Moody algebra whose root lattice is the even Lorentzian unimodular lattice II9,1 of dimension 10. En for n≥12 is an infinite-dimensional Kac–Moody algebra that has not been studied much. Landsberg and Manivel extended the definition of En for n to include the n = 7 1/2. 1/2 is not a simple Lie algebra: it contains a 57 dimensional Heisenberg algebra as its nilradical.
E10 (mathematics)
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E 3 = A 2 A 1
206.
F4 (mathematics)
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In mathematics, F4 is the name of a Lie group and also its Lie algebra f4. It is one of the five exceptional simple Lie groups. F4 has rank 4 and dimension 52. Its outer group is the trivial group. Its fundamental representation is 26-dimensional. The real form of F4 is the isometry group of a 16-dimensional Riemannian manifold known as the projective plane OP2. This can be seen systematically using a construction known as the magic square, due to Hans Freudenthal and Jacques Tits. There are 3 real forms: a third one. They are the isometry groups of the three real Albert algebras. In older books and papers, F4 is sometimes denoted by E4. The Dynkin diagram for F4 is. G = W is the group of the 24-cell: it is a solvable group of order 1152. It has minimal faithful degree μ = 24, realized by the action on the 24-cell. The F4 lattice is a cubic lattice. They form a ring called the Hurwitz quaternion ring.
F4 (mathematics)
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Algebraic structure → Group theory Group theory
207.
G2 (mathematics)
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In mathematics, G2 is the name of three simple Lie groups, their Lie algebras g 2, as well as some algebraic groups. They are the smallest of the five exceptional simple Lie groups. G2 has rank 2 and dimension 14. It has two fundamental representations, with dimension 7 and 14. In the same year, in the same journal, Engel noticed the same thing. Later it was discovered that the 2-dimensional distribution is closely related to a ball rolling on another ball. In 1908 Cartan mentioned that the group of the octonions is a simple group. In 1914 he stated that this is the compact real form of G2. In older books and papers, G2 is sometimes denoted by E2. There are 3 real Lie algebras associated with this system: The underlying real algebra of the complex Lie algebra G2 has dimension 28. It has complex conjugation as an outer automorphism and is simply connected. The maximal compact subgroup of its associated group is the compact form of G2. The Lie algebra of the compact form is 14-dimensional. The associated group is simply compact. The Lie algebra of the non-compact form has dimension 14.
G2 (mathematics)
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Algebraic structure → Group theory Group theory
208.
H4 (mathematics)
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In mathematics, a Coxeter group, named after H. S. M. Coxeter, is an abstract group that admits a formal description in terms of reflections. Indeed, the finite Coxeter groups are precisely the finite Euclidean reflection groups; the symmetry groups of regular polyhedra are an example. However, not all Coxeter groups are finite, not all can be described in terms of Euclidean reflections. Finite Coxeter groups were classified in 1935. Coxeter groups find applications in many areas of mathematics. Examples of finite Coxeter groups include the Weyl groups of simple Lie algebras. Standard references include and. M i j = ∞ means no relation of the form m should be imposed. The pair where W is a Coxeter group with generators S= is called Coxeter system. Note that in general S is not uniquely determined by W. For example, the Coxeter groups of type B3 and A1xA3 are isomorphic but the Coxeter systems are not equivalent. A number of conclusions can be drawn immediately from the above definition. The mi i = 1 means that 1 = 2 = 1 for all i; the generators are involutions. If j = 2, then the generators ri and rj commute. This follows by observing that xx = yy = 1, together with xyxy = 1 implies that xy = xy = yx = yx.
H4 (mathematics)
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Coxeter graphs of the finite Coxeter groups.
209.
Uniform polyhedron
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A uniform polyhedron is a polyhedron which has regular polygons as faces and is vertex-transitive. It follows that all vertices are congruent. Uniform polyhedra may be semi-regular. The vertices need not be convex, so many of the uniform polyhedra are also polyhedra. There are two infinite classes of uniform polyhedra together with 75 others. Dual polyhedra to uniform polyhedra are generally classified in parallel with their dual polyhedron. The dual of a regular polyhedron is regular, while the dual of an Archimedean solid is a Catalan solid. The concept of uniform polyhedron is a special case of the concept of uniform polytope, which also applies to shapes in higher-dimensional space. Coxeter, Longuet-Higgins & Miller define uniform polyhedra to be vertex-transitive polyhedra with regular faces. By a polygon they implicitly mean a polygon in Euclidean space; these are allowed to intersect each other. There are some generalizations of the concept of a uniform polyhedron. If we drop the condition that the realization of the polyhedron is non-degenerate, then we get the degenerate polyhedra. These require a more general definition of polyhedra. Some of the ways they can be degenerate are as follows: Hidden faces. Some polyhedra have faces that are hidden, in the sense that no points of their interior can be seen from the outside.
Uniform polyhedron
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Uniform polyhedron
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Platonic solid: Tetrahedron
Uniform polyhedron
Uniform polyhedron
210.
Tetrahedron
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In geometry, a tetrahedron is a polyhedron composed of four triangular faces, six straight edges, four vertex corners. The tetrahedron is the simplest of all the ordinary convex polyhedra and the only one that has fewer than 5 faces. The tetrahedron is the three-dimensional case of the more general concept of a Euclidean simplex. The tetrahedron is one kind of pyramid, a polyhedron with a flat polygon base and triangular faces connecting the base to a common point. In the case of a tetrahedron the base is a triangle, so a tetrahedron is also known as a "triangular pyramid". Like all convex polyhedra, a tetrahedron can be folded from a single sheet of paper. It has two such nets. For any tetrahedron there exists a sphere on which all four vertices lie, another sphere tangent to the tetrahedron's faces. A regular tetrahedron is one in which all four faces are equilateral triangles. It is one of the five regular Platonic solids, which have been known since antiquity. In a regular tetrahedron, not only are all its faces the same shape but so are all its edges. If alternated with regular octahedra they form the cubic honeycomb, a tessellation. The regular tetrahedron is self-dual, which means that its dual is another regular tetrahedron. The compound figure comprising two dual tetrahedra form a stellated octahedron or octangula. This form has Schläfli h.
Tetrahedron
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(Click here for rotating model)
Tetrahedron
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4-sided die
211.
Regular dodecahedron
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It is one of the five Platonic solids. It has 20 vertices, 160 diagonals. In projection, viewed above a pentagonal face, the regular dodecahedron can be seen as a linear-edged schlegel diagram, or stereographic projection as a spherical polyhedron. These projections are also used in showing the four-dimensional 120-cell, a regular 4-dimensional polytope, projecting it down to 3-dimensions. The regular dodecahedron can also be represented as a spherical tiling. The length is 2 / ϕ = √ 5 − 1. The containing sphere has a radius of √3. A137218 If the regular dodecahedron has edge length 1, its dual icosahedron has edge length ϕ. If the five Platonic solids are built with same volume, the regular dodecahedron has the shortest edges. It has 43,380 nets. The map-coloring number of a regular dodecahedron's faces is 4. The distance between the vertices on the same face not connected by an edge is ϕ times the length. If two edges share a common vertex, then the midpoints of those edges form an equilateral triangle with the center. The stellations of the regular dodecahedron make up three of the four Kepler–Poinsot polyhedra. A regular dodecahedron forms an icosidodecahedron.
Regular dodecahedron
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(Click here for rotating model)
Regular dodecahedron
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Roman dodecahedron
Regular dodecahedron
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Ho-Mg-Zn quasicrystal
Regular dodecahedron
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A climbing wall consisting of three dodecahedral pieces
212.
Regular icosahedron
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In geometry, a regular icosahedron is a convex polyhedron with 20 faces, 30 edges and 12 vertices. It is also the one with the most sides. It has five equilateral triangular faces meeting at each vertex. It is represented as 3.3.3.3.3 or 35. It is the dual of the dodecahedron, represented by, having three pentagonal faces around each vertex. A regular icosahedron is a biaugmented pentagonal antiprism in any of six orientations. The name comes from Greek εἴκοσι, ἕδρα, meaning "seat". The plural can be either "icosahedrons" or "icosahedra". Note that these vertices form five sets of three concentric, mutually orthogonal golden rectangles, whose edges form Borromean rings. If the original icosahedron has length 1, its dual dodecahedron has edge length √ 5 − 1/2 = 1 / ϕ = ϕ − 1. The 12 edges of a regular octahedron can be subdivided in the golden ratio so that the resulting vertices define a regular icosahedron. The locations of the vertices of a regular icosahedron can be described using spherical coordinates, for instance as longitude. If two vertices are taken to be at south poles, then the other ten vertices are at latitude ± arctan ≈ ± 26.57 °. These ten vertices are at evenly spaced longitudes, alternating between south latitudes. This projection is conformal, preserving angles but not lengths.
Regular icosahedron
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(Click here for rotating model)
Regular icosahedron
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Model of an icosahedron made with metallic spheres and magnetic connectors
Regular icosahedron
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Gold nanoparticle viewed in electron microscope.
Regular icosahedron
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Structure of γ-boron.
213.
Uniform 4-polytope
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In geometry, a uniform 4-polytope is a 4-polytope, vertex-transitive and whose cells are uniform polyhedra, faces are regular polygons. 47 non-prismatic convex uniform 4-polytopes, two infinite sets of convex prismatic forms have been described. There are also an unknown number of non-convex star forms. Regular star 4-polytopes 1852: Ludwig Schläfli also found 4 of the 10 regular star 4-polytopes, discounting 6 with cells or vertex figures and. This construction enumerated 45 semiregular 4-polytopes. Convex uniform polytopes: 1940: The search was expanded systematically by H.S.M. Coxeter in his publication Regular and Semi-Regular Polytopes. 1998-2000: The 4-polytopes were systematically named by Norman Johnson, given by George Olshevsky's online indexed enumeration. Johnson choros. 2004: A proof that the Conway-Guy set is complete was published by Marco Möller in his dissertation, Vierdimensionale Archimedische Polytope. Möller reproduced Johnson's system in his listing. Regular 4-polytopes are a subset of the uniform 4-polytopes, which satisfy additional requirements. Regular 4-polytopes can be expressed with Schläfli symbol have cells of faces of type, edge figures, vertex figures. The existence of a regular 4-polytope is constrained by the existence of the regular polyhedra which becomes the vertex figure. 10 regular star 4-polytopes:, and.
Uniform 4-polytope
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Schlegel diagram for the truncated 120-cell with tetrahedral cells visible
214.
5-cell
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In geometry, the 5-cell is a four-dimensional object bounded by 5 tetrahedral cells. It is also known as a C5, pentachoron, pentatope, pentahedroid, or pyramid. It is the simplest possible convex regular 4-polytope, is analogous to the tetrahedron in three dimensions and the triangle in two dimensions. The pentachoron is a four pyramid with a tetrahedral base. The regular 5-cell is one of the six regular convex 4-polytopes, represented by Schläfli symbol. Its vertex figure is a tetrahedron. Its maximal intersection with 3-dimensional space is the prism. Its dihedral angle is cos approximately 75.52 °. The 5-cell can be constructed by adding a 5th vertex such that it is equidistant from all the other vertices of the tetrahedron. The simplest set of coordinates is:, with length 2 √ 2, where τ is the golden ratio. A 5-cell can be constructed as a Boerdijk–Coxeter helix of five chained tetrahedra, folded into a 4-dimensional ring. The purple edges represent the Petrie polygon of the 5-cell. The A4 Coxeter plane projects the 5-cell into pentagram. The four sides of the pyramid are made of tetrahedron cells. Many uniform 5-polytopes have tetrahedral vertex figures: Other uniform 5-polytopes have irregular 5-cell vertex figures.
5-cell
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Vertex figure: tetrahedron
5-cell
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Schlegel diagram (vertices and edges)
215.
16-cell
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In four-dimensional geometry, a 16-cell is a regular convex 4-polytope. It is one of the six regular convex first described by the Swiss mathematician Ludwig Schläfli in the mid-19th century. It is also called C16, hexdecahedroid. It is a part of an infinite family of polytopes, orthoplexes. The dual polytope is the tesseract. Conway's name for a cross-polytope is orthoplex, for orthant complex. The 16-cell has 16 cells as the tesseract has 16 vertices. It is bounded by 16 cells, all of which are regular tetrahedra. It has 32 triangular faces, 8 vertices. The 24 edges bound 6 squares lying in the 6 coordinate planes. The eight vertices of the 16-cell are. All vertices are connected by edges except opposite pairs. The Schläfli symbol of the 16-cell is. Its figure is a regular octahedron. There are 8 tetrahedra, 6 edges meeting at every vertex.
16-cell
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Schlegel diagram (vertices and edges)
216.
Tesseract
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In geometry, the tesseract is the four-dimensional analog of the cube; the tesseract is to the cube as the cube is to the square. Just as the surface of the cube consists of six square faces, the hypersurface of the tesseract consists of eight cubical cells. The tesseract is one of the six convex regular 4-polytopes. The tesseract tetracube. It is the four-dimensional hypercube, or 4-cube as a part of the dimensional family of "measure polytopes". In this publication, well as some of Hinton's later work, the word was occasionally spelled "tessaract". The tesseract can be constructed in a number of ways. As a regular polytope with three cubes folded together around every edge, it has Schläfli symbol with hyperoctahedral symmetry of order 384. Constructed as a 4D hyperprism made of two parallel cubes, it can be named with symmetry order 96. As a Cartesian product of two squares, it can be named by a composite Schläfli symbol ×, with symmetry order 64. As an orthotope it can be represented with symmetry order 16. Since each vertex of a tesseract is adjacent to four edges, the figure of the tesseract is a regular tetrahedron. The dual polytope of the tesseract is called 16-cell, with Schläfli symbol. The standard tesseract in Euclidean 4-space is given as the convex hull of the points. That is, it consists of the points: A tesseract is bounded by eight hyperplanes.
Tesseract
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Schlegel diagram
217.
120-cell
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In geometry, the 120-cell is the convex regular 4-polytope with Schläfli symbol. It is also called a C120, hecatonicosachoron, hecatonicosahedroid. The boundary of the 120-cell is composed at each vertex. It has been called a dodecaplex, hyperdodecahedron, polydodecahedron. There are 120 cells, 720 pentagonal faces, 600 vertices. There are 4 dodecahedra, 4 edges meeting at every vertex. There are 3 pentagons meeting every edge. The dual polytope of the 120-cell is the 600-cell. The figure of the 120-cell is a tetrahedron. The 120-cell consists of 120 dodecahedral cells. For visualization purposes, it is convenient that the dodecahedron has opposing parallel faces. One can stack dodecahedrons face to face into a great circle with a circumference of 10 cells. Starting from this initial ten cell construct there are two common visualizations one can use: a layered stereographic projection, a structure of intertwining rings. The cell locations lend themselves to a hyperspherical description. Label it the "North Pole".
120-cell
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Two intertwining rings of the 120-cell.
120-cell
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Schlegel diagram (vertices and edges)
218.
600-cell
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In geometry, the 600-cell is the convex regular 4-polytope with Schläfli symbol. It is also called a C600, hexacosidedroid. It is also called a polytetrahedron, being bounded by tetrahedral cells. Its boundary is composed at each vertex. Together they form 1200 triangular faces, 120 vertices. The edges form 72 regular decagons. Each vertex of the 600-cell is a vertex of six such decagons. References: S.L. van Oss; F. Buekenhout and M. Parker. Its dual polytope is the 120-cell. It has a dihedral angle of cos 1 = ~ 164.48 °. Each cell touches, in some manner, 56 other cells. The remaining 96 vertices are obtained by taking even permutations of ½. When interpreted as quaternions, the 120 vertices of the 600-cell form a group under quaternionic multiplication. This group is often called the binary icosahedral group and denoted by 2I as it is the double cover of the ordinary icosahedral group I. Each rotational symmetry of the 600-cell is generated by specific elements of 2IL and 2IR; the pair of opposite elements generate the same element of RSG.
600-cell
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100 tetrahedra in a 10x10 array forming a clifford torus boundary in the 600 cell.
600-cell
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Schlegel diagram, vertex-centered (vertices and edges)
219.
Uniform 5-polytope
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In geometry, a uniform 5-polytope is a five-dimensional uniform polytope. By definition, a uniform 5-polytope is vertex-transitive and constructed from uniform 4-polytope facets. Most can be made from a small set of symmetry groups. These construction operations are represented by the permutations of rings of the Coxeter diagrams. Regular polytopes: 1852: Ludwig Schläfli proved in his manuscript Theorie der vielfachen Kontinuität that there are exactly 3 regular polytopes in 5 or more dimensions. Convex uniform polytopes: 1940-1988: The search was expanded systematically by H.S.M. Coxeter in his publication Regular and Semi-Regular Polytopes I, II, III. 1966: Norman W. Johnson completed his Ph.D. There are exactly three regular polytopes - 5-simplex - 5-cube - 5-orthoplex There are no nonconvex regular polytopes in 5 or more dimensions. There are 104 known convex uniform 5-polytopes, plus a number of infinite families of duoprism prisms, polygon-polyhedron duoprisms. All except the grand prism are based on Wythoff constructions, symmetry generated with Coxeter groups. The 5-simplex is the regular form in the A5 family. The 5-cube and 5-orthoplex are the regular forms in the B5 family. The bifurcating graph of the D6 family contains the pentacross, as well as a 5-demicube, an alternated 5-cube. One non-Wythoffian - The grand antiprism prism is the only known non-Wythoffian convex uniform 5-polytope, constructed from two grand antiprisms connected by polyhedral prisms.
Uniform 5-polytope
220.
5-orthoplex
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In five-dimensional geometry, a 5-orthoplex, or 5-cross polytope, is a five-dimensional polytope with 10 vertices, 40 edges, 80 triangle faces, 80 tetrahedron cells, 32 5-cell 4-faces. It has two constructed forms, the second with alternately labeled facets, with Schläfli symbol or Coxeter symbol 211. It is a part of an infinite family of polytopes, orthoplexes. The dual polytope is the 5-cube. Pentacross, derived from combining the name cross polytope with pente for five in Greek. Triacontaditeron - as a 32-facetted 5-polytope. This polytope is one of 31 5-polytopes generated from the B5 Coxeter plane, including the regular 5-cube and 5-orthoplex. H.S.M. Coxeter: H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973 Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6 H.S.M. Coxeter, Regular and Semi Regular Polytopes I, H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, Norman Johnson Uniform Polytopes, Manuscript N.W.
5-orthoplex
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Orthogonal projection inside Petrie polygon
221.
5-cube
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It is represented by Schläfli symbol or, constructed as 3 tesseracts, around each cubic ridge. It can be called a penteract, a portmanteau of tesseract and pente in Greek. It can also be called a regular decateron, being a 5-dimensional polytope constructed from 10 regular facets. It is a part of an infinite family. The dual of a 5-cube is the 5-orthoplex, of the infinite family of orthoplexes. The 5-cube can be seen as an tesseractic honeycomb on a 4-sphere. It is related to the Euclidean 4-space tesseractic honeycomb and paracompact hyperbolic honeycomb tesseractic honeycomb. This polytope is one of 31 5-polytopes generated from the regular 5-cube or 5-orthoplex. H.S.M. Coxeter: Coxeter, Regular Polytopes, Dover edition, ISBN 0-486-61480-8, p. 296, Table I: Regular Polytopes, three regular polytopes in n-dimensions Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6 H.S.M. Coxeter, Regular and Semi Regular Polytopes I, H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, Norman Johnson Uniform Polytopes, Manuscript N.W.
5-cube
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Graph
222.
5-demicube
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In five-dimensional geometry, a demipenteract or 5-demicube is a semiregular 5-polytope, constructed from a 5-hypercube with alternated vertices truncated. It was discovered by Thorold Gosset. Since it was the only 5-polytope, he called it a 5-ic semi-regular. E. L. Elte identified it as a semiregular polytope labeling it as HM5 for a 5-dimensional half measure polytope. It exists in the k21 family as 121 with the Gosset polytopes: 221, 321, 421. It is a part of a dimensional family of uniform polytopes called demihypercubes for being alternation of the hypercube family. The 5-demicube is third in a dimensional series of semiregular polytopes. Each progressive polytope is constructed vertex figure of the previous polytope. Thorold Gosset identified this series as containing all regular polytope facets containing all simplexes and orthoplexes. In Coxeter's notation the 5-demicube is given the 121. T. Gosset: On the Regular and Semi-Regular Figures in Space of n Dimensions, Messenger of Mathematics, Macmillan, 1900 H.S.M. Coxeter: Coxeter, Regular Polytopes, Dover edition, ISBN 0-486-61480-8, p. 296, Table I: Regular Polytopes, three regular polytopes in n-dimensions H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973, p. 296, Table I: Regular Polytopes, three regular polytopes in n-dimensions Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6 H.S.M.
5-demicube
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Petrie polygon projection
223.
Uniform 6-polytope
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In six-dimensional geometry, a uniform polypeton is a six-dimensional uniform polytope. All facets are 5-polytopes. Most can be made from a small set of symmetry groups. These construction operations are represented by the permutations of rings of the Coxeter-Dynkin diagrams. Each combination of at least one ring on every connected group of nodes in the diagram produces a uniform 6-polytope. The simplest polypeta are regular polytopes: the 6-simplex, the 6-orthoplex. Regular polytopes: 1852: Ludwig Schläfli proved in his manuscript Theorie der vielfachen Kontinuität that there are exactly 3 regular polytopes in 5 or more dimensions. Convex uniform polytopes: 1940: The search was expanded systematically by H.S.M. Coxeter in his publication Regular and Semi-Regular Polytopes. Nonregular uniform star polytopes: Ongoing: Thousands of nonconvex uniform polypeta are known, but mostly unpublished. Participating researchers include Norman Johnson. Uniform 6-polytopes with reflective symmetry can be generated by these four Coxeter groups, represented by permutations of rings of the Coxeter-Dynkin diagrams. There are four fundamental reflective symmety groups which generate 153 unique uniform 6-polytopes. Uniform prism There are 6 categorical uniform prisms based on the uniform 5-polytopes. Uniform duoprism There are 11 duoprismatic families of polytopes based on Cartesian products of lower-dimensional uniform polytopes.
Uniform 6-polytope
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6-simplex
224.
6-simplex
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In geometry, a 6-simplex is a self-dual regular 6-polytope. It has 7 vertices, 21 edges, 35 triangle faces, 35 tetrahedral cells, 7 5-simplex 5-faces. Its dihedral angle is cos approximately 80.41 °. It can also be called hepta-6-tope, as a 7-facetted polytope in 6-dimensions. The heptapeton is derived from hepta for seven facets in Greek and - peta for having five-dimensional facets, - on. Jonathan Bowers gives the acronym hop. The regular 6-simplex is one of 35 6-polytopes based on the Coxeter group, all shown here in A6 Coxeter plane orthographic projections. H.S.M. Coxeter: Coxeter, Regular Polytopes, Dover edition, ISBN 0-486-61480-8, p. 296, Table I: Regular Polytopes, three regular polytopes in n-dimensions H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973, p. 296, Table I: Regular Polytopes, three regular polytopes in n-dimensions Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6 H.S.M. Coxeter, Regular and Semi Regular Polytopes I, H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, H.S.M. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D.
6-simplex
–
Graph
225.
6-orthoplex
–
It has two constructed forms, the first being regular with Schläfli symbol, the second with alternately labeled facets, with Schläfli symbol or Coxeter symbol 311. It is a part of an infinite family of polytopes, orthoplexes. The dual polytope is hexeract. Hexacross, derived from combining the name cross polytope with hex for six in Greek. Hexacontitetrapeton as a 64-facetted 6-polytope. A lowest construction is based on a dual of a 6-orthotope, called a 6-fusil. Cartesian coordinates for the vertices of a 6-orthoplex, centered at the origin are, Every pair is connected by an edge, except opposites. This polytope is one of 63 6-polytopes generated from the B6 Coxeter plane, including the regular 6-cube or 6-orthoplex. H.S.M. Coxeter: H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973 Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6 H.S.M. Coxeter, Regular and Semi Regular Polytopes I, H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, H.S.M.
6-orthoplex
–
Orthogonal projection inside Petrie polygon
226.
6-cube
–
In geometry, a 6-cube is a six-dimensional hypercube with 64 vertices, 192 edges, 240 square faces, 160 cubic cells, 60 tesseract 4-faces, 12 5-cube 5-faces. It has Schläfli symbol, being composed around each 4-face. It can be called a hexeract, a portmanteau of tesseract in Greek. It can also be called a regular dodecapeton, being a 6-dimensional polytope constructed from 12 regular facets. It is a part of an infinite family of called hypercubes. The dual of a 6-cube is a part of the infinite family of cross-polytopes. Deleting alternating vertices of the 6-cube, creates another uniform polytope, called a 6-demicube, which has 12 5-demicube and 32 5-simplex facets. This polytope is one of 63 Uniform 6-polytopes generated from the B6 Coxeter plane, including the regular 6-cube or 6-orthoplex. Coxeter, H.S.M. Regular Polytopes, Dover edition, ISBN 0-486-61480-8 p. 296, Table I: Regular Polytopes, three regular polytopes in n-dimensions Klitzing, Richard. "6D uniform polytopes o3o3o3o3o4x - ax". Weisstein, Eric W. "Hypercube". MathWorld. Olshevsky, George. "Measure polytope".
6-cube
–
Hexeract Quasicrystal structure orthographically projected to 3D using the Golden Ratio.
6-cube
–
Orthogonal projection inside Petrie polygon Orange vertices are doubled, and the center yellow has 4 vertices
227.
6-demicube
–
In geometry, a 6-demicube or demihexteract is a uniform 6-polytope, constructed from a 6-cube with alternated vertices truncated. It is part of a dimensionally infinite family of uniform polytopes called demihypercubes. E. L. Elte identified it as a semiregular polytope labeling it as HM6 for a 6-dimensional half measure polytope. Coxeter named this polytope as 131 with a ring on one of the 1-length branches. It can named similarly by a exponential Schläfli symbol or. Cartesian coordinates for the vertices of a demihexeract centered at the origin are alternate halves of the hexeract: with an odd number of plus signs. The fifth figure is a Euclidean honeycomb, 331, the final is a noncompact hyperbolic honeycomb, 431. Each progressive polytope is constructed from the previous as its vertex figure. It is also the second in a dimensional series of uniform honeycombs, expressed by Coxeter as 13k series. The next figure is the Euclidean honeycomb 133 and the final is a noncompact hyperbolic honeycomb, 134. H.S.M. Coxeter: Coxeter, Regular Polytopes, Dover edition, ISBN 0-486-61480-8, p.296, Table I: Regular Polytopes, three regular polytopes in n-dimensions H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973, p.296, Table I: Regular Polytopes, three regular polytopes in n-dimensions Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6 H.S.M.
6-demicube
–
Petrie polygon projection
228.
1 22 polytope
–
In 6-dimensional geometry, the 122 polytope is a uniform polytope, constructed from the E6 group. It was first published in E. L. Elte's 1912 listing of semiregular polytopes, named as V72. Its Coxeter symbol is 122, describing its Coxeter-Dynkin diagram, with a single ring on the end of the 1-node sequence. There are two rectifications of the 122, construcated by points on the elements of 122. The rectified 122 is constructed by points at the mid-edges of the 122. The birectified 122 is constructed by points at the triangle face centers of the 122. The 1_22 polytope contains 54 5-demicubic facets. It has a 5-simplex vertex figure. Its 72 vertices represent the root vectors of the simple Lie E6. Pentacontatetra-peton - 54-facetted polypeton It is created by a Wythoff construction upon a set of 6 hyperplane mirrors in 6-dimensional space. The information can be extracted from its Coxeter-Dynkin diagram. Removing the node on either of 2-length branches leaves the 5-demicube, 131. The figure is determined by removing the ringed node and ringing the neighboring node. This makes 052. The regular complex polyhedron 332, in C 2 has a real representation as the 122 polytope in 4-dimensional space.
1 22 polytope
–
1 22
229.
2 21 polytope
–
In 6-dimensional geometry, the 221 polytope is a uniform 6-polytope, constructed within the symmetry of the E6 group. It was discovered by Thorold Gosset, published in his 1900 paper. He called an 6-ic semi-regular figure. It is also called the Schläfli polytope. Its Coxeter symbol is 221, describing its Coxeter-Dynkin diagram, with a single ring on the end of one of the 2-node sequences. He also studied its connection on the cubic surface, which are naturally in correspondence with the vertices of 221. The rectified 221 is constructed by points at the mid-edges of the 221. The birectified 241 is the same as the rectified 142. The 221 has 27 vertices, 99 facets: 72 5-simplices. Its figure is a 7-demicube. For visualization this 6-dimensional polytope is often displayed in a special skewed orthographic direction that fits its 27 vertices within a 12-gonal regular polygon. 3 vertices projected into the center. Higher elements can also be drawn on this projection. The Schläfli graph contains the 1-skeleton of this polytope. E. L. Elte named V126 in his 1912 listing of semiregular polytopes.
2 21 polytope
–
2 21
230.
Uniform 7-polytope
–
In seven-dimensional geometry, a 7-polytope is a polytope contained by 6-polytope facets. Each 5-polytope ridge being shared by exactly two 6-polytope facets. A uniform 7-polytope is one, vertex-transitive, constructed from uniform 6-polytope facets. Regular 7-polytopes are represented with 6-polytopes facets around each 4-face. There are exactly three such convex regular 7-polytopes: - 7-simplex - 7-cube - 7-orthoplex There are no nonconvex regular 7-polytopes. The topology of any given 7-polytope is defined by its Betti numbers and torsion coefficients. The value of the Euler characteristic used to characterise polyhedra does not generalize usefully to higher dimensions, whatever their underlying topology. This inadequacy of the Euler characteristic to reliably distinguish between different topologies in higher dimensions led to the discovery of the more sophisticated Betti numbers. There are 71 forms based on all permutations of the Coxeter-Dynkin diagrams with one or more rings. All 71 are enumerated below. Norman Johnson's truncation names are given. Bowers names and acronym are also given for cross-referencing. See also a list of A7 polytopes for symmetric Coxeter plane graphs of these polytopes. The B7 family has symmetry of order 645120. There are 127 forms based on all permutations of the Coxeter-Dynkin diagrams with one or more rings.
Uniform 7-polytope
–
7-simplex
231.
7-simplex
–
In 7-dimensional geometry, a 7-simplex is a self-dual regular 7-polytope. It has 8 vertices, 28 edges, 56 triangle faces, 8 6-simplex 7-faces. Its dihedral angle is cos approximately 81.79 °. It can also be called octa-7-tope, as an 8-facetted polytope in 7-dimensions. The octaexon is derived from octa for eight facets in Greek and - ex for having six-dimensional facets, - on. Jonathan Bowers gives the acronym oca. This construction is based on facets of the 8-orthoplex. This polytope is a facet in the uniform 331 with Coxeter-Dynkin diagram: This polytope is one of 71 uniform 7-polytopes with A7 symmetry. Glossary for hyperspace, George Olshevsky. Polytopes of Various Dimensions Multi-dimensional Glossary
7-simplex
–
Model created using straws (edges) and plasticine balls (vertices) in triakis tetrahedral envelope
7-simplex
–
Orthogonal projection inside Petrie polygon
232.
7-orthoplex
–
It has two constructed forms, the first being regular with Schläfli symbol, the second with alternately labeled facets, with Schläfli symbol or Coxeter symbol 411. It is a part of an infinite family of polytopes, orthoplexes. The dual polytope is hepteract. Heptacross, derived from combining the name cross polytope with hept for seven in Greek. Hecatonicosoctaexon as a 128-facetted 7-polytope. A lowest construction is based on a dual of a 7-orthotope, called a 7-fusil. Cartesian coordinates for the vertices of a 7-orthoplex, centered at the origin are, Every pair is connected by an edge, except opposites. Rectified 7-orthoplex Truncated 7-orthoplex H.S.M. Coxeter: H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973 Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6 H.S.M. Coxeter, Regular and Semi Regular Polytopes I, H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, Norman Johnson Uniform Polytopes, Manuscript N.W.
7-orthoplex
–
Orthogonal projection inside Petrie polygon
233.
7-cube
–
It can be named by its Schläfli symbol, being composed around each 5-face. It can be called a hepteract, a portmanteau of tesseract and hepta in Greek. It can also be called a regular tetradecaexon, being a 7 dimensional polytope constructed from 14 regular facets. It is a part of an infinite family of called hypercubes. The dual of a 7-cube is a part of the infinite family of cross-polytopes. Deleting alternating vertices of the hepteract, creates another uniform polytope, called a demihepteract, which has 14 demihexeractic and 64 6-simplex 6-faces. Hepteract 7D simple rotation through 2Pi with 7D projection to 3D. H.S.M. Coxeter: Coxeter, Regular Polytopes, Dover edition, ISBN 0-486-61480-8, p. 296, Table I: Regular Polytopes, three regular polytopes in n-dimensions H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973, p. 296, Table I: Regular Polytopes, three regular polytopes in n-dimensions Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6 H.S.M. Coxeter, Regular and Semi Regular Polytopes I, H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, Norman Johnson Uniform Polytopes, Manuscript N.W.
7-cube
–
Orthogonal projection inside Petrie polygon The central orange vertex is doubled
234.
7-demicube
–
In geometry, a demihepteract or 7-demicube is a uniform 7-polytope, constructed from the 7-hypercube with alternated vertices truncated. It is part of a dimensionally infinite family of uniform polytopes called demihypercubes. E. L. Elte identified it as a semiregular polytope labeling it as HM7 for a 7-dimensional half measure polytope. Coxeter named this polytope as 141 with a ring on one of the 1-length branches, Schläfli symbol or. Cartesian coordinates for the vertices of a demihepteract centered at the origin are alternate halves of the hepteract: with an odd number of plus signs. There are 95 uniform polytopes with D6 symmetry, 63 are shared by the B6 symmetry, 32 are unique: H.S.M. Coxeter: Coxeter, Regular Polytopes, Dover edition, ISBN 0-486-61480-8, p.296, Table I: Regular Polytopes, three regular polytopes in n-dimensions H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973, p.296, Table I: Regular Polytopes, three regular polytopes in n-dimensions Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6 H.S.M. Coxeter, Regular and Semi Regular Polytopes I, H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 Klitzing, Richard. "7D uniform polytopes x3o3o *b3o3o3o3o - hesa". Olshevsky, George.
7-demicube
–
Petrie polygon projection
235.
1 32 polytope
–
In 7-dimensional geometry, 132 is a uniform polytope, constructed from the E7 group. Its Coxeter symbol is 132, describing its Coxeter-Dynkin diagram, with a single ring on the end of one of the 1-node sequences. The rectified 132 is constructed by points at the mid-edges of the 132. This polytope can tessellate 7-dimensional space, with Coxeter-Dynkin diagram. It is the Voronoi cell of the dual E7* lattice. E. L. Elte named V576 in his 1912 listing of semiregular polytopes. Coxeter called it 132 for its bifurcating Coxeter-Dynkin diagram, on the end of the 1-node branch. Pentacontihexa-hecatonicosihexa-exon - 56-126 facetted polyexon It is created by a Wythoff construction upon a set of 7 hyperplane mirrors in 7-dimensional space. This makes 032, The 132 is third in a dimensional series of uniform polytopes and honeycombs, expressed by Coxeter as 13k series. The next figure is the Euclidean honeycomb 133 and the final is a noncompact hyperbolic honeycomb, 134. The rectified 132 is a rectification of the 132 polytope, creating new vertices on the center of edge of the 132. Its figure is a duoprism prism, the product of a regular tetrahedra and triangle, doubled into a prism: × ×. Rectified pentacontihexa-hecatonicosihexa-exon for rectified 56-126 facetted polyexon It is created upon a set of 7 hyperplane mirrors in 7-dimensional space. The ring represents the position of the active mirror. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C.
1 32 polytope
–
321
236.
2 31 polytope
–
In 7-dimensional geometry, 231 is a uniform polytope, constructed from the E7 group. Its Coxeter symbol is 231, describing its Coxeter-Dynkin diagram, with a single ring on the end of the 2-node branch. The rectified 231 is constructed by points at the mid-edges of the 231. The 231 is composed of 126 vertices, 2016 edges, 20160 cells, 16128 4-faces, 4788 5-faces, 632 6-faces. Its vertex figure is a 6-demicube. Its 126 vertices represent the root vectors of the simple Lie E7. This polytope is the figure for a uniform tessellation of 7-dimensional space, 331. E. L. Elte named it V126 in his 1912 listing of semiregular polytopes. It was called 231 by Coxeter with a single ring on the end of the 2-node sequence. Pentacontihexa-pentacosiheptacontihexa-exon - 56-576 facetted polyexon It is created by a Wythoff construction upon a set of 7 hyperplane mirrors in 7-dimensional space. The facet information can be extracted from its Coxeter-Dynkin diagram. Removing the node on the short branch leaves the 6-simplex. There are 576 of these facets. These facets are centered on the locations of the vertices of the 321 polytope. Removing the node on the end of the 3-length branch leaves the 221.
2 31 polytope
–
321
237.
3 21 polytope
–
In 7-dimensional geometry, the 321 polytope is a uniform 7-polytope, constructed within the symmetry of the E7 group. Removing was discovered by Thorold Gosset, published in his 1900 paper. He called an 7-ic semi-regular figure. Its Coxeter symbol is 321, describing its Coxeter-Dynkin diagram, with a single ring on the end of one of the 3-node sequences. The rectified 321 is constructed by points at the mid-edges of the 321. The birectified 321 is constructed by points at the triangle face centers of the 321. The trirectified 321 is the same as the rectified 132. In 7-dimensional geometry, the 321 is a uniform polytope. Removing has 56 vertices, 702 facets: 576 6-simplexes. For visualization this 7-dimensional polytope is often displayed in a special skewed orthographic direction that fits its 56 vertices within an 18-gonal regular polygon. Its 756 edges are drawn between 2 vertices in the center. Specific higher elements can also be drawn on this projection. The 1-skeleton of the 321 polytope is called a Gosset graph. This polytope, along with the 7-simplex, can tessellate 7-dimensional space, represented by Coxeter-Dynkin diagram:. Removing is also called the Hess polytope for Edmund Hess who first discovered it.
3 21 polytope
–
3 21
238.
Uniform 8-polytope
–
In eight-dimensional geometry, an eight-dimensional polytope or 8-polytope is a polytope contained by 7-polytope facets. Each 6-polytope ridge being shared by exactly two 7-polytope facets. A uniform 8-polytope is one, vertex-transitive, constructed from uniform 7-polytope facets. Regular 8-polytopes can be represented with 7-polytope facets around each peak. There are exactly three such convex regular 8-polytopes: - 8-simplex - 8-cube - 8-orthoplex There are no nonconvex regular 8-polytopes. The topology of any given 8-polytope is defined by its Betti numbers and torsion coefficients. This inadequacy of the Euler characteristic to reliably distinguish between different topologies in higher dimensions led to the discovery of the more sophisticated Betti numbers. There are 135 forms based on all permutations of the Coxeter-Dynkin diagrams with one or more rings. These are all enumerated below. Bowers-style acronym names are given in parentheses for cross-referencing. See also a list of 8-simplex polytopes for symmetric Coxeter plane graphs of these polytopes. The B8 family has symmetry of order 10321920. There are 255 forms based on all permutations of the Coxeter-Dynkin diagrams with one or more rings. See also a list of B8 polytopes for symmetric Coxeter plane graphs of these polytopes. The D8 family has symmetry of order 5,160,960.
Uniform 8-polytope
239.
8-simplex
–
In geometry, an 8-simplex is a self-dual regular 8-polytope. It has 9 vertices, 36 edges, 84 triangle faces, 126 tetrahedral cells, 9 7-simplex 7-faces. Its dihedral angle is cos approximately 82.82 °. It can also be called ennea-8-tope, as a 9-facetted polytope in eight-dimensions. The enneazetton is derived from ennea for nine facets in Greek and - zetta for having seven-dimensional facets, - on. This construction is based on facets of the 9-orthoplex. H.S.M. Coxeter: Coxeter, Regular Polytopes, Dover edition, ISBN 0-486-61480-8, p.296, Table I: Regular Polytopes, three regular polytopes in n-dimensions H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973, p.296, Table I: Regular Polytopes, three regular polytopes in n-dimensions Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6 H.S.M. Coxeter, Regular and Semi Regular Polytopes I, H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, H.S.M. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. Klitzing, Richard.
8-simplex
–
Orthogonal projection inside Petrie polygon
240.
8-orthoplex
–
The second with alternately labeled facets, with Schläfli symbol or Coxeter symbol 511. It is a part of an infinite family of polytopes, orthoplexes. The dual polytope is octeract. Cartesian coordinates for the vertices of an 8-cube, centered at the origin are, Every pair is connected by an edge, except opposites. It is used with the 8-simplex to form the 521 honeycomb. H.S.M. Coxeter: H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973 Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6 H.S.M. Coxeter, Regular and Semi Regular Polytopes I, H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, Norman Johnson Uniform Polytopes, Manuscript N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. Klitzing, Richard.
8-orthoplex
–
Orthogonal projection inside Petrie polygon
241.
8-cube
–
In geometry, an 8-cube is an eight-dimensional hypercube. It has 256 vertices, 1024 edges, 1792 square faces, 1792 cubic cells, 16 7-cube 7-faces. It is represented by Schläfli symbol, being composed around each 6-face. It is called an octeract, a portmanteau of tesseract and oct in Greek. It can also be called a regular hexadecazetton, being an 8-dimensional polytope constructed from 16 regular facets. It is a part of an infinite family of called hypercubes. The dual of an 8-cube is a part of the infinite family of cross-polytopes. Deleting alternating vertices of the octeract, creates another uniform polytope, called a 8-demicube, which has 16 demihepteractic and 128 8-simplex facets. H.S.M. Coxeter: Coxeter, Regular Polytopes, Dover edition, ISBN 0-486-61480-8, p. 296, Table I: Regular Polytopes, three regular polytopes in n-dimensions Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6 H.S.M. Coxeter, Regular and Semi Regular Polytopes I, H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, Norman Johnson Uniform Polytopes, Manuscript N.W.
8-cube
–
Orthogonal projection inside Petrie polygon
242.
8-demicube
–
In geometry, a demiocteract or 8-demicube is a uniform 8-polytope, constructed from the 8-hypercube, octeract, with alternated vertices truncated. It is part of a dimensionally infinite family of uniform polytopes called demihypercubes. E. L. Elte identified it as a semiregular polytope labeling it as HM8 for an 8-dimensional half measure polytope. Coxeter named this polytope as 151 with a ring on one of the 1-length branches, Schläfli symbol or. Cartesian coordinates for the vertices of an 8-demicube centered at the origin are alternate halves of the 8-cube: with an odd number of plus signs. This polytope is the vertex figure for the uniform tessellation, 251 with Coxeter-Dynkin diagram: H.S.M. Coxeter: Coxeter, Regular Polytopes, Dover edition, ISBN 0-486-61480-8, p. 296, Table I: Regular Polytopes, three regular polytopes in n-dimensions H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973, p. 296, Table I: Regular Polytopes, three regular polytopes in n-dimensions Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6 H.S.M. Coxeter, Regular and Semi Regular Polytopes I, H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 Olshevsky, George. "Demiocteract". Glossary for Hyperspace.
8-demicube
–
Petrie polygon projection
243.
1 42 polytope
–
In 8-dimensional geometry, the 142 is a uniform 8-polytope, constructed within the symmetry of the E8 group. Its Coxeter symbol is 142, describing its Coxeter-Dynkin diagram, with a single ring on the end of the 1-node sequences. The rectified 142 is the same as the birectified 241, the quadrirectified 421. The 142 is composed of 2160 7-demicubes. Its figure is a birectified 7-simplex. This polytope, along with the demiocteract, can tessellate 8-dimensional space, represented by Coxeter-Dynkin diagram:. Coxeter named 142 for its bifurcating Coxeter-Dynkin diagram with a single ring on the end of the 1-node branch. It is created by a Wythoff construction upon a set of 8 hyperplane mirrors in 8-dimensional space. The facet information can be extracted from its Coxeter-Dynkin diagram:. Removing the node on the end of the 2-length branch leaves the 7-demicube, 141. Removing the node on the end of the 4-length branch leaves the 132. The vertex figure is determined by removing the ringed node and ringing the neighboring node. This makes the birectified 7-simplex, 042. Vertices are shown as circles, colored in each projective plane. The rectified 142 is named with vertices positioned at the mid-edges of the 142.
1 42 polytope
–
421
244.
2 41 polytope
–
In 8-dimensional geometry, the 241 is a uniform 8-polytope, constructed within the symmetry of the E8 group. Its Coxeter symbol is 241, describing its Coxeter-Dynkin diagram, with a single ring on the end of the 2-node sequences. The rectified 241 is constructed by points at the mid-edges of the 241. The birectified 241 is constructed by points at the triangle face centers of the 241, is the same as the rectified 142. The 241 is composed of 17,520 facets, 144,960 6-faces, 544,320 5-faces, 1,209,600 4-faces, 1,209,600 cells, 483,840 faces, 2160 vertices. Its vertex figure is a 7-demicube. This polytope is a facet in the uniform tessellation, 251 with Coxeter-Dynkin diagram: E. L. Elte named V2160 in his 1912 listing of semiregular polytopes. It is named 241 by Coxeter with a single ring on the end of the 2-node sequence. The facet information can be extracted from its Coxeter-Dynkin diagram:. Removing the node on the short branch leaves the 7-simplex:. There are 17280 of these facets Removing the node on the end of the 4-length branch leaves the 231. There are 240 of these facets. They are centered in the 421 polytope. The figure is determined by removing the ringed node and ringing the neighboring node. This makes the 7-demicube, 141.
2 41 polytope
–
421
245.
4 21 polytope
–
In 8-dimensional geometry, the 421 is a semiregular uniform 8-polytope, constructed within the symmetry of the E8 group. It was discovered by Thorold Gosset, published in his 1900 paper. He called an 8-ic semi-regular figure. Its Coxeter symbol is 421, describing its Coxeter-Dynkin diagram, with a single ring on the end of the 4-node sequences. The rectified 421 is constructed by points at the mid-edges of the 421. The birectified 421 is constructed by points at the triangle face centers of the 421. The trirectified 421 is the same as the rectified 142. The 421 is composed of 2,160 7-orthoplex facets. Its figure is the 321 polytope. For visualization this 8-dimensional polytope is often displayed in a special skewed orthographic direction that fits its 240 vertices within a regular triacontagon. Its 6720 edges are drawn between the 240 vertices. Specific higher elements can also be drawn on this projection. As its 240 vertices represent the root vectors of the simple Lie E8, the polytope is sometimes referred to as the E8 polytope. The vertices of this polytope can be obtained by taking the 240 integral octonions of norm 1. This polytope was discovered by Thorold Gosset, who described it as an 8-ic semi-regular figure.
4 21 polytope
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The 4 21 graph created as string art.
4 21 polytope
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4 21
4 21 polytope
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The 4 21 polytope can be projected into 3-space as a physical vertex-edge model. Pictured here as 2 concentric 600-cells (at the golden ratio) using Zome tools. (Not all of the 3360 edges of length √2(√5-1) are represented.)
246.
Uniform 9-polytope
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In nine-dimensional geometry, a nine-dimensional polytope or 9-polytope is a polytope contained by 8-polytope facets. Each 7-polytope ridge being shared by exactly two 8-polytope facets. A uniform 9-polytope is one, vertex-transitive, constructed from uniform 8-polytope facets. Regular 9-polytopes can be represented with 8-polytope facets around each peak. There are exactly three such convex regular 9-polytopes: - 9-simplex - 9-cube - 9-orthoplex There are no nonconvex regular 9-polytopes. The topology of any given 9-polytope is defined by its Betti numbers and torsion coefficients. The value of the Euler characteristic used to characterise polyhedra does not generalize usefully to higher dimensions, whatever their underlying topology. This inadequacy of the Euler characteristic to reliably distinguish between different topologies in higher dimensions led to the discovery of the more sophisticated Betti numbers. - 9-orthoplex, 611 - The A9 family has symmetry of order 3628800. There are 256+16-1=271 forms based on all permutations of the Coxeter-Dynkin diagrams with one or more rings. These are all enumerated below. Bowers-style acronym names are given in parentheses for cross-referencing. There are 511 forms based on all permutations of the Coxeter-Dynkin diagrams with one or more rings. Eleven cases are shown below: Nine rectified forms and 2 truncations. Bowers-style acronym names are given in parentheses for cross-referencing.
Uniform 9-polytope
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9-simplex
247.
9-simplex
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In geometry, a 9-simplex is a self-dual regular 9-polytope. Its dihedral angle is cos approximately 83.62 °. It can also be called deca-9-tope, as a 10-facetted polytope in 9-dimensions.. The decayotton is derived from deca for ten facets in Greek and - yott, having 8-dimensional facets, - on. This construction is based on facets of the 10-orthoplex. H.S.M. Coxeter: Coxeter, Regular Polytopes, Dover edition, ISBN 0-486-61480-8, p.296, Table I: Regular Polytopes, three regular polytopes in n-dimensions H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973, p.296, Table I: Regular Polytopes, three regular polytopes in n-dimensions Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6 H.S.M. Coxeter, Regular and Semi Regular Polytopes I, H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, H.S.M. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. Klitzing, Richard. "9D uniform polytopes x3o3o3o3o3o3o3o3o - day".
9-simplex
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Orthogonal projection inside Petrie polygon
248.
9-orthoplex
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It has two constructed forms, the second with alternately labeled facets, with Schläfli symbol or Coxeter symbol 611. It is one of an infinite family of polytopes, orthoplexes. The dual polytope is the enneract. Cartesian coordinates for the vertices of a 9-orthoplex, centered at the origin, are, Every pair is connected by an edge, except opposites. H.S.M. Coxeter: H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973 Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6 H.S.M. Coxeter, Regular and Semi Regular Polytopes I, H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, Norman Johnson Uniform Polytopes, Manuscript N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. Klitzing, Richard. "9D uniform polytopes x3o3o3o3o3o3o3o4o - vee".
9-orthoplex
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Orthogonal projection inside Petrie polygon
249.
9-cube
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It can be named by its Schläfli symbol, being composed around each 7-face. It is also called an enneract, a portmanteau of tesseract and enne in Greek. It can also be called a regular octadecayotton, as a nine-dimensional polytope constructed with 18 regular facets. It is a part of an infinite family of called hypercubes. The dual of a 9-cube is a part of the infinite family of cross-polytopes. Deleting alternating vertices of the 9-cube, creates another uniform polytope, called a 9-demicube, which has 18 8-demicube and 256 8-simplex facets. H.S.M. Coxeter: Coxeter, Regular Polytopes, Dover edition, ISBN 0-486-61480-8, p. 296, Table I: Regular Polytopes, three regular polytopes in n-dimensions H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973, p. 296, Table I: Regular Polytopes, three regular polytopes in n-dimensions Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6 H.S.M. Coxeter, Regular and Semi Regular Polytopes I, H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, Norman Johnson Uniform Polytopes, Manuscript N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D.
9-cube
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Orthogonal projection inside Petrie polygon Orange vertices are doubled, yellow have 4, and the green center has 8
250.
9-demicube
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In geometry, a demienneract or 9-demicube is a uniform 9-polytope, constructed from the 9-cube, with alternated vertices truncated. It is part of a dimensionally infinite family of uniform polytopes called demihypercubes. E. L. Elte identified it as a semiregular polytope labeling it as HM9 for a 9-dimensional half measure polytope. Coxeter named this polytope as 161 with a ring on one of the 1-length branches, Schläfli symbol or. Cartesian coordinates for the vertices of a demienneract centered at the origin are alternate halves of the enneract: with an odd number of plus signs. H.S.M. Coxeter: Coxeter, Regular Polytopes, Dover edition, ISBN 0-486-61480-8, p. 296, Table I: Regular Polytopes, three regular polytopes in n-dimensions H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973, p. 296, Table I: Regular Polytopes, three regular polytopes in n-dimensions Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6 H.S.M. Coxeter, Regular and Semi Regular Polytopes I, H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 Klitzing, Richard. "9D uniform polytopes x3o3o *b3o3o3o3o3o3o - henne". Olshevsky, George.
9-demicube
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Petrie polygon
251.
Uniform 10-polytope
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In ten-dimensional geometry, a 10-polytope is a 10-dimensional polytope whose boundary consists of 9-polytope facets, exactly two such facets meeting at each 8-polytope ridge. A uniform 10-polytope is one, vertex-transitive, constructed from uniform facets. Regular 10-polytopes can be represented with x 9-polytope facets around each peak. There are exactly three such convex 9-polytopes: - 9-simplex - 9-cube - 9-orthoplex There are no nonconvex regular 9-polytopes. The topology of any given 10-polytope is defined by its Betti numbers and torsion coefficients. This inadequacy of the Euler characteristic to reliably distinguish between different topologies in higher dimensions led to the discovery of the more sophisticated Betti numbers. There are 512+16-1=527 forms based on all permutations of the Coxeter-Dynkin diagrams with one or more rings. 31 are shown below: all one and two ringed forms, the final omnitruncated form. Bowers-style acronym names are given in parentheses for cross-referencing. There are 1023 forms based on all permutations of the Coxeter-Dynkin diagrams with one or more rings. Twelve cases are shown below: two truncations. Bowers-style acronym names are given in parentheses for cross-referencing. The D10 family has symmetry of order 1,857,945,600. This family has × 256 − 1 = 767 Wythoffian uniform polytopes, generated by marking one or more nodes of the D10 Coxeter-Dynkin diagram. Of these, 256 are unique to this family, with 2 listed below.
Uniform 10-polytope
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10-simplex
252.
10-simplex
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In geometry, a 10-simplex is a self-dual regular 10-polytope. Its dihedral angle is cos approximately 84.26 °. It can also be called hendeca-10-tope, as an 11-facetted polytope in 10-dimensions. The hendecaxennon is derived from hendeca for 11 facets in Greek and - xenn, having 9-dimensional facets, - on. This construction is based on facets of the 11-orthoplex. H.S.M. Coxeter: Coxeter, Regular Polytopes, Dover edition, ISBN 0-486-61480-8, p.296, Table I: Regular Polytopes, three regular polytopes in n-dimensions H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973, p.296, Table I: Regular Polytopes, three regular polytopes in n-dimensions Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6 H.S.M. Coxeter, Regular and Semi Regular Polytopes I, H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, H.S.M. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. Klitzing, Richard. "10D uniform polytopes x3o3o3o3o3o3o3o3o3o - ux".
10-simplex
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Orthogonal projection inside Petrie polygon
253.
10-orthoplex
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It has two constructed forms, the second with alternately labeled facets, with Schläfli symbol or Coxeter symbol 711. It is one of an infinite family of polytopes, orthoplexes. The dual polytope is the 10-cube. Decacross is derived from combining the family name cross polytope with deca for ten as a 1024-facetted 10-polytope. Cartesian coordinates for the vertices of a 10-orthoplex, centred at the origin are, Every pair is connected by an edge, except opposites. H.S.M. Coxeter: H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973 Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6 H.S.M. Coxeter, Regular and Semi Regular Polytopes I, H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, Norman Johnson Uniform Polytopes, Manuscript N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. Klitzing, Richard.
10-orthoplex
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Orthogonal projection inside Petrie polygon
254.
10-cube
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In geometry, a 10-cube is a ten-dimensional hypercube. It can be named by its Schläfli symbol, being composed around each 8-face. It is a part of an infinite family of called hypercubes. The dual of a dekeract can be called a decacross, is a part of the infinite family of cross-polytopes. Deleting alternating vertices of the dekeract, creates another uniform polytope, called a 10-demicube, which has 20 demienneractic and 512 enneazettonic facets. H.S.M. Coxeter: Coxeter, Regular Polytopes, Dover edition, ISBN 0-486-61480-8, p.296, Table I: Regular Polytopes, three regular polytopes in n-dimensions H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973, p. 296, Table I: Regular Polytopes, three regular polytopes in n-dimensions Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6 H.S.M. Coxeter, Regular and Semi Regular Polytopes I, H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, Norman Johnson Uniform Polytopes, Manuscript N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. Klitzing, Richard.
10-cube
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Orthogonal projection inside Petrie polygon Orange vertices are doubled, and central yellow one has four
255.
10-demicube
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In geometry, a 10-demicube or demidekeract is a uniform 10-polytope, constructed from the 10-cube with alternated vertices truncated. It is part of a dimensionally infinite family of uniform polytopes called demihypercubes. E. L. Elte identified it as a semiregular polytope labeling it as HM10 for a ten-dimensional half measure polytope. Coxeter named this polytope as 171 with a ring on one of the 1-length branches, Schläfli symbol or. Cartesian coordinates for the vertices of a demidekeract centered at the origin are alternate halves of the dekeract: with an odd number of plus signs. H.S.M. Coxeter: Coxeter, Regular Polytopes, Dover edition, ISBN 0-486-61480-8, p.296, Table I: Regular Polytopes, three regular polytopes in n-dimensions H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973, p.296, Table I: Regular Polytopes, three regular polytopes in n-dimensions Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6 H.S.M. Coxeter, Regular and Semi Regular Polytopes I, H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, H.S.M. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. Klitzing, Richard. "10D uniform polytopes x3o3o *b3o3o3o3o3o3o3o - hede".
10-demicube
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Petrie polygon projection
256.
Polytope
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For example, a three-dimensional polyhedron is a 3-polytope. Polytopes in more than three dimensions were first discovered by Ludwig Schläfli. The German polytop was coined by the mathematician Reinhold Hoppe, was introduced to English mathematicians as polytope by Alicia Boole Stott. Different definitions are attested in mathematical literature. Many of these definitions are not equivalent, resulting in different sets of objects being called polytopes. They represent different approaches to generalizing the convex polytopes to include other objects with similar properties. In this approach, a polytope may be regarded as a decomposition of some given manifold. An example of this approach defines a polytope as a set of points that admits a simplicial decomposition. However this definition does not allow star polytopes with interior structures, so is restricted to certain areas of mathematics. The discovery of other unusual constructions led to the idea of a polyhedron as a bounding surface, ignoring its interior. A polyhedron is understood as a surface whose faces are a 4-polytope as a hypersurface whose facets are polyhedra, so forth. This approach is used in the theory of abstract polytopes. This terminology is typically confined to polyhedra that are convex. A polytope comprises elements of different dimensionality such as vertices, edges, faces, so on. Terminology for these is not fully consistent across different authors.
Polytope
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A polygon is a 2-dimensional polytope.
257.
Simplex
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In geometry, a simplex is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions. Specifically, a k-simplex is a k-dimensional polytope, the convex hull of its k + 1 vertices. Then, the simplex determined by them is the set of C =. For example, a 2-simplex is a triangle, a 4-simplex is a 5-cell. A line segment may be considered a 1-simplex. A simplex may be defined as the smallest convex set containing the given vertices. A regular simplex is a simplex, also a regular polytope. A regular n-simplex may be constructed by connecting a new vertex to all original vertices by the common edge length. In topology and combinatorics, it is common to “glue together” simplices to form a simplicial complex. The associated combinatorial structure is called an simplicial complex, in which context the word "simplex" simply means any finite set of vertices. A 0-simplex is a point. A 1-simplex is a segment. A 2-simplex is a triangle. A 3-simplex is a tetrahedron. The convex hull of any nonempty subset of the n+1 points that define an n-simplex is called a face of the simplex.
Simplex
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A regular 3-simplex or tetrahedron
258.
Cross-polytope
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In geometry, a cross-polytope, orthoplex, hyperoctahedron, or cocube is a regular, convex polytope that exists in n-dimensions. A 2-orthoplex is a square, a 3-orthoplex is a regular octahedron, a 4-orthoplex is a 16-cell. Its facets are simplexes of the previous dimension, while the cross-polytope's figure is another cross-polytope from the previous dimension. The vertices of a cross-polytope are all the permutations of. The cross-polytope is the convex hull of its vertices. The n-dimensional cross-polytope can also be defined as the closed ball in the ℓ1-norm on Rn:. In 1 dimension the cross-polytope is simply the segment, in 2 dimensions it is a square with vertices. In 3 dimensions it is an octahedron—one of the five convex regular polyhedra known as the Platonic solids. Higher-dimensional cross-polytopes are generalizations of these. The cross-polytope is the dual polytope of the hypercube. The 1-skeleton of a n-dimensional cross-polytope is a Turán graph T. The 4-dimensional cross-polytope also goes by the name 16-cell. It is one of six regular 4-polytopes. These 4-polytopes were first described in the mid-19th century. The infinite tessellations of hypercubes, he labeled as δn.
Cross-polytope
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2 dimensions square
259.
Hypercube
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In geometry, a hypercube is an n-dimensional analogue of a square and a cube. A hypercube's longest diagonal in n-dimensions is equal to n. An n-dimensional hypercube is also called an n-dimensional cube. It has now been superseded. The hypercube is the special case of a hyperrectangle. A hypercube is a hypercube whose side has length one unit. Often, the hypercube whose corners are the 2n points in Rn with coordinates equal to 1 is called "the" unit hypercube. A hypercube can be defined by increasing the numbers of dimensions of a shape: 0 – A point is a hypercube of dimension zero. 1 – If one moves this point one unit length, it will sweep out a line segment, a unit hypercube of dimension one. 2 -- If one moves its length in a perpendicular direction from itself; it sweeps out a 2-dimensional square. 3 – If one moves the square one unit length in the direction perpendicular to the plane it lies on, it will generate a 3-dimensional cube. 4 – If one moves the cube one unit length into the fourth dimension, it generates a 4-dimensional unit hypercube. This can be generalized to any number of dimensions. The 1-skeleton of a hypercube is a hypercube graph. A hypercube of n dimensions is the convex hull of the points given by all sign permutations of the Cartesian coordinates.
Hypercube
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Perspective projections
260.
Demihypercube
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In geometry, demihypercubes are a class of n-polytopes constructed from alternation of an n-hypercube, labeled as hγn for being half of the hypercube family, γn. New facets are formed. The 2n facets become 2n - 2n - simplex facets are formed in place of the deleted vertices. They have been named with a demi - prefix to each hypercube name: demitesseract, etc.. The demitesseract is identical to the regular 16-cell. The demipenteract is considered semiregular for having only regular facets. Higher forms are all uniform polytopes. The edges of a demihypercube form two copies of the halved cube graph. Thorold Gosset described the demipenteract in his 1900 publication listing all of the regular and semiregular figures in n-dimensions above 3. He called a 5-ic semi-regular. It also exists within the k21 polytope family. The demihypercubes can be represented as half the vertices of. The vertex figures of demihypercubes are rectified n-simplexes. They are represented by Coxeter-Dynkin diagrams of three constructive forms:... S...
Demihypercube
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Alternation of the n -cube yields one of two n -demicubes, as in this 3-dimensional illustration of the two tetrahedra that arise as the 3-demicubes of the 3-cube.
261.
Uniform 1 k2 polytope
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In geometry, 1k2 polytope is a uniform polytope in n-dimensions constructed from the En Coxeter group. The family was named with a single ring on the end of the 1-node sequence. It can be named by an extended Schläfli symbol. The family can be extended backwards to include the 4-simplex in 4-dimensions. Each polytope is constructed from 1k-1,2 and -demicube facets. Each has a vertex figure of a polytope is a birectified n-simplex, t2. The sequence ends as an infinite tessellation of hyperbolic space. "Geometrical Deduction of Semiregular from Regular Polytopes and Space Fillings." Verhandelingen der Koninklijke Akad. Wetenschappen Amsterdam 11, 3-24, 1910. Alicia Boole Stott, "Geometrical deduction of semiregular from regular polytopes and space fillings," Verhandelingen der Koninklijke Akademie van Wetenschappen te Amsterdam, Vol. 11, No. 1, pp. 1–24 plus 3 plates, 1910. Stott, A. B. 1910. "Geometrical Deduction of Semiregular from Regular Polytopes and Space Fillings." Verhandelingen der Koninklijke Akad.
Uniform 1 k2 polytope
262.
Uniform 2 k1 polytope
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In geometry, 2k1 polytope is a uniform polytope in n dimensions constructed from the En Coxeter group. The family was named by their Coxeter symbol as 2k1 with a single ring on the end of the 2-node sequence. It can be named by an extended Schläfli symbol. The family can be extended backwards to include the 5-orthoplex in 5-dimensions, the 4-simplex in 4-dimensions. Each polytope is constructed from - 2k-1,1 - polytope facets, each has a vertex figure as an - demicube. The sequence ends as an infinite hyperbolic tessellation of 9-space. "Geometrical Deduction of Semiregular from Regular Polytopes and Space Fillings." Verhandelingen der Koninklijke Akad. Wetenschappen Amsterdam 11, 3-24, 1910. Alicia Boole Stott, "Geometrical deduction of semiregular from regular polytopes and space fillings," Verhandelingen der Koninklijke Akademie van Wetenschappen te Amsterdam, Vol. 11, No. 1, pp. 1–24 plus 3 plates, 1910. Stott, A. B. 1910. "Geometrical Deduction of Semiregular from Regular Polytopes and Space Fillings." Verhandelingen der Koninklijke Akad. H. S. M. Coxeter: Regular and Semi-Regular Polytopes, Part I, Mathematische Zeitschrift, Springer, Berlin, 1940 N.W.
Uniform 2 k1 polytope
263.
Uniform k 21 polytope
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In geometry, a uniform k21 polytope is a polytope in k + 4 dimensions constructed from the En Coxeter group, having only regular polytope facets. The family was named by their Coxeter symbol k21 with a single ring on the end of the k-node sequence. Gosset named them by their dimension for example the 5-ic semiregular figure. The sequence as identified by Gosset ends as an infinite tessellation in 8-space, called the E8 lattice. (A final form was not discovered by Gosset and is called the E9 lattice: 621. It is a tessellation of constructed of The family starts uniquely as 6-polytopes. Rectified 5-cell are included at the beginning for completeness. The demipenteract also exists in the demihypercube family. They are also sometimes named like E6 polytope, although there are many uniform polytopes within the E6 symmetry. The orthoplex faces have a Schläfli symbol of rather than the regular. This construction is an implication of two "facet types". The others are attached to a simplex. In contrast, every ridge is attached to an orthoplex. Each has a figure as the previous form. For example, the rectified 5-cell has a figure as a triangular prism.
Uniform k 21 polytope
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3-ic
264.
Pentagonal polytope
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In geometry, a pentagonal polytope is a regular polytope in n dimensions constructed from the Hn Coxeter group. The family was named by H. S. M. Coxeter, because the pentagonal polytope is a pentagon. It can be named by its Schläfli symbol as or. The family ends with n = 5 as infinite tessellations of 4-dimensional hyperbolic space. There are two types of pentagonal polytopes; they may be termed the icosahedral types, by their three-dimensional members. The two types are duals of each other. Their vertex figures are the simplices of one less dimension. Their vertex figures are pentagonal polytopes of one less dimension. The pentagonal polytopes can be stellated to form new regular polytopes: In three dimensions, this forms the four Kepler -- Poinsot polyhedra. In four dimensions, this forms the ten Schläfli–Hess polychora:, and. In hyperbolic space there are four regular star-honeycombs:. Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6 H.S.M. Coxeter, Star Polytopes and the Schlafli Function f Coxeter, Regular Polytopes, 3rd.
Pentagonal polytope
265.
Polytope families
Polytope families
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Triangle
266.
List of regular polytopes and compounds
–
This page lists the regular polytopes and regular polytope compounds in Euclidean, spherical and hyperbolic spaces. The Schläfli symbol describes every regular tessellation of an n-sphere, Euclidean and hyperbolic spaces. A Schläfli symbol describing an n-polytope equivalently describes a tessellation of a -sphere. For example, the cube has Schläfli symbol, with its octahedral symmetry, or, is represented by Coxeter diagram. The regular polytopes are grouped by dimension and subgrouped by convex, nonconvex and infinite forms. Nonconvex forms use the same vertices as the convex forms, but have intersecting facets. Infinite forms tessellate a one-lower-dimensional Euclidean space. Infinite forms can be extended to tessellate a hyperbolic space. Hyperbolic space is like normal space at a small scale, but parallel lines diverge at a distance. This allows vertex figures to have negative angle defects, like making a vertex with seven equilateral triangles and allowing it to lie flat. It cannot be done in a regular plane, but can be at the right scale of a hyperbolic plane. This table shows a summary of regular polytope counts by dimension. There are no Euclidean regular star tessellations in any number of dimensions. A one-dimensional polytope or 1-polytope is a closed line segment, bounded by its two endpoints. A 1-polytope is regular by definition and is represented by Schläfli symbol, or a Coxeter diagram with a single ringed node.
List of regular polytopes and compounds
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Small stellated dodecahedron
List of regular polytopes and compounds
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{5}
List of regular polytopes and compounds
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Great dodecahedron
List of regular polytopes and compounds
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Great stellated dodecahedron
267.
Hexagon
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In geometry, a hexagon is a six sided polygon or 6-gon. The total of the internal angles of any hexagon is 720°. A regular hexagon can also be constructed as a truncated equilateral triangle, t, which alternates two types of edges. A regular hexagon is defined as a hexagon, both equiangular. It meaning that it is both cyclic and tangential. The common length of the sides equals the radius of the circumscribed circle, which equals 3 3 times the apothem. All internal angles are 120 degrees. A regular hexagon has 6 reflection symmetries, making up the dihedral group D6. The longest diagonals of a regular hexagon, connecting opposite vertices, are twice the length of one side. Like equilateral triangles, regular hexagons fit together without any gaps to tile the plane, so are useful for constructing tessellations. The cells of a beehive honeycomb are hexagonal for this reason and because the shape makes efficient use of building materials. The Voronoi diagram of a regular lattice is the honeycomb tessellation of hexagons. It is not usually considered a triambus, although it is equilateral. The maximal diameter, D is twice R, which equals the side length, t. The minimal diameter or the diameter of the inscribed circle, d, is twice the minimal radius or inradius, r.
Hexagon
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Giants causeway closeup
Hexagon
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The ideal crystalline structure of graphene is a hexagonal grid.
Hexagon
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Assembled E-ELT mirror segments
Hexagon
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A beehive honeycomb