1.
Regular polygon
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In Euclidean geometry, a regular polygon is a polygon that is equiangular and equilateral. Regular polygons may be convex or star, in the limit, a sequence of regular polygons with an increasing number of sides becomes a circle, if the perimeter is fixed, or a regular apeirogon, if the edge length is fixed. These properties apply to all regular polygons, whether convex or star, a regular n-sided polygon has rotational symmetry of order n. All vertices of a regular polygon lie on a common circle and that is, a regular polygon is a cyclic polygon. Together with the property of equal-length sides, this implies that every regular polygon also has a circle or incircle that is tangent to every side at the midpoint. Thus a regular polygon is a tangential polygon, a regular n-sided polygon can be constructed with compass and straightedge if and only if the odd prime factors of n are distinct Fermat primes. The symmetry group of a regular polygon is dihedral group Dn, D2, D3. It consists of the rotations in Cn, together with reflection symmetry in n axes that pass through the center, if n is even then half of these axes pass through two opposite vertices, and the other half through the midpoint of opposite sides. If n is odd then all pass through a vertex. All regular simple polygons are convex and those having the same number of sides are also similar. An n-sided convex regular polygon is denoted by its Schläfli symbol, for n <3 we have two degenerate cases, Monogon, degenerate in ordinary space. Digon, a line segment, degenerate in ordinary space. In certain contexts all the polygons considered will be regular, in such circumstances it is customary to drop the prefix regular. For instance, all the faces of uniform polyhedra must be regular, for n >2 the number of diagonals is n 2, i. e.0,2,5,9. for a triangle, square, pentagon, hexagon. The diagonals divide the polygon into 1,4,11,24, for a regular n-gon inscribed in a unit-radius circle, the product of the distances from a given vertex to all other vertices equals n. For a regular simple n-gon with circumradius R and distances di from a point in the plane to the vertices. For a regular n-gon, the sum of the distances from any interior point to the n sides is n times the apothem. This is a generalization of Vivianis theorem for the n=3 case, the sum of the perpendiculars from a regular n-gons vertices to any line tangent to the circumcircle equals n times the circumradius
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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For edge in graph theory, see Edge In geometry, an edge is a particular type of line segment joining two vertices in a polygon, polyhedron, or higher-dimensional polytope. In a polygon, an edge is a segment on the boundary. 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 interior or exterior is not an edge but instead is called a diagonal. In graph theory, an edge is an abstract object connecting two vertices, unlike polygon and polyhedron edges which have a concrete geometric representation as a line segment. However, any polyhedron can be represented by its skeleton or edge-skeleton, conversely, the graphs that are skeletons of three-dimensional polyhedra can be characterized by Steinitzs theorem as being exactly the 3-vertex-connected planar graphs. Any convex polyhedrons surface has Euler characteristic V − E + F =2, where V is the number of vertices, E is the number of edges and this equation is known as Eulers polyhedron formula. Thus the number of edges is 2 less than the sum of the numbers of vertices and faces, for example, a cube has 8 vertices and 6 faces, and hence 12 edges. In a polygon, two edges meet at each vertex, more generally, by Balinskis theorem, at least d edges meet at every vertex of a convex polytope. Similarly, in a polyhedron, exactly two faces meet at every edge, while in higher dimensional polytopes three or more two-dimensional faces meet at every edge. Thus, the edges of a polygon are its facets, the edges of a 3-dimensional convex polyhedron are its ridges, archived from the original on 4 February 2007
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 an angle and the corners of polygons and polyhedra are vertices. A vertex is a point of a polygon, polyhedron, or other higher-dimensional polytope. However, in theory, vertices may have fewer than two incident edges, which is usually not allowed for geometric vertices. However, a smooth approximation to a polygon will also have additional vertices. A polygon vertex xi of a simple polygon P is a principal polygon vertex if the diagonal intersects the boundary of P only at x and x, there are two types of principal vertices, ears and mouths. A principal vertex 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 vertex xi of a simple polygon P is called a mouth if the diagonal lies outside the boundary of P. Any convex polyhedrons surface has Euler characteristic V − E + F =2, where V is the number of vertices, E is the number of edges and this equation is known as Eulers polyhedron 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 12 edges and 6 faces, and hence 8 vertices
Vertex (geometry)
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A vertex of an angle is the endpoint where two line segments or rays come together.
4.
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 groups, and they play an important role in group theory, geometry. The notation for the group of order n differs in geometry. In geometry, Dn or Dihn refers to the symmetries of the n-gon, 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, 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, the following Cayley table shows the effect of composition in the group D3. R0 denotes the identity, r1 and r2 denote counterclockwise rotations by 120° and 240° respectively, for example, s2s1 = r1, because the reflection s1 followed by the reflection s2 results in a rotation of 120°. The order of elements denoting the composition is right to left, the composition operation is not commutative. In all cases, addition and subtraction of subscripts are to be performed using modular arithmetic with modulus n, if we center the regular polygon at the origin, then elements of the dihedral group act as linear transformations of the plane. This lets us represent elements of Dn as matrices, with composition being matrix multiplication and this is an example of a group representation. For example, the elements of the group D4 can be represented by the eight matrices. In general, the matrices for elements of Dn have the following form, rk is a rotation matrix, expressing a counterclockwise rotation through an angle of 2πk/n. Sk is a reflection across a line makes an angle of πk/n with the x-axis. Further equivalent definitions of Dn are, D1 is isomorphic to Z2, D2 is isomorphic to K4, the Klein four-group. D1 and D2 are exceptional in that, D1 and D2 are the only abelian dihedral groups, Dn is a subgroup of the symmetric group Sn for n ≥3
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.
5.
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. For a simple polygon, regardless of whether it is convex or non-convex, a polygon has exactly one internal angle per vertex. If every internal angle of a polygon is less than 180°. In contrast, an angle is an angle formed by one side of a simple polygon. The sum of the angle and the external angle on the same vertex is 180°. The sum of all the angles of a simple polygon is 180° where n is the number of sides. The formula can be proved using induction and starting with a triangle for which the angle sum is 180°. The sum of the angles of any simple convex or non-convex polygon is 360°. 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. For example, for convex and concave polygons k =1, since the exterior angle sum is 360°
Internal angle
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Internal and External angles
6.
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. It is not an SI unit, as the SI unit of measure is the radian. 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 rotations and angles is unknown. One theory states that it is related to the fact that 360 is approximately the number of days in a year. Ancient astronomers noticed that the sun, which follows through the path over the course of the 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 the Babylonian astronomers and their Greek successors, was based on chords of a circle, a chord of length equal to the radius made a natural base quantity. One sixtieth of this, using their standard sexagesimal divisions, was a degree, Aristarchus of Samos and Hipparchus seem to have been among the first Greek scientists to exploit Babylonian astronomical knowledge and techniques systematically. Timocharis, Aristarchus, Aristillus, Archimedes, and Hipparchus were the first Greeks known to divide the circle in 360 degrees of 60 arc minutes, eratosthenes used a simpler sexagesimal system dividing a circle into 60 parts. Furthermore, it is divisible by every number from 1 to 10 except 7 and this property has many useful applications, such as dividing the world into 24 time zones, each of which is nominally 15° of longitude, to correlate with the established 24-hour day convention. Finally, it may be the case more than one of these factors has come into play. For many practical purposes, a degree is a small enough angle that whole degrees provide sufficient precision. When this is not the case, as in astronomy or for geographic coordinates, degree measurements may be written using decimal degrees, with the symbol behind the decimals. Alternatively, the sexagesimal unit subdivisions can be used. One degree is divided into 60 minutes, and one minute into 60 seconds, use of degrees-minutes-seconds is also called DMS notation. These subdivisions, also called the arcminute and arcsecond, are represented by a single and double prime. For example,40. 1875° = 40° 11′ 15″, or, using quotation mark characters, additional precision can be provided using decimals for the arcseconds component. The older system of thirds, fourths, etc. which continues the sexagesimal unit subdivision, was used by al-Kashi and other ancient astronomers, but is rarely used today
Degree (angle)
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One degree (shown in red) and eighty nine (shown in blue)
7.
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 polygon whose interior is a convex set. In a convex polygon, all angles are less than or equal to 180 degrees. A simple polygon which is not convex is called concave, the following properties of a simple polygon are all equivalent to convexity, Every internal angle is less than or equal to 180 degrees. Every point on line segment between two points inside or on the boundary of the polygon remains inside or on the boundary. The polygon is contained in a closed half-plane defined by each of its edges. For each edge, the points are all on the same side of the line that the edge defines. The angle at each vertex contains all vertices in its edges. The polygon is the hull of its edges. Additional properties of convex polygons include, The intersection of two convex polygons is a convex polygon, a convex polygon may br triangulated in linear time through a fan triangulation, consisting in adding diagonals from one vertex to all other vertices. Hellys theorem, For every collection of at least three convex polygons, if the intersection of three of them is nonempty, then the whole collection has a nonempty intersection. Krein–Milman theorem, A convex polygon is the hull of its vertices. Thus it is defined by the set of its vertices. Hyperplane separation theorem, Any two convex polygons with no points in common have a separator line, if the polygons are closed and 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, inscribing triangle property, every convex polygon with area A can be inscribed in a triangle of area at most equal to 2A. Equality holds for a parallelogram.5 × Area ≤ Area ≤2 × Area, the mean width of a convex polygon is equal to its perimeter divided by pi. So its width is the diameter of a circle with the perimeter as the polygon. Every polygon inscribed in a circle, if not self-intersecting, is convex, however, not every convex polygon can be inscribed in a circle
Convex polygon
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An example of a convex polygon: a regular pentagon
8.
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. The center of circle is called the circumcenter and its radius is called the circumradius. A polygon which has a circle is called a cyclic polygon. All regular simple polygons, all isosceles trapezoids, all triangles, a related notion is the one of a minimum bounding circle, which is the smallest circle that completely contains the polygon within it. All triangles are cyclic, i. e. every triangle has a circumscribed circle and this can be proven on the grounds that the general equation for a circle with center and radius r in the Cartesian coordinate system is 2 +2 = r 2. 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, and 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, and the radius is the length to any of the three vertices. This is because the circumcenter is equidistant from any pair of the triangles vertices, in coastal navigation, a triangles 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, in the Euclidean plane, it is possible to give explicitly an equation of the circumcircle in terms of the Cartesian coordinates of the vertices of the inscribed triangle. Suppose that A = B = C = are the coordinates of points A, B, 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, a similar approach allows one to deduce the equation of the circumsphere of a tetrahedron. A unit vector perpendicular to the containing the circle is given by n ^ = × | × |. An equation for the circumcircle in trilinear coordinates x, y, z is a/x + b/y + c/z =0, an equation for the circumcircle in barycentric coordinates x, y, z is a2/x + b2/y + c2/z =0. The isogonal conjugate of the circumcircle is the line at infinity, given in coordinates by ax + by + cz =0. Additionally, the circumcircle of a triangle embedded in d dimensions can be using a generalized method. Let A, B, and C be d-dimensional points, which form the vertices of a triangle and we start by transposing the system to place C at the origin, a = A − C, b = B − C. The circumcenter, p0, is given by p 0 = ×2 ∥ a × b ∥2 + C, the Cartesian coordinates of the circumcenter are U x =1 D U y =1 D with D =2. Without loss of generality this can be expressed in a form after translation of the vertex A to the origin of the Cartesian coordinate systems
Cyclic polygon
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Circumscribed circle, C, and circumcenter, O, of a cyclic polygon, P
9.
Isogonal figure
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In geometry, a polytope is isogonal or vertex-transitive if, loosely speaking, all its vertices are equivalent. That implies that each vertex is surrounded by the kinds of face in the same or reverse order. Technically, we say that for any two vertices there exists a symmetry of the polytope mapping the first isometrically onto the second. Other ways of saying this are that the group of automorphisms of the polytope is transitive on its vertices, all vertices of a finite n-dimensional isogonal figure exist on an -sphere. The term isogonal has long used for polyhedra. Vertex-transitive is a synonym borrowed from modern ideas such as symmetry groups, all regular polygons, apeirogons and regular star polygons are isogonal. The dual of a polygon is an isotoxal polygon. Some even-sided polygons and apeirogons which alternate two edge lengths, for example a rectangle, are isogonal, all planar isogonal 2n-gons have dihedral symmetry with reflection lines across the mid-edge points. An isogonal polyhedron and 2D tiling has a kind of vertex. An isogonal polyhedron with all faces is also a uniform polyhedron. Geometrically distorted variations of uniform polyhedra and tilings can also be given the vertex configuration, isogonal polyhedra and 2D tilings may be further classified, Regular if it is also isohedral and isotoxal, this implies that every face is the same kind of regular polygon. Quasi-regular if it is also isotoxal but not isohedral, semi-regular if every face is a regular polygon but it is not isohedral or isotoxal. Uniform if every face is a polygon, i. e. it is regular, quasiregular or semi-regular. Noble if it is also isohedral and these definitions can be extended to higher-dimensional polytopes and tessellations. Most generally, all uniform polytopes are isogonal, for example, the dual of an isogonal polytope is called an isotope which is transitive on its facets. A polytope or tiling may be called if its vertices form k transitivity classes. A more restrictive term, k-uniform is defined as a figure constructed only from regular polygons. They can be represented visually with colors by different uniform colorings, edge-transitive Face-transitive Peter R. Cromwell, Polyhedra, Cambridge University Press 1997, ISBN 0-521-55432-2, p.369 Transitivity Grünbaum, Branko, Shephard, G. C
Isogonal figure
10.
Isotoxal figure
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In geometry, a polytope, or a tiling, is isotoxal or edge-transitive if its symmetries act transitively on its edges. The term isotoxal is derived from the Greek 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 a polygon with D2 symmetry. All regular polygons are isotoxal, having double the symmetry order. A regular 2n-gon is a polygon and can be marked with alternately colored vertices. An isotoxal polyhedron or tiling must be either isogonal or isohedral or both, regular polyhedra are isohedral, isogonal and isotoxal. Quasiregular polyhedra are isogonal and isotoxal, but not isohedral, their duals are isohedral and isotoxal, not every polyhedron or 2-dimensional tessellation constructed from regular polygons is isotoxal. An isotoxal polyhedron has the dihedral angle for all edges. There are nine convex isotoxal polyhedra formed from the Platonic solids,8 formed by the Kepler–Poinsot polyhedra, cS1 maint, Multiple names, authors list Coxeter, Harold Scott MacDonald, Longuet-Higgins, M. S. Miller, J. C. P. Uniform polyhedra, Philosophical Transactions of the Royal Society of London, mathematical and Physical Sciences,246, 401–450, doi,10. 1098/rsta.1954.0003, ISSN 0080-4614, JSTOR91532, MR0062446
Isotoxal figure
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A rhombic dodecahedron is an isohedral and isotoxal polyhedron
Isotoxal figure
11.
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 divided into the Archaic period, Classical period. It is antedated in the second millennium BC by Mycenaean Greek, the language of the Hellenistic phase is known as Koine. Koine is regarded as a historical stage of its own, although in its earliest form it closely resembled Attic Greek. Prior to the Koine period, Greek of the classic and earlier periods included several regional dialects, Ancient Greek was the language of Homer and of fifth-century Athenian historians, playwrights, and philosophers. It has contributed many words to English vocabulary and has been a subject of study in educational institutions of the Western world since the Renaissance. This article primarily contains information about the Epic and Classical phases of the language, Ancient Greek was a pluricentric language, divided into many dialects. The main dialect groups are Attic and Ionic, Aeolic, Arcadocypriot, some dialects are found in standardized literary forms used in literature, while others are attested only in inscriptions. There are also several historical forms, homeric Greek is a literary form of Archaic Greek used in the epic poems, the Iliad and Odyssey, and in later poems by other authors. Homeric Greek had significant differences in grammar and pronunciation from Classical Attic, the origins, early form and development of the Hellenic language family are not well understood because of a lack of contemporaneous evidence. Several theories exist about what Hellenic dialect groups may have existed between the divergence of early Greek-like speech from the common Proto-Indo-European language and the Classical period and they have the same general outline, but differ in some of the detail. The invasion would not be Dorian unless the invaders had some relationship to the historical Dorians. The invasion is known to have displaced population to the later Attic-Ionic regions, the Greeks of this period believed there were three major divisions of all Greek people—Dorians, Aeolians, and Ionians, each with their own defining and distinctive dialects. Often non-west is called East Greek, Arcadocypriot apparently descended more closely from the Mycenaean Greek of the Bronze Age. Boeotian had come under a strong Northwest Greek influence, and can in some respects be considered a transitional dialect, thessalian likewise had come under Northwest Greek influence, though to a lesser degree. Most of the dialect sub-groups listed above had further subdivisions, generally equivalent to a city-state and its surrounding territory, Doric notably had several intermediate divisions as well, into Island Doric, Southern Peloponnesus Doric, and Northern Peloponnesus Doric. The Lesbian dialect was Aeolic Greek and this dialect slowly replaced most of the older dialects, although Doric dialect has survived in the Tsakonian language, which is spoken in the region of modern Sparta. Doric has also passed down its aorist terminations into most verbs of Demotic Greek, by about the 6th century AD, the Koine had slowly metamorphosized into Medieval Greek
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
12.
Polygon
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In elementary geometry, a polygon /ˈpɒlɪɡɒn/ is a plane figure that is bounded by a finite chain of straight line segments closing in a loop to form a closed polygonal chain or circuit. These segments are called its edges or sides, and the points where two edges meet are the vertices or corners. The interior of the polygon is called its body. An n-gon is a polygon with n sides, for example, a polygon is a 2-dimensional example of the more general polytope in any number of dimensions. The basic geometrical notion of a polygon has been adapted in various ways to suit particular purposes, mathematicians are often concerned only with the bounding closed polygonal chain and with simple polygons which do not self-intersect, and they often define a polygon accordingly. A polygonal boundary may be allowed to intersect itself, creating star polygons and these and other generalizations of polygons are described below. The word polygon derives from the Greek adjective πολύς much, many and it has been suggested that γόνυ knee may be the origin of “gon”. Polygons are primarily classified by the number of sides, Polygons may be characterized by their convexity or 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 line segment with endpoints on the boundary passes through only interior points between its endpoints. Non-convex, a line may be found which meets its boundary more than twice, equivalently, there exists a line segment between two boundary points that passes outside the polygon. Simple, the boundary of the polygon does not cross itself, there is at least one interior angle greater than 180°. Star-shaped, the interior is visible from at least one point. The polygon must be simple, and may be convex or concave, self-intersecting, the boundary of the polygon crosses itself. Branko Grünbaum calls these coptic, though this term does not seem to be widely used, star polygon, a polygon which self-intersects in a regular way. A polygon cannot be both a star and star-shaped, equiangular, all corner angles are equal. Cyclic, all lie on a single circle, called the circumcircle. Isogonal or vertex-transitive, all lie within the same symmetry orbit. The polygon is cyclic and equiangular
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
13.
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 and they are regular polygons, and 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, every triangle center of an equilateral triangle coincides with its centroid, which implies that the equilateral triangle is the only triangle with no Euler line connecting some of the centers. 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 the circumcenter, incenter, centroid, or orthocenter coincide. It is also equilateral if its circumcenter coincides with the Nagel point, for any triangle, the three medians partition the triangle into six smaller triangles. A triangle is equilateral if and only if any three of the triangles have either the same perimeter or the same inradius. A triangle is equilateral if and only if the circumcenters of any three of the triangles have the same distance from the centroid. Morleys 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, and PC satisfy the inequality that any two of them sum to at least as great as the third. By Eulers inequality, the triangle has the smallest ratio R/r of the circumradius to the inradius of any triangle, specifically. The triangle of largest area of all those inscribed in a circle is equilateral. The ratio of the area of the incircle to the area of an equilateral triangle, the ratio of the area to the square of the perimeter of an equilateral triangle,1123, is larger than that for any other triangle. If a segment splits an equilateral triangle into two regions with equal perimeters and with areas A1 and A2, then 79 ≤ A1 A2 ≤97, in no other triangle is there a point for which this ratio is as small as 2. For any point P in the plane, with p, q, and t from the vertices A, B. For any point P on the circle of an equilateral triangle, with distances p, q. There are numerous triangle inequalities that hold with equality if and only if the triangle is equilateral, an equilateral triangle is the most symmetrical triangle, having 3 lines of reflection and rotational symmetry of order 3 about its center. Its symmetry group is the group of order 6 D3
Equilateral triangle
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A regular tetrahedron is made of four equilateral triangles.
Equilateral triangle
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Equilateral triangle
14.
Compass and straightedge
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The idealized ruler, known as a straightedge, is assumed to be infinite in length, and has no markings on it and only one edge. The compass is assumed to collapse when lifted from the page, more formally, the only permissible constructions are those granted by Euclids first three postulates. It turns out to be the case that every point constructible using straightedge, the ancient Greek mathematicians first conceived compass-and-straightedge constructions, and a number of ancient problems in plane geometry impose this restriction. The ancient Greeks developed many constructions, but in 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, in terms of algebra, a length is constructible if and only if it represents a constructible number, and an angle is constructible if and only if its cosine is a constructible number. A number is constructible if and only if it can be using the four basic arithmetic operations. Circles can only be starting from two given points, the centre and a point on the circle. The compass may or may not collapse when its 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 segment between two points or to extend an existing segment. The modern compass generally does not collapse and several modern constructions use this feature and it would appear that the modern compass is a more powerful instrument than the ancient collapsing compass. However, by Proposition 2 of Book 1 of Euclids Elements, although the proposition is correct, its proofs have a long and checkered history. Eyeballing it and getting close does not count as a solution and that is, it must have a finite number of steps, and not be the limit of ever closer approximations. One of the purposes of Greek mathematics was to find exact constructions for various lengths, for example. The Greeks could not find constructions for these three problems, among others, Squaring the circle, Drawing a square the same area as a given circle, doubling the cube, Drawing a cube with twice the volume of a given cube. Trisecting the angle, Dividing a given angle into three smaller angles all of the same size, for 2000 years people tried to find constructions within the limits set above, and failed. All three have now been proven under mathematical rules to be generally impossible, the ancient Greek mathematicians first attempted compass-and-straightedge constructions, and they discovered how to construct sums, differences, products, ratios, and square roots of given lengths. They could also construct half of an angle, a square whose area is twice that of another square, a square having the same area as a given polygon
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
15.
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 circle is called the circumcenter and its radius is called the circumradius. A polygon which has a circle is called a cyclic polygon. All regular simple polygons, all isosceles trapezoids, all triangles, a related notion is the one of a minimum bounding circle, which is the smallest circle that completely contains the polygon within it. All triangles are cyclic, i. e. every triangle has a circumscribed circle and this can be proven on the grounds that the general equation for a circle with center and radius r in the Cartesian coordinate system is 2 +2 = r 2. 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, and 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, and the radius is the length to any of the three vertices. This is because the circumcenter is equidistant from any pair of the triangles vertices, in coastal navigation, a triangles 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, in the Euclidean plane, it is possible to give explicitly an equation of the circumcircle in terms of the Cartesian coordinates of the vertices of the inscribed triangle. Suppose that A = B = C = are the coordinates of points A, B, 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, a similar approach allows one to deduce the equation of the circumsphere of a tetrahedron. A unit vector perpendicular to the containing the circle is given by n ^ = × | × |. An equation for the circumcircle in trilinear coordinates x, y, z is a/x + b/y + c/z =0, an equation for the circumcircle in barycentric coordinates x, y, z is a2/x + b2/y + c2/z =0. The isogonal conjugate of the circumcircle is the line at infinity, given in coordinates by ax + by + cz =0. Additionally, the circumcircle of a triangle embedded in d dimensions can be using a generalized method. Let A, B, and C be d-dimensional points, which form the vertices of a triangle and we start by transposing the system to place C at the origin, a = A − C, b = B − C. The circumcenter, p0, is given by p 0 = ×2 ∥ a × b ∥2 + C, the Cartesian coordinates of the circumcenter are U x =1 D U y =1 D with D =2. Without loss of generality this can be expressed in a form after translation of the vertex A to the origin of the Cartesian coordinate systems
Circumscribed circle
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Circumscribed circle, C, and circumcenter, O, of a cyclic polygon, P
16.
Line segment
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In geometry, a line segment is a part of a line that is bounded by two distinct end points, and contains every point on the line between its endpoints. A closed line segment includes both endpoints, while a line segment excludes both endpoints, a half-open line segment includes exactly one of the endpoints. Examples of line include the sides of a triangle or square. More generally, when both of the end points are vertices of a polygon or polyhedron, the line segment is either an edge if they are adjacent vertices. When the end points both lie on a such as a circle, a line segment is called a chord. Sometimes one needs to distinguish between open and closed line segments, thus, the line segment can be expressed as a convex combination of the segments two end points. In geometry, it is defined that a point B is between two other points A and C, if the distance AB added to the distance BC is equal to the distance AC. Thus in R2 the line segment with endpoints A = and C = is the collection of points. A line segment is a connected, non-empty set, if V is a topological vector space, then a closed line segment is a closed set in V. However, an open line segment is an open set in V if and only if V is one-dimensional. More generally than above, the concept of a segment can be defined in an ordered geometry. A pair of segments can be any one of the following, intersecting, parallel, skew. The last possibility is a way that line segments differ from lines, in an axiomatic treatment of geometry, the notion of betweenness is either assumed to satisfy a certain number of axioms, or else be defined in terms of an isometry of a line. 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 sets to the analysis of a line segment. The Segment Addition Postulate can be used to add congruent segment or segments with equal lengths and consequently substitute other segments into another statement to make segments congruent. A line segment can be viewed as a case of an ellipse in which the semiminor axis goes to zero, the foci go to the endpoints. A complete orbit of this ellipse traverses the line segment twice, as a degenerate orbit this is a radial elliptic trajectory. In addition to appearing as the edges and diagonals of polygons and polyhedra, some very frequently considered segments in a triangle include the three altitudes, the three medians, the perpendicular bisectors of the sides, and the internal angle bisectors
Line segment
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historical image – create a line segment (1699)
17.
Bicentric polygon
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In geometry, a bicentric polygon is a tangential polygon which is also cyclic — that is, inscribed in an outer circle that passes through each vertex of the polygon. All triangles and 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. This is one version of Eulers triangle formula and this condition is known as Fuss theorem. A complicated general formula is known for any number n of sides for the relation among the circumradius R, the r. The radius of the circle is the apothem. For any regular polygon, the relations between the edge length a, the radius r of the incircle, and the radius R of the circumcircle are. The fact that it will always do so is implied by Poncelets closure theorem, which more generally applies for inscribed and circumscribed conics
Bicentric polygon
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An equilateral triangle
18.
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 that is tangent to each of the polygons 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 rhombi and kites. A convex polygon has an incircle if and only if all of its internal angle bisectors are concurrent and this common point is the incenter. There exists a tangential polygon of n sequential sides a1. An if and only if the system of equations x 1 + x 2 = a 1, x 2 + x 3 = a 2, …, x n + x 1 = a n has a solution in positive reals. If such a solution exists, then x1, xn are the tangent lengths of the polygon. If the number n of sides is odd, then for any set of sidelengths a 1, …. But if n is even there are an infinitude of them, for example, in the quadrilateral case where all sides are equal we can have a rhombus with any value of the acute angles, and all rhombi are tangential to an incircle. If the n sides of a polygon are a1. An, the inradius is r = K s =2 K ∑ i =1 n a i where K is the area of the polygon, for a tangential polygon with an odd number of sides, all sides are equal if and only if all angles are equal. A tangential polygon with an number of sides has all sides equal if. In a tangential polygon with an number of sides, the sum of the odd numbered sides lengths is equal to the sum of the even numbered sides lengths. A tangential polygon has an area than any other polygon with the same perimeter. In a tangential hexagon ABCDEF, the main diagonals AD, BE, and CF are concurrent according to Brianchons theorem
Tangential polygon
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A tangental trapezoid
19.
Inscribed figure
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In geometry, an inscribed planar shape or solid is one that is enclosed by and fits snugly inside another geometric shape or solid. To say that figure F is inscribed in figure G means precisely the same thing as figure G is circumscribed about figure F, a circle or ellipse inscribed in a convex polygon is tangent to every side or face of the outer figure. Familiar examples of inscribed figures include circles inscribed in triangles or regular polygons, a circle inscribed in any polygon is called its incircle, in which case the polygon is said to be a tangential polygon. A polygon inscribed in a circle is said to be a polygon. The inradius or filling radius of a given outer figure is the radius of the circle or sphere. The definition given above assumes that the objects concerned are embedded in two- or three-dimensional Euclidean space, but can easily be generalized to higher dimensions and other metric spaces. For an alternative usage of the term inscribed, see the inscribed square problem, every circle has an inscribed triangle with any three given angle measures, and every triangle can be inscribed in some circle. Every triangle has a circle, called the incircle. Every circle has a regular polygon of n sides, for any n≥3. Every regular polygon has a circle, and every circle can be inscribed in some regular polygon of n sides. Not every polygon with more than three sides has a circle, those polygons that do are called tangential polygons. Not every polygon with more than three sides is a 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 triangles centroid, every triangle has an infinitude of inscribed ellipses. One of them is a circle, and one of them is the Steiner inellipse which is tangent to the triangle at the midpoints of the sides, every acute triangle has three inscribed squares. In a right triangle two of them are merged and coincide with other, so there are only two distinct inscribed squares. An obtuse triangle has an inscribed square, with one side coinciding with part of the triangles longest side. A Reuleaux triangle, or more generally any curve of constant width, circumconic and inconic Cyclic quadrilateral Inscribed and circumscribed figures
Inscribed figure
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Inscribed circles of various polygons
20.
Angle
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In planar geometry, an angle is the figure formed by two rays, called the sides of the angle, sharing a common endpoint, called the vertex of the angle. Angles formed by two rays lie in a plane, but this plane does not have to be a Euclidean plane, Angles are also formed by the intersection of two planes in Euclidean and other spaces. Angles formed by the intersection of two curves in a plane are defined as the angle determined by the tangent rays at the point of intersection. Similar statements hold in space, for example, the angle formed by two great circles on a sphere is the dihedral angle between the planes determined by the great circles. Angle is also used to designate the measure of an angle or of a rotation and this measure is the ratio of the length of a circular arc to its radius. In the case of an angle, the arc is centered at the vertex. In the case of a rotation, the arc is centered at the center of the rotation and delimited by any other point and its image by the rotation. The word angle comes from the Latin word angulus, meaning corner, cognate words are the Greek ἀγκύλος, meaning crooked, curved, both are connected with the Proto-Indo-European root *ank-, meaning to bend or bow. Euclid defines a plane angle as the inclination to each other, in a plane, according to Proclus an angle must be either a quality or a quantity, or a relationship. In mathematical expressions, it is common to use Greek letters to serve as variables standing for the size of some angle, lower case 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, sometimes, where there is no risk of confusion, the angle may be referred to simply by its vertex. However, in geometrical situations it is obvious from context that the positive angle less than or equal to 180 degrees is meant. Otherwise, a convention may be adopted so that ∠BAC always refers to the angle from B to C. Angles smaller than an angle are called acute angles. An angle equal to 1/4 turn is called a right angle, two lines that form a right angle are said to be normal, orthogonal, or perpendicular. Angles larger than an angle and smaller than a straight angle are called obtuse angles. An angle equal to 1/2 turn is called a straight angle, Angles larger than a straight angle but less than 1 turn are called reflex angles
Angle
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An angle enclosed by rays emanating from a vertex.
21.
Rotational symmetries
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Rotational symmetry, also known as radial symmetry in biology, is the property a shape has when it looks the same after some rotation by a partial turn. An objects degree of symmetry is the number of distinct orientations in which it looks the same. Formally the rotational symmetry is symmetry with respect to some or all rotations in m-dimensional Euclidean space, rotations are direct isometries, i. e. isometries preserving orientation. 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 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 Noethers theorem, rotational symmetry of a physical system is equivalent to the angular momentum conservation 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, for each point or axis of symmetry, the abstract group type is cyclic group of order n, Zn. The fundamental domain is a sector of 360°/n, if there is e. g. rotational symmetry with respect to an angle of 100°, then also with respect to one of 20°, the greatest common divisor of 100° and 360°. A typical 3D object with rotational symmetry but no mirror symmetry is a propeller and this is the rotation group of a regular prism, or regular bipyramid. 4×3-fold and 3×2-fold axes, the rotation group T of order 12 of a regular tetrahedron, the group is isomorphic to alternating group A4. 3×4-fold, 4×3-fold, and 6×2-fold axes, the rotation group O of order 24 of a cube, the group is isomorphic to symmetric group S4. 6×5-fold, 10×3-fold, and 15×2-fold axes, the rotation group I of order 60 of a dodecahedron, the group is isomorphic to alternating group A5. The group contains 10 versions of D3 and 6 versions of D5, in the case of the Platonic solids, the 2-fold axes are through the midpoints of opposite edges, the number of them is half the number of edges. Rotational symmetry with respect to any angle is, in two dimensions, circular symmetry, the fundamental domain is a half-line. In three dimensions we can distinguish cylindrical symmetry and spherical symmetry and that is, no dependence on the angle using cylindrical coordinates and no dependence on either angle using spherical coordinates. The fundamental domain is a half-plane through the axis, and a radial half-line, axisymmetric or axisymmetrical are adjectives which refer to an object having cylindrical symmetry, or axisymmetry
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.
22.
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. Dihedral groups are among the simplest examples of groups, and they play an important role in group theory, geometry. The notation for the group of order n differs in geometry. In geometry, Dn or Dihn refers to the symmetries of the n-gon, 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, 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, the following Cayley table shows the effect of composition in the group D3. R0 denotes the identity, r1 and r2 denote counterclockwise rotations by 120° and 240° respectively, for example, s2s1 = r1, because the reflection s1 followed by the reflection s2 results in a rotation of 120°. The order of elements denoting the composition is right to left, the composition operation is not commutative. In all cases, addition and subtraction of subscripts are to be performed using modular arithmetic with modulus n, if we center the regular polygon at the origin, then elements of the dihedral group act as linear transformations of the plane. This lets us represent elements of Dn as matrices, with composition being matrix multiplication and this is an example of a group representation. For example, the elements of the group D4 can be represented by the eight matrices. In general, the matrices for elements of Dn have the following form, rk is a rotation matrix, expressing a counterclockwise rotation through an angle of 2πk/n. Sk is a reflection across a line makes an angle of πk/n with the x-axis. Further equivalent definitions of Dn are, D1 is isomorphic to Z2, D2 is isomorphic to K4, the Klein four-group. D1 and D2 are exceptional in that, D1 and D2 are the only abelian dihedral groups, Dn is a subgroup of the symmetric group Sn for n ≥3
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.
23.
Triangle
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A triangle is a polygon with three edges and three vertices. It is one of the shapes in geometry. A triangle with vertices A, B, and C is denoted △ A B C, in Euclidean geometry any three points, when non-collinear, determine a unique triangle and 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 all sides the same length. An equilateral triangle is also a 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 triangles isosceles triangles, the 45–45–90 right triangle, which appears in the tetrakis 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, a side can be marked with a pattern of ticks, short line segments in the form of tally marks, two sides have equal lengths if they are both marked with the same pattern. In a triangle, the pattern is no more than 3 ticks. Similarly, patterns of 1,2, or 3 concentric arcs inside the angles are used to indicate equal angles, triangles can also be classified according to their internal angles, measured here in degrees. A right triangle has one of its interior angles measuring 90°, the side opposite to the right angle is the hypotenuse, the longest side of the triangle. The other two sides are called the legs or catheti of the triangle, special right triangles are right triangles with additional properties that make calculations involving them easier. One of the two most famous is the 3–4–5 right triangle, where 32 +42 =52, in this situation,3,4, and 5 are a Pythagorean triple. The other one is a triangle that has 2 angles that each measure 45 degrees. Triangles that do not have an angle measuring 90° are called oblique triangles, a triangle with all interior angles measuring less than 90° is an acute triangle or acute-angled triangle. If c is the length of the longest side, then a2 + b2 > c2, a triangle with one interior angle measuring more than 90° is an obtuse triangle or obtuse-angled triangle. If c is the length of the longest side, then a2 + b2 < c2, a triangle with an interior angle of 180° is degenerate
Triangle
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The Flatiron Building in New York is shaped like a triangular prism
Triangle
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A triangle
24.
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. Though the word beehive is commonly used to describe the nest of any bee colony, scientific, nest is used to discuss colonies which house themselves in natural or artificial cavities or are hanging and exposed. Hive is used to structures used by humans to house a honey bee nest. Several species of Apis live in colonies, but only the honey bee. A bees nest is comparable to a birds nest built with a purpose to protect the dweller, the beehives internal structure is a densely packed group of hexagonal prismatic cells made of beeswax, called a honeycomb. The bees use the cells to store food and to house the brood, Beehives serve several purposes, production of honey, pollination of nearby crops, housing supply bees for apitherapy treatment, and to try to mitigate the effects of colony collapse disorder. In America, 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 caves, rock cavities and hollow trees as nesting sites. In warmer climates they may occasionally build exposed hanging nests as pictured, members of other subgenera have exposed aerial combs. The nest is composed of multiple honeycombs, parallel to each other and it usually has a single entrance. Western honey bees prefer nest cavities approximately 45 litres in volume, bees usually occupy nests for several years. The bees often smooth the bark surrounding the nest entrance, honeycombs are attached to the walls along the cavity tops and sides, but small passageways are left along the comb edges. The peanut-shaped queen cells are built at the lower edge of the comb. Bees were kept in hives in Egypt in antiquity. The walls of the Egyptian sun temple of Nyuserre Ini from the 5th Dynasty, dated earlier than 2422 BC, inscriptions detailing the production of honey are found on the tomb of Pabasa from the 26th Dynasty, and describe honey stored in jars, and cylindrical hives. The archaeologist Amihai Mazar cites 30 intact hives that were discovered in the ruins of the city of Rehov and this is evidence that an advanced honey industry existed in Israel, approximately 4,000 years ago. The beehives, made of straw and unbaked clay, were found in rows, with a total of 150 hives. Ezra Marcus from the University of Haifa said the discovery provided a glimpse of ancient beekeeping seen in texts, an altar decorated with fertility figurines was found alongside the hives and may indicate religious practices associated with beekeeping
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
25.
Honeycomb
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A honeycomb is a mass of hexagonal prismatic 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 harvest honey, honey bees consume about 8.4 lb of honey to secrete 1 lb of wax, so it makes economic sense to return the wax to the hive after harvesting the honey. The structure of the comb may be left intact when honey is extracted from it by uncapping and spinning in a centrifugal machine—the honey extractor. If the honeycomb is too worn out, the wax can be reused in a number of ways, such foundation sheets allow the bees to build the comb with less effort, and the hexagonal pattern of worker-sized cell bases discourages the bees from building the larger drone cells. Fresh, new comb is sometimes sold and used intact as comb honey, broodcomb becomes dark over time, because of the cocoons embedded in the cells and the tracking of many feet, called travel stain by beekeepers when seen on frames of comb honey. 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 axes of honeycomb cells are always quasihorizontal, and the rows of honeycomb cells are always horizontally aligned. Thus, each cell has two vertically oriented walls, with the upper and lower parts of the composed of two angled walls. The open end of a cell is referred to as the top of the cell. The cells slope slightly upwards, between 9 and 14°, 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, known in geometry 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 irregular and lumpy with no apparent attempt at efficiency. The closed ends of the cells are also an example of geometric efficiency. The ends are trihedral sections of rhombic dodecahedra, with the angles of all adjacent surfaces measuring 120°. The shape of the cells is such that two opposing honeycomb layers nest into each other, with each facet of the ends being shared by opposing cells. Individual cells do not show this geometric perfection, in a regular comb, in transition zones between the larger cells of drone comb and the smaller cells of worker comb, or when the bees encounter obstacles, the shapes are often distorted. Cells are also angled up about 13° from horizontal to prevent honey from dripping out, in 1965, László Fejes Tóth discovered the trihedral pyramidal shape used by the honeybee is not the theoretically optimal three-dimensional geometry. A cell end composed of two hexagons and two smaller rhombuses would actually be. 035% more efficient, bees used their antennae, mandibles and legs to manipulate the wax during comb construction, while actively warming the wax
Honeycomb
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Honeycomb
Honeycomb
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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
26.
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. The center of circle is called the circumcenter and its radius is called the circumradius. A polygon which has a circle is called a cyclic polygon. All regular simple polygons, all isosceles trapezoids, all triangles, a related notion is the one of a minimum bounding circle, which is the smallest circle that completely contains the polygon within it. All triangles are cyclic, i. e. every triangle has a circumscribed circle and this can be proven on the grounds that the general equation for a circle with center and radius r in the Cartesian coordinate system is 2 +2 = r 2. 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, and 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, and the radius is the length to any of the three vertices. This is because the circumcenter is equidistant from any pair of the triangles vertices, in coastal navigation, a triangles 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, in the Euclidean plane, it is possible to give explicitly an equation of the circumcircle in terms of the Cartesian coordinates of the vertices of the inscribed triangle. Suppose that A = B = C = are the coordinates of points A, B, 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, a similar approach allows one to deduce the equation of the circumsphere of a tetrahedron. A unit vector perpendicular to the containing the circle is given by n ^ = × | × |. An equation for the circumcircle in trilinear coordinates x, y, z is a/x + b/y + c/z =0, an equation for the circumcircle in barycentric coordinates x, y, z is a2/x + b2/y + c2/z =0. The isogonal conjugate of the circumcircle is the line at infinity, given in coordinates by ax + by + cz =0. Additionally, the circumcircle of a triangle embedded in d dimensions can be using a generalized method. Let A, B, and C be d-dimensional points, which form the vertices of a triangle and we start by transposing the system to place C at the origin, a = A − C, b = B − C. The circumcenter, p0, is given by p 0 = ×2 ∥ a × b ∥2 + C, the Cartesian coordinates of the circumcenter are U x =1 D U y =1 D with D =2. Without loss of generality this can be expressed in a form after translation of the vertex A to the origin of the Cartesian coordinate systems
Circumradius
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Circumscribed circle, C, and circumcenter, O, of a cyclic polygon, P
27.
Inscribed
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In geometry, an inscribed planar shape or solid is one that is enclosed by and fits snugly inside another geometric shape or solid. To say that figure F is inscribed in figure G means precisely the same thing as figure G is circumscribed about figure F, a circle or ellipse inscribed in a convex polygon is tangent to every side or face of the outer figure. Familiar examples of inscribed figures include circles inscribed in triangles or regular polygons, a circle inscribed in any polygon is called its incircle, in which case the polygon is said to be a tangential polygon. A polygon inscribed in a circle is said to be a polygon. The inradius or filling radius of a given outer figure is the radius of the circle or sphere. The definition given above assumes that the objects concerned are embedded in two- or three-dimensional Euclidean space, but can easily be generalized to higher dimensions and other metric spaces. For an alternative usage of the term inscribed, see the inscribed square problem, every circle has an inscribed triangle with any three given angle measures, and every triangle can be inscribed in some circle. Every triangle has a circle, called the incircle. Every circle has a regular polygon of n sides, for any n≥3. Every regular polygon has a circle, and every circle can be inscribed in some regular polygon of n sides. Not every polygon with more than three sides has a circle, those polygons that do are called tangential polygons. Not every polygon with more than three sides is a 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 triangles centroid, every triangle has an infinitude of inscribed ellipses. One of them is a circle, and one of them is the Steiner inellipse which is tangent to the triangle at the midpoints of the sides, every acute triangle has three inscribed squares. In a right triangle two of them are merged and coincide with other, so there are only two distinct inscribed squares. An obtuse triangle has an inscribed square, with one side coinciding with part of the triangles longest side. A Reuleaux triangle, or more generally any curve of constant width, circumconic and inconic Cyclic quadrilateral Inscribed and circumscribed figures
Inscribed
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Inscribed circles of various polygons
28.
Reflection symmetry
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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 object or figure which is indistinguishable from its image is called mirror symmetric. The set of operations that preserve a property of the object form a group. Two objects are symmetric to each other with respect to a group of operations if one is obtained from the other by some of the operations. Another way to think about the function is that if the shape were to be folded in half over the axis. 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 reflection symmetry are kites, deltoids, rhombuses, and isosceles trapezoids. All even-sided polygons have two simple reflective forms, one with lines of reflections through vertices, and 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 shapes with reflection symmetry, and between 2/3 and 1 for any convex shape, for each line or 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-plane or half-space, in certain contexts there is rotational as well as reflection symmetry. Then mirror-image symmetry is equivalent to inversion symmetry, in contexts in modern physics the term parity or P-symmetry is used for both. For more general types of reflection there are more general types of reflection symmetry. For example, with respect to a non-isometric affine involution with respect to circle inversion, most animals are bilaterally symmetric, likely because this supports forward movement and streamlining. Mirror symmetry is used in architecture, as in the facade of Santa Maria Novella. It is also found in the design of ancient structures such as Stonehenge, Symmetry was a core element in some styles of architecture, such as Palladianism. Patterns in nature Point reflection symmetry Stewart, Ian, weidenfeld & Nicolson. is potty Weyl, Hermann. Mapping with symmetry - source in Delphi Reflection Symmetry Examples from Math Is Fun
Reflection symmetry
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Many animals, such as this spider crab Maja crispata, are bilaterally symmetric.
Reflection symmetry
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Figures with the axes of symmetry drawn in. The figure with no axes is asymmetric.
Reflection symmetry
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Mirror symmetry is often used in architecture, as in the facade of Santa Maria Novella, Venice, 1470.
29.
Rhombus
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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 parallelogram and a kite. A rhombus with right angles is a square, the word rhombus comes from Greek ῥόμβος, meaning something that spins, which derives from the verb ῥέμβω, meaning to turn round and round. The word was used both by Euclid and Archimedes, who used the term solid rhombus for two right 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. This is a case of the superellipse, with exponent 1. Every rhombus has two diagonals connecting pairs of vertices, and two pairs of parallel sides. Using congruent triangles, one can prove that the rhombus is symmetric across each of these diagonals and 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, that is, a rhombus is an orthodiagonal quadrilateral, the first property implies that every rhombus is a parallelogram. Thus denoting the common side as a and the diagonals as p and q, not every parallelogram is a rhombus, though any parallelogram with perpendicular diagonals is a rhombus. In general, any quadrilateral with perpendicular diagonals, one of which is a line of symmetry, is a kite, every rhombus is a kite, and any quadrilateral that is both a kite and parallelogram is a rhombus. A rhombus is a tangential quadrilateral and that is, it has an inscribed circle that is tangent to all four sides. As for all parallelograms, the area K of a rhombus is the product of its base, the base is simply any side length a, K = a ⋅ h. The inradius, denoted by r, can be expressed in terms of the p and q as. The dual polygon of a rhombus is a rectangle, A rhombus has all sides equal, a rhombus has opposite angles equal, while a rectangle has opposite sides equal. A rhombus has a circle, while a rectangle has a circumcircle. A rhombus has an axis of symmetry through each pair of opposite vertex angles, the diagonals of a rhombus intersect at equal angles, while the diagonals of a rectangle are equal in length. The figure formed by joining the midpoints of the sides of a rhombus is a rectangle, a rhombohedron is a three-dimensional figure like a cube, except that its six faces are rhombi instead of squares. The rhombic dodecahedron is a polyhedron with 12 congruent rhombi as its faces
Rhombus
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Some polyhedra with all rhombic faces
Rhombus
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Two rhombi.
Rhombus
30.
Directed edge
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In mathematics, and more specifically in graph theory, a directed graph is a graph that is a set of vertices connected by edges, where the edges have a direction associated with them. It differs from an ordinary or undirected graph, in that the latter is defined in terms of unordered pairs of vertices, more specifically, these entities are addressed as directed multigraphs. On the other hand, the definition allows a directed graph to have loops. More specifically, directed graphs without loops are addressed as directed graphs. Symmetric directed graphs are directed graphs where all edges are bidirected, simple directed graphs are directed graphs that have no loops and no multiple arrows with same source and target nodes. As already introduced, in case of arrows the entity is usually addressed as directed multigraph. Some authors describe digraphs with loops as loop-digraphs. Complete directed graphs are directed graphs where each pair of vertices is joined by a symmetric pair of directed arrows. It follows that a complete digraph is symmetric, oriented graphs are directed graphs having no bidirected edges. It follows that a graph is an oriented graph iff it hasnt any 2-cycle. Tournaments are oriented graphs obtained by choosing a direction for each edge in undirected complete graphs. Directed acyclic graphs are directed graphs with no directed cycles, multitrees are DAGs in which no two directed paths from a single starting vertex meet back at the same ending vertex. Oriented trees or polytrees are DAGs formed by orienting the edges of undirected 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 in computer science as a representation of the paths that might be traversed through a program during its execution. Signal-flow graphs are directed graphs in which nodes represent system variables and branches represent functional connections between pairs of nodes, flow graphs are digraphs associated with a set of linear algebraic or differential equations. State diagrams are directed multigraphs that represent finite state machines, representations of a quiver label its vertices with vector spaces and its edges compatibly with linear transformations between them, and transform via natural transformations. If a path leads from x to y, then y is said to be a successor of x and reachable from x, the arrow is called the inverted arrow of. The adjacency matrix of a graph is unique up to identical permutation of rows. Another matrix representation for a graph is its incidence matrix. For a vertex, the number of head ends adjacent to a vertex is called the indegree of the vertex, the indegree of v is denoted deg− and its outdegree is denoted deg+
Directed edge
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A directed graph.
31.
Hexagonal tiling
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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, english mathematician 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 and 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 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, 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 chicken wire, tubular graphene sheets have been synthesised, these are known as carbon nanotubes. They have many applications, due to their high tensile strength. Chicken wire consists of a lattice of wires. The hexagonal tiling appears in many crystals, in three dimensions, the face-centered cubic and hexagonal close packing are common crystal structures. They are the densest known sphere packings in three dimensions, and are believed to be optimal, structurally, they comprise parallel layers of hexagonal tilings, similar to the structure of graphite. They differ in the way that the layers are staggered from each other, pure copper, amongst other materials, forms a face-centered cubic lattice. There are three distinct uniform colorings of a tiling, all generated from reflective symmetry of Wythoff constructions. The represent the periodic repeat of one colored tile, counting hexagonal distances as h first, the 3-color tiling is a tessellation generated by the order-3 permutohedrons. A chamferred hexagonal tiling replacing edges with new hexagons and transforms into another hexagonal tiling, in the limit, the original faces disappear, and the new hexagons degenerate into rhombi, and it becomes a rhombic tiling. The hexagons can be dissected into sets of 6 triangles and this is similar to the relation of the rhombic dodecahedron and the rhombo-hexagonal dodecahedron tessellations in 3 dimensions. This tiling is related to regular polyhedra with vertex figure n3. It is similarly related to the uniform truncated polyhedra with vertex figure n.6.6 and this tiling is also a part of a sequence of truncated rhombic polyhedra and tilings with Coxeter group symmetry
Hexagonal tiling
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Chicken wire fencing
Hexagonal tiling
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Graphene
Hexagonal tiling
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Periodic
Hexagonal tiling
32.
Stellation
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In geometry, stellation is the process of extending a polygon, polyhedron in three dimensions, or, in general, a polytope in n dimensions to form a new figure. The new figure is a stellation of the original, the word stellation comes from the Latin stellātus, starred, which in turn comes from Latin stella, star. In 1619 Kepler defined stellation for polygons and polyhedra, as the process of extending edges or faces until they meet to form a new polygon or polyhedron and he stellated the regular dodecahedron to obtain two regular star polyhedra, the small stellated dodecahedron and great stellated dodecahedron. He also stellated the regular octahedron to obtain the stella octangula, stellating a regular polygon 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 one end of a given edge to the other. A regular star polygon is represented by its Schläfli symbol, where n is the number of vertices, m is the used in sequencing the edges around it. Making m =1 gives the convex, if n and m do have a common divisor, then the figure is a regular compound. For example is the compound of two triangles or hexagram, while is a compound of two pentagrams. Some authors use the Schläfli symbol for such regular compounds, others regard the symbol as indicating a single path which is wound m times around n/m vertex points, such that one edge is superimposed upon another and each vertex point is visited m times. In this case a modified symbol may be used for the compound, a regular n-gon has /2 stellations if n is even, and /2 stellations if n is odd. Like the heptagon, the octagon also has two octagrammic stellations, one, being a star polygon, and the other, being the compound of two squares. A polyhedron is stellated by extending the edges or face planes of a polyhedron until they meet again to form a new polyhedron or compound, the interior of the new polyhedron is divided by the faces into a number of cells. The face planes of a polyhedron may divide space into many such cells, for a symmetrical polyhedron, these cells will fall into groups, or sets, of congruent cells - we say that the cells in such a congruent set are of the same type. A common method of finding stellations involves selecting one or more cell types and this can lead to a huge number of possible forms, so further criteria are often imposed to reduce the set to those stellations that are significant and unique in some way. A set of cells forming a layer around its core is called a shell. For a symmetrical polyhedron, a shell may be made up of one or more cell types, based on such ideas, several restrictive categories of interest have been identified. Adding successive shells to the core leads to the set of main-line stellations
Stellation
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Magnus Wenninger with some of his models of stellated polyhedra in 2009
Stellation
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Construction of a stellated dodecagon: a regular polygon with Schläfli symbol {12/5}.
Stellation
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Marble floor mosaic by Paolo Uccello, Basilica of St Mark, Venice, c. 1430
33.
Hexagram
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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 in historical, religious and cultural contexts, for example in Hanafism, Jewish identity, in mathematics, the root system for the simple Lie group G2 is in the form of a hexagram, with six long roots and six short roots. A six-pointed star, like a hexagon, can be created using a compass. Without changing the radius of the compass, set its pivot on the circles circumference, with the pivot on the last point found, similarly find a third point on the circumference, and repeat until six such points have been marked. With a straight edge, join alternate points on the circumference to form two overlapping equilateral triangles. It is possible that as a geometric shape, like for example the triangle, circle, or square. The hexagram is a symbol called satkona yantra or sadkona yantra found on ancient South Indian Hindu temples. It symbolizes the nara-narayana, or perfect meditative state of balance achieved between Man and God, and if maintained, results in moksha, or nirvana, some researchers have theorized that the hexagram represents the astrological chart at the time of Davids birth or anointment as king. The hexagram is also known as the Kings Star in astrological circles, in antique papyri, pentagrams, together with stars and other signs, are frequently found on amulets bearing the Jewish names of God, and used to guard against fever and other diseases. Curiously the hexagram is not found among these signs, in the Greek Magical Papyri at Paris and London there are 22 signs side by side, and a circle with twelve signs, but neither a pentagram nor a hexagram. Six-pointed stars have also found in cosmological diagrams in Hinduism, Buddhism. The reasons behind this symbols common appearance in Indic religions and the West are unknown, one possibility is that they have a common origin. The other possibility is that artists and religious people from several cultures independently created the hexagram shape, within Indic lore, the shape is generally understood to consist of two triangles—one pointed up and the other down—locked in harmonious embrace. The two components are called Om and the Hrim in Sanskrit, and symbolize mans position between earth and sky, the downward triangle symbolizes Shakti, the sacred embodiment of femininity, and the upward triangle symbolizes Shiva, or Agni Tattva, representing the focused aspects of masculinity. The mystical union of the two triangles represents Creation, occurring through the union of male and female. The two locked triangles are known as Shanmukha—the six-faced, representing the six faces of Shiva & Shaktis progeny Kartikeya. This symbol is also a part of several yantras and has significance in Hindu ritual worship
Hexagram
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Diagram showing the two mystic syllables Om and Hrim
Hexagram
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A regular hexagram
Hexagram
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The Star of David in the oldest surviving complete copy of the Masoretic text, the Leningrad Codex, dated 1008.
Hexagram
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Star of David on the Salt Lake Assembly Hall
34.
Square
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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. e. A rhombus with equal diagonals a convex quadrilateral with sides a, b, c, d whose area is A =12 =12. Opposite sides of a square are both parallel and 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-hypercubes and n-orthoplexes, a truncated square, t, is an octagon. An alternated square, h, is a digon, the perimeter of a square whose four sides have length ℓ is P =4 ℓ and the area A is A = ℓ2. In classical times, the power was described in terms of the area of a square. 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 22. In terms of the circumradius R, the area of a square is A =2 R2, since the area of the circle is π R2, in terms of the inradius r, the area of the square is A =4 r 2. Because it is a polygon, a square is the quadrilateral of least perimeter enclosing a given area. Dually, a square is the quadrilateral containing the largest area within a given perimeter. Indeed, if A and P are the area and perimeter enclosed by a quadrilateral, then the isoperimetric inequality holds,16 A ≤ P2 with equality if. The diagonals of a square are 2 times the length of a side of the square and this value, known as the square root of 2 or Pythagoras constant, was the first number proven to be irrational. A square can also be defined as a parallelogram with equal diagonals that bisect the angles, if a figure is both a rectangle and a rhombus, then it is a square. If a circle is circumscribed around a square, the area of the circle is π /2 times the area of the square, if a circle is inscribed in the square, the area of the circle is π /4 times the area of the square. A square has an area than any other quadrilateral with the same perimeter
Square
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A regular quadrilateral (tetragon)
35.
Star figure
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A regular polygram can either be in a set of regular polygons or in a set of regular polygon compounds. The polygram names combine a numeral prefix, such as penta-, the prefix is normally a Greek cardinal, but synonyms using other prefixes exist. The -gram suffix derives from γραμμῆς meaning a line, a regular polygram, as a general regular polygon, is denoted by its Schläfli symbol, where p and q are relatively prime and q ≥2. For integers p and q, it can be considered as being constructed by connecting every qth point out of p points regularly spaced in a circular placement. In other cases where n and m have a factor, a polygram is interpreted as a lower polygon, with k = gcd. These figures are called regular compound polygons, list of regular polytopes and compounds#Stars Cromwell, P. Polyhedra, CUP, Hbk. P.175 Grünbaum, B. and G. C, shephard, Tilings and Patterns, New York, W. H. Freeman & Co. Polyhedra with Hollow Faces, Proc of NATO-ASI Conference on Polytopes, etc. ed T. Bisztriczky et al. John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 Robert Lachlan, 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
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Regular polygrams {n/d}, with red lines showing constant d, and blue lines showing compound sequences k{n/d}
36.
Star polygon
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In geometry, a star polygon is a type of non-convex polygon. Only the regular polygons have been studied in any depth. The first usage is included in polygrams which includes polygons like the pentagram, star polygon names combine a numeral prefix, such as penta-, with the Greek suffix -gram. The prefix is normally a Greek cardinal, but synonyms using other prefixes exist, for example, a nine-pointed polygon or enneagram is also known as a nonagram, using the ordinal nona from Latin. The -gram suffix derives from γραμμή meaning a line, alternatively for integers p and q, it can be considered as being constructed by connecting every qth point out of p points regularly spaced in a circular placement. A regular star polygon is denoted by its Schläfli symbol, where p and q are relatively prime, the symmetry group of is dihedral group Dn of order 2n, independent of k. A regular star polygon can also be obtained as a sequence of stellations of a regular core polygon. Regular star polygons were first studied systematically by Thomas Bradwardine, if p and q are not coprime, a degenerate polygon will result with coinciding vertices and edges. For example will appear as a triangle, but can be labeled with two sets of vertices 1-6 and this should be seen not as two overlapping triangles, but a double-winding of a single unicursal hexagon. For |n/d|, the vertices have an exterior angle, β. These polygons are often seen in tiling patterns, the parametric angle α can be chosen to match internal angles of neighboring polygons in a tessellation pattern. The interior of a polygon may be treated in different ways. Three such treatments are illustrated for a pentagram, branko Grunbaum and Geoffrey Shephard consider two of them, as regular star polygons and concave isogonal 2n-gons. These include, Where a side occurs, one side is treated as outside and this is shown in the left hand illustration and commonly occurs in computer vector graphics rendering. The number of times that the polygonal curve winds around a given region determines its density, the exterior is given a density of 0, and any region of density >0 is treated as internal. This is shown in the illustration and commonly occurs in the mathematical treatment of polyhedra. Where a line may be drawn between two sides, the region in which the line lies is treated as inside the figure and this is shown in the right hand illustration and commonly occurs when making a physical model. When the area of the polygon is calculated, each of these approaches yields a different answer, star polygons feature prominently in art and culture
Star polygon
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{5/2}
37.
Cube
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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-magnetic and non-sparking qualities and it has excellent metalworking, forming and machining properties. It has many specialized applications in tools for hazardous environments, musical instruments, precision measurement devices, bullets, beryllium alloys present a toxic inhalation hazard during manufacture. Beryllium copper is a ductile, weldable, and machinable alloy and it is resistant to non-oxidizing acids, to plastic decomposition products, to abrasive wear, and to galling. It can be heat-treated for increased strength, durability, and electrical conductivity, beryllium copper attains the greatest strength of any copper-based alloy. In solid form and as finished objects, beryllium copper presents no known health hazard, however, inhalation of dust, mist, or 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 lungs and 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, load cells, and other parts that must retain their shape under repeated stress and strain. It has high electrical conductivity, and is used in low-current contacts for batteries, beryllium copper is non-sparking but physically tough and nonmagnetic, fulfilling the requirements of ATEX directive for Zones 0,1, and 2. Beryllium copper screwdrivers, pliers, wrenches, cold chisels, knives, and hammers are available for environments with explosive hazards, such oil rigs, coal mines, an alternative metal sometimes used for non-sparking tools is aluminium bronze. Compared to steel tools, beryllium copper tools are more expensive, not as strong, and less durable, beryllium copper is frequently used for percussion instruments for its consistent tone and resonance, especially tambourines and triangles. Beryllium copper has been used for armour piercing bullets, though usage is unusual because bullets made from steel alloys are much less expensive and have similar properties. Beryllium copper is used for measurement-while-drilling tools in the drilling industry. A non-magnetic alloy is required, as magnetometers are used for field-strength data received from the tool, beryllium copper gaskets are used to create an RF-tight, electronic seal on doors used with EMC testing and anechoic chambers. For a time, beryllium copper was used in the manufacture of clubs, particularly wedges. Though some golfers prefer the feel of BeCu club heads, regulatory issues, kiefer Plating of Elkhart, Indiana built some beryllium-copper trumpet bells for the Schilke Music Co. of Chicago. These light-weight bells produce a sound preferred by some musicians, beryllium copper wire is produced in many forms, round, square, flat and shaped, in coils, on spools and in straight lengths. Beryllium copper valve seats and guides are used in high performance engines with coated titanium valves
Cube
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Example of a non-sparking tool made of beryllium copper
Cube
38.
Giant's Causeway
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The Giants 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 bhFomhórach in Irish and it is located in County Antrim on the north coast of Northern Ireland, about three miles northeast of the town of Bushmills. It was declared a World Heritage Site by UNESCO in 1986, in a 2005 poll of Radio Times readers, the Giants Causeway was named as the fourth greatest natural wonder in the United Kingdom. The tops of the columns form stepping stones that lead from the cliff foot, most of the columns are hexagonal, although there are also some with four, five, seven or eight sides. The tallest are about 12 metres high, and the lava in the cliffs is 28 metres thick in places. Much of the Giants Causeway and Causeway Coast World Heritage Site is today owned and managed by the National Trust, the remainder of the site is owned by the Crown Estate and a number of private landowners. As the lava cooled, contraction occurred, in many cases the horizontal fracture has resulted in a bottom face that is convex while the upper face of the lower segment is concave, producing what are called ball and socket joints. The size of the columns is primarily determined by the speed at which lava from a volcanic eruption cools, the extensive fracture 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 built by a giant. The story goes that the Irish giant Fionn mac Cumhaill, from the Fenian Cycle of Gaelic mythology, was challenged to a fight by the Scottish giant Benandonner, Fionn accepted the challenge and built the causeway across the North Channel so that the two giants could meet. In one version of the story, Fionn defeats Benandonner, in another, Fionn hides from Benandonner when he realises that his foe is much bigger than he. Fionns wife, Oonagh, disguises Fionn as a baby and tucks him in a cradle, when Benandonner sees the size of the baby, he reckons that its father, Fionn, must be a giant among giants. He flees back to Scotland in fright, destroying the causeway behind him so that Fionn could not follow, across the sea, there are identical basalt columns at Fingals Cave on the Scottish isle of Staffa, and it is possible that the story was influenced by this. In overall Irish mythology, Fionn mac Cumhaill is not a giant, the Fomhóraigh are a race of supernatural beings in Irish mythology who were sometimes described as giants and who may have originally been part of a pre-Christian pantheon. In the caption to the plates French geologist Nicolas Desmarest suggested, for the first time in print, visitors can walk over the basalt columns which are at the edge of the sea, a half-mile walk from the entrance to the site. The Causeway was without a permanent visitors centre between 2000 and 2012, as the building, built in 1986, burned down in 2000. A privately financed proposal was given approval in 2007 by the Environment Minister
Giant's Causeway
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The Giant's Causeway
Giant's Causeway
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Engraving of Susanna Drury's A View of the Giant's Causeway: East Prospect, 1768
Giant's Causeway
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Red basaltic prisms
Giant's Causeway
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Giant's Causeway at sunset
39.
Wax
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Waxes are a diverse class of organic compounds that are hydrophobic, malleable solids near ambient temperatures. They include higher alkanes and lipids, typically with melting points above about 40 °C, waxes are insoluble in water but soluble in organic, nonpolar solvents. Natural waxes of different types are produced by plants and animals, waxes are organic compounds that characteristically consist of long alkyl chains. They may also include various groups such as fatty acids, primary and secondary long chain alcohols, unsaturated bonds, aromatics, amides, ketones. They frequently contain fatty acid esters as well, synthetic waxes are often long-chain hydrocarbons that lack functional groups. Waxes are synthesized by plants and animals. Those of animal origin typically consist of wax esters derived from a variety of carboxylic acids, in waxes of plant origin characteristic mixtures of unesterified hydrocarbons may predominate over esters. The composition depends not only on species, but also on location of the organism. The most commonly known animal wax is beeswax, but other insects secrete waxes, a major component of the beeswax used in constructing honeycombs is the ester myricyl palmitate which is an ester of triacontanol and palmitic acid. Its melting point is 62-65 °C, spermaceti occurs in large amounts in the head oil of the sperm whale. One of its constituents is cetyl palmitate, another ester of a fatty acid. Lanolin is a wax obtained from wool, consisting of esters of sterols, plants secrete waxes into and on the surface of their cuticles as a way to control evaporation, wettability and hydration. From the commercial perspective, the most important plant wax is carnauba wax, containing the ester myricyl cerotate, it has many applications, such as confectionery and other food coatings, car and furniture polish, floss coating, and surfboard wax. Other more specialized vegetable waxes include candelilla wax and ouricury wax, plant and animal based waxes or oils can undergo selective chemical modifications to produce waxes with more desirable properties than are available in the unmodified starting material. This approach has relied on green chemistry approaches including olefin metathesis and enzymatic reactions, although many natural waxes contain esters, paraffin waxes are hydrocarbons, mixtures of alkanes usually in a homologous series of chain lengths. These materials represent a significant fraction of petroleum and they are refined by vacuum distillation. Paraffin waxes are mixtures of saturated n- and iso- alkanes, naphthenes, a typical alkane paraffin wax chemical composition comprises hydrocarbons with the general formula CnH2n+2, such as Hentriacontane, C31H64. The degree of branching has an important influence on the properties, millions of tons of paraffin waxes are produced annually
Wax
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Commercial honeycomb foundation, made by pressing beeswax between patterned metal rollers.
Wax
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Ceroline brand wax for floors and furniture, first half of 20th century. From the Museo del Objeto del Objeto collection
Wax
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Wax candle.
Wax
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A typical modern wax sculpture of Cecilia Cheung at Madame Tussauds Hong Kong.
40.
Hexagonal prism
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In geometry, the hexagonal prism is a prism with hexagonal base. This polyhedron has 8 faces,18 edges, and 12 vertices, since it has eight faces, it is an octahedron. However, the octahedron is primarily used to refer to the regular octahedron. Because of the ambiguity of the octahedron and the dissimilarity of the various eight-sided figures. Before sharpening, many take the shape of a long 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, and represented by the product ×. The dual of a prism is a hexagonal bipyramid. The symmetry group of a hexagonal prism is D6h of order 24. The rotation 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, 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. Hexagonal Prism Interactive Model -- works in your web browser
Hexagonal prism
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Uniform Hexagonal prism
41.
Pascal's theorem
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The theorem is also valid in the Euclidean plane, but the statement needs to be adjusted to deal with the special cases when opposite sides are parallel. The most natural setting for Pascals theorem is in a plane since all lines meet. However, with the interpretation of what happens when some opposite sides of the hexagon are parallel. If two pairs of sides are parallel, then all three pairs of opposite sides form pairs of parallel lines and there is no Pascal line in the Euclidean plane. This theorem is a generalization of Pappuss theorem – Pappuss theorem is the case of a degenerate conic of two lines. Pascals theorem is the reciprocal and projective dual of Brianchons theorem. It was formulated by Blaise Pascal in a written in 1639 when he was 16 years old. Par B. P. Pascals theorem is a 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. The Braikenridge–Maclaurin theorem may be applied in the Braikenridge–Maclaurin construction, which is a construction of the conic defined by five points. Then if 2n of those points lie on a common line, if six unordered points are given on a conic section, they can be connected into a hexagon in 60 different ways, resulting in 60 different instances of Pascals theorem and 60 different Pascal lines. This configuration of 60 lines is called the Hexagrammum Mysticum, as Thomas Kirkman proved in 1849, these 60 lines can be associated with 60 points in such a way that each point is on three lines and each line contains three points. 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 a Steiner point, 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, Pascals original note has no proof, but there are various modern proofs of the theorem. It is sufficient to prove the theorem when the conic is a circle and this was realised by Pascal, whose first lemma states the theorem for a circle. His second lemma states that what is true in one plane remains true upon projection to another plane, a short elementary proof of Pascals theorem in the case of a circle was found by van Yzeren, based on the proof in. This proof proves the theorem for circle and then generalizes it to conics, a short elementary computational proof in the case of the real projective plane was found by Stefanovic We can infer the proof from existence of isogonal conjugate too
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.
42.
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 hyperbola, the parabola, and the ellipse, the circle is a special case of the ellipse, and is of sufficient interest in its own right that it was sometimes called a fourth type of conic section. The conic sections have been studied by the ancient Greek mathematicians with this work culminating around 200 BC, the conic sections of the Euclidean plane have various distinguishing properties. Many of these have used as the basis for a definition of the conic sections. The type of conic is determined by the value of the eccentricity, in analytic geometry, a conic may be defined as a plane algebraic curve of degree 2, that is, as the set of points whose coordinates satisfy a quadratic equation in two variables. This equation may be written in form, and some geometric properties can be studied as algebraic conditions. In the Euclidean plane, the conic sections appear to be different from one another. By extending the geometry to a projective plane this apparent difference vanishes, further extension, by expanding the real coordinates to admit complex coordinates, provides the means to see this unification algebraically. The conic sections have been studied for thousands of years and have provided a source of interesting. A conic is the curve obtained as the intersection of a plane, called the cutting plane and we shall assume that the cone is a right circular cone for the purpose of easy description, but this is not required, any double cone with some circular cross-section will suffice. Planes that pass through the vertex of the cone will intersect the cone in a point and these are called degenerate conics and some authors do not consider them to be conics at all. Unless otherwise stated, we assume that conic refers to a non-degenerate conic. There are three types of conics, the ellipse, parabola, and hyperbola, the circle is a special kind of ellipse, although historically it had been considered as a fourth type. The circle and the ellipse arise when the intersection of the cone and plane is a closed curve, if the cutting plane is parallel to exactly one generating line of the cone, then the conic is unbounded and is called a parabola. In the remaining case, the figure is a hyperbola, in this case, the plane will intersect both halves of the cone, producing two separate unbounded curves. A property that the conic sections share is often presented as the following definition, a conic section is the locus of all points P whose distance to a fixed point F is a constant multiple of the distance from P to a fixed line L. For 0 < e <1 we obtain an ellipse, for e =1 a parabola, a circle is a limiting case and is not defined by a focus and directrix, in the plane. The eccentricity of a circle is defined to be zero and its focus is the center of the circle, an ellipse and a hyperbola each have two foci and distinct directrices for each of them
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
43.
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 defined in two ways, first as a simple hexagon with vertices at the intersections as defined before. The second is a hexagon with the lines going through the symmedian point as three of the edges and the other three edges join pairs of adjacent vertices. Nevertheless, the Lemoine hexagon is a polygon, meaning that its vertices all lie on a common circle. The circumcircle of the Lemoine hexagon is known as the first Lemoine circle, sur quelques propriétés dun point remarquable dun triangle, Association francaise pour l’avancement des sciences, Congrès, pp. 90–95. Mackay, J. S. Symmedians of a triangle and their concomitant circles, Proceedings of the Edinburgh Mathematical Society,14, 37–103, doi,10. 1017/S0013091500031758
Lemoine hexagon
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The Lemoine hexagon, shown with self-intersecting connectivity, circumscribed by the first Lemoine circle
44.
Symmedian point
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In geometry, symmedians are three particular geometrical lines associated with every triangle. They are constructed by taking a median of the triangle, the angle formed by the symmedian and the angle bisector has the same measure as the angle between the median and the angle bisector, but it is on the other side of the angle bisector. The three symmedians meet at a center called the Lemoine point. Ross Honsberger called its one of the crown jewels of modern geometry. For instance, if three cevians of a triangle intersect at a point P, then their isogonal lines also intersect at a point, 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 and this point is called the triangles symmedian point, or alternatively the Lemoine point or Grebe point. The green lines are the angle bisectors, the symmedians and medians are symmetric about the angle bisectors The concept of a symmedian point extends to tetrahedra. Given a tetrahedron ABCD two planes P and Q through AB are isogonal conjugates if they form equal angles with the planes ABC, let M be the midpoint of the side CD. The plane containing the side AB that is isogonal to the plane ABM is called a plane of the tetrahedron. The symmedian planes can be shown to intersect at a point and this is also the point that minimizes the squared distance from the faces of the tetrahedron
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.
45.
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 running from each vertex of the triangle and 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 of the midpoints of each side of a triangle at 90 degree angles. The three perpendicular bisectors meet at the circumcenter, other sets of lines associated with a triangle are concurrent as well. For example, Any median is concurrent with two other area bisectors each of which is parallel to a side, a cleaver of a triangle is a line segment that bisects the perimeter of the triangle and has one endpoint at the midpoint of one of the three sides. The three cleavers concur at the center of the Spieker circle, which is the incircle of the medial triangle, a splitter of a triangle is a line 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, Any line through a triangle that splits both the triangles area and its perimeter in half goes through the triangles incenter, and each triangle has one, two, or three of these lines. Thus if there are three of them, they concur at the incenter, the Tarry point of a triangle is the point of concurrency of the lines through the vertices of the triangle perpendicular to the corresponding sides of the triangles first Brocard triangle. The Napoleon points and generalizations of them are points of concurrency, a generalization of this notion is the Jacobi point. The de Longchamps point is the point of concurrence of several lines with the Euler line, in the case in which the original triangle has no angle greater than 120°, this point is also the Fermat point. The two bimedians of a quadrilateral and the segment joining the midpoints of the diagonals are concurrent and are all bisected by their point of intersection. In a tangential quadrilateral, the four angle bisectors concur at the center of the incircle, other concurrencies of a tangential quadrilateral are given here. In a cyclic quadrilateral, four segments, each perpendicular to one side. These line segments are called the maltitudes, which is an abbreviation for midpoint altitude and their common point is called the anticenter. If the successive sides of a hexagon are a, b, c, d, e, f. If a hexagon has a conic, then by Brianchons theorem its principal diagonals are concurrent. Concurrent lines arise in the dual of Pappuss hexagon theorem, for each side of a cyclic hexagon, extend the adjacent sides to their intersection, forming a triangle exterior to the given side
Concurrent lines
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46.
Brianchon's theorem
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In geometry, Brianchons 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 reciprocal and projective dual of this theorem give Pascals theorem. Brianchons theorem is true in both the plane and the real projective plane. However, its statement in the plane is in a sense less informative. Consider, for example, five tangent lines to a parabola and these may be considered sides of a hexagon whose sixth side is the line at infinity, but there is no line at infinity in the affine plane. In two instances, a line from a vertex to the vertex would be a line parallel to one of the five tangent lines. Brianchons theorem stated only for the plane would therefore have to be stated differently in such a situation. The projective dual of Brianchons theorem has exceptions in the affine plane, Brianchons theorem can be proved by the idea of radical axis or reciprocation
Brianchon's theorem
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Contents
47.
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,18 edges,15 faces, in 6 triangular prism cells and it has Coxeter diagram, and symmetry, order 72. There are three constructions for the honeycomb with two lower symmetries, the regular complex polytope 32, in C2 has a real representation as a 3-3 duoprism in 4-dimensional space. 32 has 9 vertices, and 6 3-edges and its symmetry is 32, order 18. It also has a lower construction, or 3×3, with symmetry 33. This is the if the red and blue 3-edges are considered distinct. The dual of a 3-3 duoprism is called a 3-3 duopyramid and it has 9 tetragonal disphenoid cells,18 triangular faces,15 edges, and 6 vertices. It can be seen in orthogonal projection as a 6-gon circle of vertices, orthogonal projection The regular complex polygon 23 has 6 vertices in C2 with a real represention in R4 matching the same vertex arrangement of the 3-3 duopyramid. It has 9 2-edges corresponding to the edges of the 3-3 duopyramid. It can be seen in a projection with 3 sets of colored edges. This arrangement of vertices and edges makes a complete graph with each vertex from one triangle is connected to every vertex on the other. It is also called a Thomsen graph or 4-cage, 3-4 duoprism Tesseract 5-5 duoprism Convex regular 4-polytope Duocylinder Regular Polytopes, H. S. M. Coxeter, Dover Publications, Inc. Coxeter, The Beauty of Geometry, Twelve Essays, Dover Publications,1999, ISBN 0-486-40919-8 Coxeter, H. S. M. Regular Skew Polyhedra in Three and Four Dimensions. John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 Norman Johnson Uniform Polytopes, Johnson, The Theory of Uniform Polytopes and Honeycombs, Ph. D. Dissertation, University of Toronto,1966 Olshevsky, George, archived from the original on 4 February 2007. Catalogue of Convex Polychora, section 6, George Olshevsky
3-3 duoprism
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3-3 duoprisms
48.
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 9 vertices,18 edges,15 faces, in 6 triangular prism cells and it has Coxeter diagram, and symmetry, order 72. There are three constructions for the honeycomb with two lower symmetries, the regular complex polytope 32, in C2 has a real representation as a 3-3 duoprism in 4-dimensional space. 32 has 9 vertices, and 6 3-edges and its symmetry is 32, order 18. It also has a lower construction, or 3×3, with symmetry 33. This is the if the red and blue 3-edges are considered distinct. The dual of a 3-3 duoprism is called a 3-3 duopyramid and it has 9 tetragonal disphenoid cells,18 triangular faces,15 edges, and 6 vertices. It can be seen in orthogonal projection as a 6-gon circle of vertices, orthogonal projection The regular complex polygon 23 has 6 vertices in C2 with a real represention in R4 matching the same vertex arrangement of the 3-3 duopyramid. It has 9 2-edges corresponding to the edges of the 3-3 duopyramid. It can be seen in a projection with 3 sets of colored edges. This arrangement of vertices and edges makes a complete graph with each vertex from one triangle is connected to every vertex on the other. It is also called a Thomsen graph or 4-cage, 3-4 duoprism Tesseract 5-5 duoprism Convex regular 4-polytope Duocylinder Regular Polytopes, H. S. M. Coxeter, Dover Publications, Inc. Coxeter, The Beauty of Geometry, Twelve Essays, Dover Publications,1999, ISBN 0-486-40919-8 Coxeter, H. S. M. Regular Skew Polyhedra in Three and Four Dimensions. John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 Norman Johnson Uniform Polytopes, Johnson, The Theory of Uniform Polytopes and Honeycombs, Ph. D. Dissertation, University of Toronto,1966 Olshevsky, George, archived from the original on 4 February 2007. Catalogue of Convex Polychora, section 6, George Olshevsky
3-3 duopyramid
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3-3 duoprisms
49.
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,15 tetrahedral cells and it has a dihedral angle of cos−1, or approximately 78. 46°. It can also be called a hexateron, or hexa-5-tope, as a 6-facetted polytope in 5-dimensions, the name hexateron is derived from hexa- for having six facets and teron for having four-dimensional facets. By Jonathan Bowers, a hexateron is given the acronym 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 6-orthoplex or rectified 6-cube respectively and it is first in a dimensional series of uniform polytopes and honeycombs, expressed by Coxeter as 13k series. A degenerate 4-dimensional case exists as 3-sphere tiling, a tetrahedral dihedron and it is first in a dimensional series of uniform polytopes and honeycombs, expressed by Coxeter as 3k1 series. A degenerate 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 the Coxeter group, the 5-simplex can also be considered a 5-cell pyramid, constructed as a 5-cell base in a 4-space hyperplane, and an apex point above the hyperplane. The five sides of the pyramid are made of 5-cell cells, 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, 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. 5D uniform polytopes x3o3o3o3o - hix, archived from the original on 4 February 2007. Polytopes of Various Dimensions, Jonathan Bowers Multi-dimensional Glossary
5-simplex
50.
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 meeting in identical vertices, excluding the 5 Platonic solids. 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 families, there are 13 Archimedean solids. All the Archimedan solids can be made via Wythoff constructions from the Platonic solids with tetrahedral, octahedral and 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, his published enumeration only includes the 13 uniform polyhedra, here the vertex configuration refers to the type of regular polygons that meet at any given vertex. For example, a configuration of means that a square, hexagon. Some definitions of semiregular polyhedron include one more figure, the square gyrobicupola or pseudo-rhombicuboctahedron. The number of vertices is 720° divided by the angle defect. The cuboctahedron and icosidodecahedron are edge-uniform and are called quasi-regular, the duals of the Archimedean solids are called the Catalan solids. Together with the bipyramids and trapezohedra, these are the face-uniform solids with regular vertices, the snub cube and snub dodecahedron are known as chiral, as they come in a left-handed form and right-handed form. When something comes in forms which are each others three-dimensional mirror image. The different Archimedean and Platonic solids can be related to each other using a handful of general constructions, starting with a Platonic solid, truncation involves cutting away of corners. To preserve symmetry, the cut is in a perpendicular to the line joining a corner to the center of the polyhedron and is the same for all corners. Depending on how much is truncated, different Platonic and Archimedean solids can be created, expansion or cantellation involves moving each face away from the center and taking the convex hull. Expansion with twisting also involves rotating the faces, thus breaking the rectangles corresponding to edges into triangles, the last construction we use here is truncation of both corners and edges. Ignoring scaling, expansion can also be viewed as truncation of corners and edges, note the duality between the cube and the octahedron, and between the dodecahedron and the icosahedron
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.
51.
Truncated tetrahedron
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In geometry, the truncated tetrahedron is an Archimedean solid. It has 4 regular hexagonal faces,4 equilateral triangle faces,12 vertices and 18 edges and 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 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 cube, with Coxeter diagram. There are two positions of this construction, and combining them creates the uniform compound of two truncated tetrahedra. The area A and the volume V of a tetrahedron of edge length a are. The densest packing of the Archimedean truncated tetrahedron is believed to be Φ = 207/208, in fact, if the truncation of the corners is slightly smaller than that of an Archimedean truncated tetrahedron, this new shape can be used to completely fill space. 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 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, Friauf and his 1927 paper The crystal structure of the intermetallic compound MgCu2. Giant truncated tetrahedra were used for the Man the Explorer and Man the Producer theme pavilions in Expo 67 and they were made of massive girders of steel bolted together in a geometric lattice. The truncated tetrahedra were interconnected with lattice steel platforms, all of these buildings were demolished after the end of Expo 67, as they had not been built to withstand the severity of the Montreal weather over the years. Their only remnants are in the Montreal city archives, the Public Archives Of Canada, the Tetraminx puzzle has a truncated tetrahedral shape. This puzzle shows a dissection of a tetrahedron into 4 octahedra and 6 tetrahedra. It contains 4 central planes of rotations, in the mathematical field of graph theory, a truncated tetrahedral graph is a Archimedean graph, the graph of vertices and edges of the truncated tetrahedron, one of the Archimedean solids. It has 12 vertices and 18 edges and it is a connected cubic graph, and connected cubic transitive graph. It is also a part of a sequence of cantic polyhedra, in this wythoff construction the edges between the hexagons represent degenerate digons
Truncated tetrahedron
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(Click here for rotating model)
Truncated tetrahedron
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Triakis tetrahedron (dual polyhedron)
52.
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,60 vertices and 90 edges and it is the Goldberg polyhedron GPV or 1,1, containing pentagonal and hexagonal faces. This geometry is associated with footballs typically patterned with white hexagons, 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 C60 molecule and it is used in the cell-transitive hyperbolic space-filling tessellation, the bitruncated order-5 dodecahedral honeycomb. This polyhedron can be constructed from an icosahedron with the 12 vertices truncated such that one third of each edge is cut off at each of both ends and this creates 12 new pentagon faces, and leaves the original 20 triangle faces as regular hexagons. Thus the length of the edges is one third of that of the original edges, cartesian coordinates for the vertices of a truncated icosahedron centered at the origin are all even permutations of, where φ =1 + √5/2 is the golden mean. Using φ2 = φ +1 one verifies that all vertices are on a sphere, centered at the origin, with the radius equal to √9φ +10. Permutations, X axis Y axis Z axis The truncated icosahedron has five special orthogonal projections, centered, on a vertex, the last two correspond to the A2 and H2 Coxeter planes. The truncated icosahedron can also be represented as a spherical tiling and this projection is conformal, preserving angles but not areas or lengths. Straight lines on the sphere are projected as circular arcs on the plane and this result is easy to get by using one of the three orthogonal golden rectangles drawn into the original icosahedron as the starting point for our considerations. The angle between the segments joining the center and the vertices connected by shared edge is approximately 23. 281446°. The area A and the volume V of the truncated icosahedron of edge length a are, with unit edges, the surface area is 21 for the pentagons and 52 for the hexagons, together 73. The truncated icosahedron easily demonstrates the Euler characteristic,32 +60 −90 =2, the balls used in association football and team handball are perhaps the best-known example of a spherical polyhedron analog to the truncated icosahedron, found in everyday life. The ball comprises the same pattern of regular pentagons and regular hexagons, but it is more due to the pressure of the air inside. This ball type was introduced to the World Cup in 1970, geodesic domes are typically based on triangular facetings of this geometry with example structures found across the world, popularized by Buckminster Fuller. A variation of the icosahedron was used as the basis of the wheels used by the Pontiac Motor Division between 1971 and 1976 on its Trans Am and Grand Prix. This shape was also the configuration of the used for focusing the explosive shock waves of the detonators in both the gadget and Fat Man atomic bombs. The truncated icosahedron can also be described as a model of the Buckminsterfullerene, or buckyball, molecule, an allotrope of elemental carbon, discovered in 1985
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
53.
Truncated cuboctahedron
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In geometry, the truncated cuboctahedron is an Archimedean solid, named by Kepler as a truncation of a cuboctahedron. It has 12 square faces,8 regular hexagonal faces,6 regular octagonal faces,48 vertices and 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, however, the resulting figure is topologically equivalent to a truncated cuboctahedron and can always be deformed until the faces are regular. The alternative name great rhombicuboctahedron refers to the fact that the 12 square faces lie in the planes as the 12 faces of the rhombic dodecahedron which is dual to the cuboctahedron. One unfortunate point of confusion, There is a uniform polyhedron by the same name. See nonconvex great rhombicuboctahedron.7551724 a 2 V = a 3 ≈41.7989899 a 3, 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, There is only one uniform coloring of the faces of this polyhedron, one color for each face type. A 2-uniform coloring, with symmetry, exists with alternately colored hexagons. The truncated cuboctahedron can also be represented as a spherical tiling and this projection is conformal, preserving angles but not areas or lengths. Straight lines on the sphere are projected as circular arcs on the plane, the truncated cuboctahedron is one of a family of uniform polyhedra related to the cube and regular octahedron. This polyhedron can be considered a member of a sequence of patterns with vertex configuration. For p <6, the members of the sequence are omnitruncated polyhedra, for p <6, they are tilings of the hyperbolic plane, starting with the truncated triheptagonal tiling. In the mathematical field of theory, a truncated cuboctahedral graph is the graph of vertices and edges of the truncated cuboctahedron. It has 48 vertices and 72 edges, and is a zero-symmetric and cubic Archimedean graph, cube Cuboctahedron Octahedron Truncated icosidodecahedron Truncated octahedron – truncated tetratetrahedron Cromwell, P. Polyhedra. Eric W. Weisstein, Great rhombicuboctahedron at MathWorld, 3D convex uniform polyhedra x3x4x - girco
Truncated cuboctahedron
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(Click here for rotating model)
Truncated cuboctahedron
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Disdyakis dodecahedron (dual polyhedron)
54.
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. It has 30 square faces,20 regular hexagonal faces,12 regular decagonal faces,120 vertices and 180 edges – more than any other convex nonprismatic uniform polyhedron, 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, however, the resulting figure is topologically equivalent to this and can always be deformed until the faces are regular. One unfortunate point of confusion is there is a nonconvex uniform polyhedron of the same name. The surface area A and the volume V of the truncated icosidodecahedron of edge length a are, V = a 3 ≈206.803399 a 3. If a set of all 13 Archimedean solids were constructed with all edge lengths equal, the truncated icosidodecahedron would be the largest. Cartesian coordinates for the vertices of a truncated icosidodecahedron with edge length 2φ −2, centered at the origin, are all the permutations of, and. The truncated icosidodecahedron has seven special orthogonal projections, centered on a vertex, the last two correspond to the A2 and H2 Coxeter planes. The truncated icosidodecahedron can also be represented as a spherical tiling and 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. Within Icosahedral symmetry there are unlimited geometric variations of the truncated icosidodecahedron with isogonal faces, the truncated dodecahedron, rhombicosidodecahedron, and truncated icosahedron as degenerate limiting cases. In the mathematical field of theory, a truncated icosidodecahedral graph is the graph of vertices and edges of the truncated icosidodecahedron. It has 120 vertices and 180 edges, and is a zero-symmetric and cubic Archimedean graph and this polyhedron can be considered a member of a sequence of uniform patterns with vertex figure and Coxeter-Dynkin diagram. For p <6, the members of the sequence are omnitruncated polyhedra, for p >6, they are tilings of the hyperbolic plane, starting with the truncated triheptagonal tiling. Wenninger, Magnus, Polyhedron Models, Cambridge University Press, ISBN 978-0-521-09859-5, MR0467493 Cromwell, the Geometrical Foundation of Natural Structure, A Source Book of Design. Cromwell, P. Polyhedra, CUP hbk, pbk, eric W. Weisstein, GreatRhombicosidodecahedron at MathWorld. 3D convex uniform polyhedra x3x5x - grid, editable printable net of a truncated icosidodecahedron with interactive 3D view The Uniform Polyhedra Virtual Reality Polyhedra The Encyclopedia of Polyhedra
Truncated icosidodecahedron
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(Click here for rotating model)
Truncated icosidodecahedron
Truncated icosidodecahedron
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Disdyakis triacontahedron (dual polyhedron)
55.
Tetrahedral symmetry
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A regular tetrahedron has 12 rotational symmetries, and a symmetry order of 24 including transformations that combine a reflection and a rotation. The set of orientation-preserving symmetries forms a group referred to as the alternating subgroup A4 of S4, chiral and full are discrete point symmetries. They are among the 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 meet at order 2 and 3 gyration points. T,332, +, or 23, of order 12 – chiral or rotational tetrahedral symmetry, there are three orthogonal 2-fold rotation axes, like chiral dihedral symmetry D2 or 222, with in addition four 3-fold axes, centered between the three orthogonal directions. This group is isomorphic to A4, the group on 4 elements, in fact it is the group of even permutations of the four 3-fold axes. The three elements of the latter are the identity, clockwise rotation, and anti-clockwise rotation, corresponding to permutations of the three orthogonal 2-fold axes, preserving orientation. Td, *332, or 43m, of order 24 – achiral or full tetrahedral symmetry and 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, see also the isometries of the regular tetrahedron. This group has the same axes as T, with mirror planes through two of the orthogonal directions. The 3-fold axes are now S6 axes, and there is an inversion symmetry. Th is isomorphic to T × Z2, every element of Th is either an element of T, apart from these two normal subgroups, there is also a normal subgroup D2h, of type Dih2 × Z2 = Z2 × Z2 × Z2. It is the product of the normal subgroup of T with Ci. The quotient group is the same as above, of type Z3, the three elements of the latter are the identity, clockwise rotation, and anti-clockwise rotation, corresponding to permutations of the three orthogonal 2-fold axes, preserving orientation. It is the symmetry of a cube with on each face a line segment dividing the face into two rectangles, such that the line segments of adjacent faces do not meet at the edge. The symmetries correspond to the permutations of the body diagonals
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.
56.
Octahedral symmetry
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A regular octahedron has 24 rotational symmetries, and a symmetry order of 48 including transformations that combine a reflection and a rotation. A cube has the set of symmetries, since it is the dual of an octahedron. Chiral and full octahedral symmetry are the point symmetries with the largest symmetry groups compatible with translational symmetry. They are among the point groups of the cubic crystal system. But as it is also the direct product S4 × S2, one can identify the elements of S4 as a ∈ [0,4. ). So e. g. the identity is represented as 0, the pairs can be seen in the six files below. Each file is denoted by the m ∈, and the position of each permutation in the file corresponds to the n ∈. A rotoreflection is a combination of rotation and reflection,7 ′ ∘4 =19 ′,7 ′ ∘22 =17 ′, The reflection 7 ′ applied on the 90° rotation 22 gives the 90° rotoreflection 17 ′. O,432, or + of order 24, is chiral octahedral symmetry or rotational octahedral symmetry. This group is like chiral tetrahedral symmetry T, but the C2 axes are now C4 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, O is the rotation group of the cube and the regular octahedron. Oh, *432, or m3m of order 48 - achiral octahedral symmetry or full octahedral symmetry and 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 isomorphic to S4. C4, and is the symmetry group of the cube. It is the group for n =3. See also the isometries of the cube, with the 4-fold axes as coordinate axes, a fundamental domain of Oh is given by 0 ≤ x ≤ y ≤ z. An object with symmetry is characterized by the part of the object in the fundamental domain, for example the cube is given by z =1. Ax + by + cz =1 gives a polyhedron with 48 faces, faces are 8-by-8 combined to larger faces for a = b =0 and 6-by-6 for a = b = c. The 9 mirror lines of full octahedral symmetry can be divided into two subgroups of 3 and 6, representing in two orthogonal subsymmetries, D2h, and Td, D2h symmetry can be doubled to D4h by restoring 2 mirrors from one of three orientations
Octahedral symmetry
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Each face of the disdyakis dodecahedron is a fundamental domain
Octahedral symmetry
57.
Icosahedral symmetry
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A regular icosahedron has 60 rotational symmetries, and a symmetry order of 120 including transformations that combine a reflection and a rotation. A regular dodecahedron has the set of symmetries, since it is the dual of the icosahedron. The set of orientation-preserving symmetries forms a group referred to as A5, the latter group is also known as the Coxeter group H3, and is also represented by Coxeter notation, and Coxeter diagram. Icosahedral symmetry is not compatible with translational symmetry, so there are no associated crystallographic point groups or space groups. Presentations corresponding to the above are, I, ⟨ s, t ∣ s 2, t 3,5 ⟩ I h, ⟨ s, t ∣ s 3 −2, t 5 −2 ⟩ and these correspond to the icosahedral groups being the triangle groups. The first presentation was given by William Rowan Hamilton in 1856, note that other presentations are possible, for instance as an alternating group. The icosahedral rotation group I is of order 60, the group I is isomorphic to A5, the alternating group of even permutations of five objects. This isomorphism can be realized by I acting on various compounds, notably the compound of five cubes, the group contains 5 versions of Th with 20 versions of D3, and 6 versions of D5. The full icosahedral group Ih has order 120 and it has I as normal subgroup of index 2. The group Ih is isomorphic to I × Z2, or A5 × Z2, with the inversion in the corresponding to element. Ih acts on the compound of five cubes and the compound of five octahedra and it acts on the compound of ten tetrahedra, I acts on the two chiral halves, and −1 interchanges the two halves. Notably, it does not act as S5, and these groups are not isomorphic, the group contains 10 versions of D3d and 6 versions of D5d. I is also isomorphic to PSL2, but Ih is not isomorphic to SL2, all of these classes of subgroups are conjugate, and admit geometric interpretations. Note that the stabilizer of a vertex/edge/face/polyhedron and its opposite are equal, stabilizers of an opposite pair of vertices can be interpreted as stabilizers of the axis they generate. Stabilizers of a pair of edges in Ih give Z2 × Z2 × Z2, there are 5 of these, stabilizers of an opposite pair of faces can be interpreted as stabilizers of the anti-prism they generate. g. Flattening selected subsets of faces to combine each subset into one face, or replacing each face by multiple faces, in aluminum, the icosahedral structure was discovered experimentally three years after this by Dan Shechtman, which earned him the Nobel Prize in 2011. Icosahedral symmetry is equivalently the projective linear group PSL, and is the symmetry group of the modular curve X. The modular curve X is geometrically a dodecahedron with a cusp at the center of each polygonal face, similar geometries occur for PSL and more general groups for other modular curves
Icosahedral symmetry
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Faces of disdyakis triacontahedron are the fundamental domain
Icosahedral symmetry
58.
Chamfered tetrahedron
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In geometry, chamfering or edge-truncation is a topological operator that modifies one polyhedron into another. It is similar to expansion, moving faces apart and outward, for polyhedra, this operation adds a new hexagonal face in place of each original edge. In Conway polyhedron notation it is represented by the letter c, a polyhedron with e edges will have a chamfered form containing 2e new vertices, 3e new edges, and 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 GP to GP. A regular polyhedron, GP, create a Goldberg polyhedra sequence, the truncated octahedron or truncated icosahedron, GP creates a Goldberg sequence, GP, GP, GP, GP. A truncated tetrakis hexahedron or pentakis dodecahedron, GP, creates a Goldberg sequence, like the expansion operation, chamfer can be applied to any dimension. For polygons, it triples the number of vertices, for polychora, new cells are created around the original edges, The cells are prisms, containing two copies of the original face, with pyramids augmented onto the prism sides. The chamfered tetrahedron is a convex polyhedron constructed as a truncated cube or chamfer operation on a tetrahedron. It is the Goldberg polyhedron GIII, containing triangular and hexagonal faces and it can look a little like a truncated tetrahedron, which has 4 hexagonal and 4 triangular faces, which is the related Goldberg polyhedron, GIII. Net In geometry, the cube is a convex polyhedron constructed from the rhombic dodecahedron by truncating the 6 vertices. The 6 vertices are truncated such that all edges are equal length, the original 12 rhombic faces become flattened hexagons, and the truncated vertices become squares. The hexagonal faces are equilateral but not regular and they are formed by a truncated rhombus, have 2 internal angles of about 109.47 degrees and 4 internal angles of about 125.26 degrees, while a regular hexagon would have all 120 degree angles. Because all its faces have a number of sides with 180 degree rotation symmetry. It is also the Goldberg polyhedron GIV, containing square and hexagonal faces, the chamfered cube is the Minkowski sum of a rhombic dodecahedron and a cube of side length 1 when eight vertices of the rhombic dodecahedron are at and its six vertices are at the permutations of. The chamfered cube can be constructed with symmetry and rectangular faces. This can be seen as a pyritohedron with 6 axial edges planned and we can construct a truncated octahedron model by twenty four chamfered cube blocks. This polyhedron looks similar to the truncated octahedron, In geometry. The 8 vertices are truncated such that all edges are equal length, the original 12 rhombic faces become flattened hexagons, and the truncated vertices become triangles
Chamfered tetrahedron
Chamfered tetrahedron
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Example chamfered pentagon (v,e) --> (v+2e,3e)
59.
Chamfered cube
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In geometry, chamfering or edge-truncation is a topological operator that modifies one polyhedron into another. It is similar to expansion, moving faces apart and outward, for polyhedra, this operation adds a new hexagonal face in place of each original edge. In Conway polyhedron notation it is represented by the letter c, a polyhedron with e edges will have a chamfered form containing 2e new vertices, 3e new edges, and 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 GP to GP. A regular polyhedron, GP, create a Goldberg polyhedra sequence, the truncated octahedron or truncated icosahedron, GP creates a Goldberg sequence, GP, GP, GP, GP. A truncated tetrakis hexahedron or pentakis dodecahedron, GP, creates a Goldberg sequence, like the expansion operation, chamfer can be applied to any dimension. For polygons, it triples the number of vertices, for polychora, new cells are created around the original edges, The cells are prisms, containing two copies of the original face, with pyramids augmented onto the prism sides. The chamfered tetrahedron is a convex polyhedron constructed as a truncated cube or chamfer operation on a tetrahedron. It is the Goldberg polyhedron GIII, containing triangular and hexagonal faces and it can look a little like a truncated tetrahedron, which has 4 hexagonal and 4 triangular faces, which is the related Goldberg polyhedron, GIII. Net In geometry, the cube is a convex polyhedron constructed from the rhombic dodecahedron by truncating the 6 vertices. The 6 vertices are truncated such that all edges are equal length, the original 12 rhombic faces become flattened hexagons, and the truncated vertices become squares. The hexagonal faces are equilateral but not regular and they are formed by a truncated rhombus, have 2 internal angles of about 109.47 degrees and 4 internal angles of about 125.26 degrees, while a regular hexagon would have all 120 degree angles. Because all its faces have a number of sides with 180 degree rotation symmetry. It is also the Goldberg polyhedron GIV, containing square and hexagonal faces, the chamfered cube is the Minkowski sum of a rhombic dodecahedron and a cube of side length 1 when eight vertices of the rhombic dodecahedron are at and its six vertices are at the permutations of. The chamfered cube can be constructed with symmetry and rectangular faces. This can be seen as a pyritohedron with 6 axial edges planned and we can construct a truncated octahedron model by twenty four chamfered cube blocks. This polyhedron looks similar to the truncated octahedron, In geometry. The 8 vertices are truncated such that all edges are equal length, the original 12 rhombic faces become flattened hexagons, and the truncated vertices become triangles
Chamfered cube
Chamfered cube
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Example chamfered pentagon (v,e) --> (v+2e,3e)
60.
Johnson solid
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In geometry, a Johnson solid is a strictly convex polyhedron, which is not uniform, and each face of which is a regular polygon. There is no requirement that each face must be the same polygon, 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, at least three faces meet at every vertex, and 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. Although there is no restriction that any given regular polygon cannot be a face of a Johnson solid, it turns out that the faces of Johnson solids always have 3,4,5,6,8. In 1966, Norman Johnson published a list which included all 92 solids and he did not prove that there were only 92, but he did conjecture that there were no others. Victor Zalgaller in 1969 proved that Johnsons list was complete, however, it is not vertex-transitive, as it has different isometry at different vertices, making it a Johnson solid rather than an Archimedean solid. The naming of Johnson Solids follows a flexible & precise descriptive formula, from there, a series of prefixes are attached to the word to indicate additions, rotations and transformations, Bi- indicates that two copies of the solid in question are joined base-to-base. For cupolae and rotundae, the solids can be joined so that like either faces or unlike faces meet, using this nomenclature, an octahedron can be described as a square bipyramid, a cuboctahedron as a triangular gyrobicupola, and an icosidodecahedron as a pentagonal gyrobirotunda. Elongated indicates a prism is joined to the base of the solid in question, a rhombicuboctahedron can thus be described as an elongated square orthobicupola. Gyroelongated indicates an antiprism is joined to the base of the solid in question or between the bases in the case of Bi- solids, an icosahedron can thus be described as a gyroelongated pentagonal bipyramid. Augmented indicates a pyramid or cupola is joined to one or more faces of the solid in question, diminished indicates a pyramid or cupola is removed from one or more faces of the solid in question. Gyrate indicates a cupola mounted on or featured in the solid in question is rotated such that different edges match up, the last three operations — augmentation, diminution, and gyration — can be performed multiple times certain large solids. Bi- & Tri- indicate a double and treble operation respectively, for example, a bigyrate solid has two rotated cupolae, and a tridiminished solid has three removed pyramids or cupolae. In in certain solids, a distinction is made between solids where altered faces are parallel and solids where altered faces are oblique. Para- indicates the former, that the solid in question has altered parallel faces, for example, a parabiaugmented solid has had two parallel faces augmented, and a metabigyrate solid has had 2 oblique faces gyrated. The last few Johnson solids have names based on certain polygon complexes from which they are assembled and these names are defined by Johnson with the following nomenclature, A lune is a complex of two triangles attached to opposite sides of a square. Spheno- indicates a complex formed by two adjacent lunes
Johnson solid
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The elongated square gyrobicupola (J 37), a Johnson solid
61.
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, the following formulae for the volume and surface area can be used if all faces are regular, with edge length a, V = a 3 ≈1.17851. A2 The dual of the cupola has 6 triangular and 3 kite faces, The triangular cupola can be augmented by 3 square pyramids. This isnt a Johnson solid because of its coplanar faces, merging those coplanar triangles into larger ones, topologically this is another triangular cupola with isosceles trapezoidal side faces. If all the triangles are retained and the hexagon is replaced by 6 triangles. The triangular cupola can form a tessellation of space with square pyramids and/or octahedra, the family of cupolae with regular polygons exists up to n=5, and higher if isosceles triangles are used in the cupolae. Eric W. Weisstein, Triangular cupola at MathWorld
Triangular cupola
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Triangular cupola
62.
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 triangular 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 following formulae for volume and surface area can be used if all faces are regular, with edge length a, V = a 3 ≈3.77659. A2 The dual of the triangular cupola has 15 faces,6 isosceles triangles,3 rhombi. The elongated triangular cupola can form a tessellation of space with tetrahedra, eric W. Weisstein, Elongated triangular cupola at MathWorld
Elongated triangular cupola
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Elongated triangular cupola
63.
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 triangular cupola and this is called gyroelongation, which means that an antiprism is joined to the base of a solid, or between the bases of more than one solid. The gyroelongated triangular cupola can also be seen as a 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 and they were named by Norman Johnson, who first listed these polyhedra in 1966. The following formulae for volume and surface area can be used if all faces are regular, a 2 The dual of the gyroelongated triangular cupola has 15 faces,6 kites,3 rhombi, and 6 pentagons. Eric W. Weisstein, Gyroelongated triangular cupola at MathWorld
Gyroelongated triangular cupola
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Gyroelongated triangular cupola
64.
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 augmenting a hexagonal prism by attaching square pyramids to two of its nonadjacent, parallel equatorial faces. Attaching the pyramids to nonadjacent, nonparallel equatorial faces yields a metabiaugmented hexagonal prism, 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, Parabiaugmented hexagonal prism at MathWorld
Parabiaugmented hexagonal prism
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Parabiaugmented hexagonal prism
65.
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 augmenting a hexagonal prism by attaching square pyramids to three of its nonadjacent equatorial faces. A Johnson solid is one of 92 strictly convex polyhedra that have regular faces but are not uniform and 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
66.
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 convex polyhedra that have regular faces but are not uniform and they were named by Norman Johnson, who first listed these polyhedra in 1966. It is one of the elementary Johnson solids, which do not arise from cut and paste manipulations of the Platonic, however, it does have a strong relationship to the icosidodecahedron, an Archimedean solid. Most evident is the cluster of three pentagons and four triangles on one side of the solid, the triangular hebesphenorotunda is the only Johnson solid with faces of 3,4,5 and 6 sides. These coordinates produce a triangular hebesphenorotunda with edge length 2, resting on the XY plane, a second, inverted, triangular hebesphenorotunda can be obtained by negating the second and third coordinates of each point. This second polyhedron will be joined to the first at their common hexagonal face, 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
67.
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 right hexagonal pyramid with a regular hexagon base has C6v symmetry. Bipyramid, prism and antiprism Weisstein, Eric W. Hexagonal Pyramid, virtual Reality Polyhedra www. georgehart. com, The Encyclopedia of Polyhedra Conway Notation for Polyhedra Try, Y6
Hexagonal pyramid
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Hexagonal pyramid
68.
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 triangles and regular hexagons, arranged so that each hexagon is surrounded by triangles. The name derives from the fact that it combines a hexagonal tiling. Two hexagons and two triangles alternate around each vertex, and its edges form an arrangement of lines. Its dual is the rhombille tiling and this pattern, and its place in the classification of uniform tilings, was already known to Johannes Kepler in his 1619 book Harmonices Mundi. The pattern has long used in Japanese basketry, where it is called kagome. The Japanese term for this pattern has taken up in physics. It occurs also in the structures of certain minerals. Conway calls it a hexadeltille, combining elements from a hexagonal tiling. Kagome is a traditional Japanese woven bamboo pattern, its name is composed from the words kago, meaning basket, and me, meaning eye, referring to the pattern of holes in a woven basket. It is an arrangement of laths composed of interlaced triangles such that each point where two laths cross has four neighboring points, forming the pattern of a trihexagonal tiling. The weaved process gives the Kagome a chiral wallpaper group symmetry, the term kagome lattice was coined by Japanese physicist Kôdi Husimi, and first appeared in a 1951 paper by his assistant Ichirō Shōji. The kagome lattice in this sense consists of the vertices and edges of the trihexagonal tiling, despite the name, these crossing points do not form a mathematical lattice. It is represented by the vertices and edges of the cubic honeycomb, filling space by regular tetrahedra. It contains four sets of planes of points and lines. A second expression in three dimensions has parallel layers of two dimensional lattices and is called an orthorhombic-kagome lattice, the trihexagonal prismatic honeycomb represents its edges and vertices. Some minerals, namely jarosites and herbertsmithite, contain two layers or three dimensional kagome lattice arrangement of atoms in their crystal structure. These minerals display novel physical properties connected with geometrically frustrated magnetism, for instance, the spin arrangement of the magnetic ions in Co3V2O8 rests in a kagome lattice which exhibits fascinating magnetic behavior at low temperatures
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
69.
2-uniform tiling
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Euclidean plane tilings by convex regular polygons have been widely used since antiquity. The first systematic mathematical treatment was that of Kepler in his Harmonices Mundi and this means that, for every pair of flags, there is a symmetry operation mapping the first flag to the second. This is equivalent to the tiling being an edge-to-edge tiling by congruent regular polygons, There must be six equilateral triangles, four squares or three regular hexagons at a vertex, yielding the three regular tessellations. Vertex-transitivity means that for pair of vertices there is a symmetry operation mapping the first vertex to the second. Note that there are two mirror forms of 34.6 tiling, only one of which is shown in the following table. All other regular and 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. Such periodic tilings may be classified by the number of orbits of vertices, edges and tiles. If there are k orbits of vertices, a tiling is known as k -uniform or k -isogonal, if there are t orbits of tiles, as t -isohedral, if there are e orbits of edges, as e -isotoxal. K-uniform tilings with the vertex figures can be further identified by their wallpaper group symmetry. 1-uniform tilings include 3 regular tilings, and 8 semiregular ones, There are 20 2-uniform tilings,61 3-uniform tilings,151 4-uniform tilings,332 5-uniform tilings and 673 6-uniform tilings. Each can be grouped by the m of distinct vertex figures. For edge-to-edge Euclidean tilings, the angles of the polygons meeting at a vertex must add to 360 degrees. A regular n -gon has internal angle 180 degrees, only eleven of these can occur in a uniform tiling of regular polygons, given in previous sections. In particular, if three polygons meet at a vertex and one has an odd number of sides, the two polygons must be the same. If they are not, they would have to alternate around the first polygon, vertex types are listed for each. If two tilings share the two vertex types, they are given subscripts 1,2. There are 61 3-uniform tilings of the Euclidean plane,39 are 3-Archimedean with 3 distinct vertex types, while 22 have 2 identical vertex types in different symmetry orbits. Chavey There are 151 4-uniform tilings of the Euclidean plane, Brian Galebachs search reproduced Krotenheerdts list of 33 4-uniform tilings with 4 distinct vertex types, as well as finding 85 of them with 3 vertex types, and 33 with 2 vertex types
2-uniform tiling
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Periodic
2-uniform tiling
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A regular tiling has one type of regular face.
2-uniform tiling
70.
E-ELT
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The European Extremely Large Telescope is an astronomical observatory and the worlds 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 observatory aims to gather 13 times more light than the largest optical telescopes existing in 2014, and be able to correct for atmospheric distortions. It has around 256 times the light gathering area of the Hubble Space Telescope and, according to the E-ELTs specifications, the facility is expected to take 11 years to construct. Construction work on the E-ELT site started in June 2014, by December 2014, ESO had secured over 90% of the total funding and authorized construction of the telescope to start, which will cost around one billion Euro for the first construction phase. First light is planned for 2024, on 26 April 2010, the European Southern Observatory Council selected Cerro Armazones, Chile, as the baseline site for the planned E-ELT. Other sites that were under discussion included Cerro Macon, Salta, in Argentina, Roque de los Muchachos Observatory, on the Canary Islands, and sites in South Africa, Morocco, and Antarctica. Early designs included a primary mirror with a diameter of 42 metres and area of about 1,300 m2. However, in 2011 a proposal was put forward to reduce its size by 13% to 978 m2, for a 39.3 m diameter primary mirror and a 4.2 m diameter secondary mirror. It reduced projected costs from 1.275 billion to 1.055 billion euros, the ESO Council endorsed the revised baseline design in June 2011 and expected a construction proposal for approval in December 2011. Funding was subsequently included in the 2012 budget for initial work to begin in early 2012, the project received preliminary approval in June 2012. ESO approved the start of construction in December 2014, with over 90% funding of the nominal budget secured, the design phase of the 5-mirror anastigmat was fully funded within the ESO budget. With the 2011 changes in the design, the construction cost was estimated to be €1.055 billion. The start of operations is planned for 2024, the ESO focused on the current design after a feasibility study concluded the proposed 100 metres diameter Overwhelmingly Large Telescope would cost €1.5 billion, and be too complex. Both current fabrication technology and road transportation constraints limit single mirrors to being roughly 8 metres in a single piece, the E-ELT will use a similar design, as well as techniques to work around atmospheric distortion of incoming light, known as adaptive optics. A 40-metre-class mirror will allow the study of the atmospheres of extrasolar planets. The E-ELT is the highest priority in the European planning activities for research infrastructures, such as the Astronet Science Vision and Infrastructure Roadmap, the first three mirrors are curved, and form a three mirror anastigmat design for excellent image quality over the 10 arcminute field of view. The fourth and fifth mirrors are flat, and provide adaptive optics correction for atmospheric distortions, the surface of the 39-metre primary mirror will be composed of 798 hexagonal segments. In January 2017, ESO awarded the contract for the fabrication of the 4608 edge sensors to the FAMES consortium and these sensors can measure relative positions to an accuracy of a few nanometres, the most accurate ever used in a telescope
E-ELT
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Engineer rendering of the 39-metre European Extremely Large Telescope (E-ELT)
E-ELT
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ESO Council meets at ESO headquarters in Garching, 2012.
E-ELT
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A real night-time panorama of Cerro Armazones, chosen in April 2010.
E-ELT
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Renderings of the 40-metre-class E-ELT at dusk (left) and from above (right)
71.
Carapace
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A carapace is a dorsal section of the exoskeleton or shell in a number of animal groups, including arthropods, such as crustaceans and arachnids, as well as vertebrates, such as turtles and tortoises. In turtles and 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 and these include Cladocera, ostracods, and isopods, but isopods only have a developed cephalic shield carapace covering the head. In arachnids, the carapace is formed by the fusion of prosomal tergites into a plate which carries the eyes, ocularium, ozopores. In a few orders, such as Solifugae and Schizomida, the carapace may be subdivided, alternative terms for the carapace of arachnids and their relatives, which avoids confusion with crustaceans, are prosomal dorsal shield and peltidium. The carapace is the convex part of the shell structure of a turtle, consisting primarily of the animals rib cage, dermal armor
Carapace
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The molted carapace of a lady crab from Long Beach, New York.
Carapace
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Diagram of a prawn, with the carapace highlighted in red.
Carapace
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Diagram of an arachnid, with the carapace highlighted in purple
Carapace
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A Greek tortoise shell opened to show the skeleton from below
72.
Saturn
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Saturn is the sixth planet from the Sun and the second-largest in the Solar System, after Jupiter. It is a gas giant with a 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 gods sickle. Saturns interior is composed of a core of iron–nickel and rock. This core is surrounded by a layer of metallic hydrogen, an intermediate layer of liquid hydrogen and liquid helium. Saturn has a yellow hue due to ammonia crystals in its upper atmosphere. Saturns magnetic field strength is around one-twentieth of Jupiters, 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, higher than on Jupiter, 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 gas giant because it is predominantly composed of hydrogen and helium. It lacks a definite surface, though it may have a solid core, Saturns rotation causes it to have the shape of an oblate spheroid, that is, it is flattened at the poles and bulges at its equator. Its equatorial and polar radii differ by almost 10%,60,268 km versus 54,364 km, Jupiter, Uranus, and Neptune, the other giant planets in the Solar System, are also oblate but to a lesser extent. Saturn is the planet of the Solar System that is less dense than water—about 30% less. Although Saturns core is considerably denser than water, the specific density of the planet is 0.69 g/cm3 due to the atmosphere. Jupiter has 318 times the Earths mass, while Saturn is 95 times the mass of the Earth, together, Jupiter and Saturn hold 92% of the total planetary mass in the Solar System. On 8 January 2015, NASA reported determining the center of the planet Saturn, the temperature, pressure, and density inside Saturn all rise steadily toward the core, which causes hydrogen to transition into a metal in the deeper layers. Standard planetary models suggest that the interior of Saturn is similar to that of Jupiter, having a rocky core surrounded by hydrogen. This core is similar in composition to the Earth, but more dense, in 2004, they estimated that the core must be 9–22 times the mass of the Earth, which corresponds to a diameter of about 25,000 km. This is surrounded by a liquid metallic hydrogen layer, followed by a liquid layer of helium-saturated molecular hydrogen that gradually transitions to a gas with increasing altitude
Saturn
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Saturn in natural color, photographed by Cassini in 2004
Saturn
Saturn
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Composite image roughly comparing the sizes of Saturn and Earth
Saturn
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Auroral lights at Saturn’s north pole.
73.
Voyager 1
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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 its twin, having operated for 39 years,6 months and 30 days, the spacecraft still communicates with the Deep Space Network to receive routine commands and return data. At a distance of 138 AU from the Sun as of March 2017, the probes primary mission objectives included flybys of Jupiter, Saturn, and Saturns large moon, Titan. It studied the weather, magnetic fields, and rings of the two planets and was the first probe to provide detailed images of their moons. After completing its mission with the flyby of Saturn on November 20,1980, Voyager 1 began an extended mission to explore the regions. On August 25,2012, Voyager 1 crossed the heliopause to become the first spacecraft to enter interstellar space, 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 Voyagers engineers design Voyager to cope effectively with the intense radiation environment around Jupiter. Initially, Voyager 1 was planned as Mariner 11 of the Mariner program, due to budget cuts, the mission was scaled back to be a flyby of Jupiter and Saturn and renamed the Mariner Jupiter-Saturn probes. As the program progressed, the name was changed to Voyager. Voyager 1 was constructed by the Jet Propulsion Laboratory and it has 16 hydrazine thrusters, three-axis stabilization gyroscopes, and referencing instruments to keep the probes radio antenna pointed toward Earth. Collectively, these instruments are part of the Attitude and Articulation Control Subsystem, the spacecraft also included 11 scientific instruments to study celestial objects such as planets as it travels through space. The radio communication system of Voyager 1 was designed to be used up to, the communication system includes a 3. 7-meter diameter parabolic dish high-gain antenna to send and receive radio waves via the three Deep Space Network stations on the Earth. The craft normally transmits data to Earth over Deep Space Network Channel 18, using a frequency of either 2.3 GHz or 8.4 GHz, while signals from Earth to Voyager are broadcast at 2.1 GHz. When Voyager 1 is unable to communicate directly with the Earth, signals from Voyager 1 take over 19 hours to reach Earth. Voyager 1 has three radioisotope thermoelectric generators mounted on a boom, each MHW-RTG contains 24 pressed plutonium-238 oxide spheres. The RTGs generated about 470 W of electric power at the time of launch, the power output of the RTGs declines over time, but the crafts RTGs will continue to support some of its operations until 2025. As of 2017-04-04, Voyager 1 has 73. 14% of the plutonium-238 that it had at launch, by 2050, it will have 56. 5% left. Since the 1990s, space probes usually have completely autonomous cameras, the computer command subsystem controls the cameras
Voyager 1
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Voyager 1, artist's impression
Voyager 1
Voyager 1
Voyager 1
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Voyager 1 lifted off with a Titan IIIE
74.
Benzene
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Benzene is an important organic chemical compound with the chemical formula C6H6. The benzene molecule is composed of 6 carbon atoms joined in a ring with 1 hydrogen atom attached to each, because it contains only carbon and hydrogen atoms, benzene is classed as a hydrocarbon. Benzene is a constituent of crude oil and is one of the elementary petrochemicals. Because of the cyclic continuous pi bond between the atoms, benzene is classed as an aromatic hydrocarbon, the second -annulene. Benzene is a colorless and highly flammable liquid with a sweet smell and it is used primarily as a precursor to the manufacture of chemicals with more complex structure, such as ethylbenzene and cumene, of which billions of kilograms are produced. Because benzene has a high number, it is an important component of gasoline. Because benzene is a carcinogen, most non-industrial applications have been limited. The word benzene derives historically from gum benzoin, a resin known to European pharmacists. An acidic material was derived from benzoin by sublimation, and named flowers of benzoin, the hydrocarbon derived from benzoic acid thus acquired the name benzin, benzol, or benzene. Michael Faraday first isolated and identified benzene in 1825 from the oily residue derived from the production of illuminating gas, in 1833, Eilhard Mitscherlich produced it by distilling benzoic acid and lime. He gave the compound the name benzin, in 1845, Charles Mansfield, working under August Wilhelm von Hofmann, isolated benzene from coal tar. Four years later, Mansfield began the first industrial-scale production of benzene, gradually, the sense developed among chemists that a number of substances were chemically related to benzene, comprising a diverse chemical family. In 1855, Hofmann used the word aromatic to designate this family relationship, in 1997, benzene was detected in deep space. The empirical formula for benzene was known, but its highly polyunsaturated structure. In 1865, the German chemist Friedrich August Kekulé published a paper in French suggesting that the structure contained a ring of six carbon atoms with alternating single and double bonds, the next year he published a much longer paper in German on the same subject. Kekulés symmetrical ring could explain these facts, as well as benzenes 1,1 carbon-hydrogen ratio. Here Kekulé spoke of the creation of the theory and he said that he had discovered the ring shape of the benzene molecule after having a reverie or day-dream of a snake seizing its own tail. This vision, he said, came to him years of studying the nature of carbon-carbon bonds
Benzene
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A bottle of benzene. The warnings show benzene is a toxic and flammable liquid.
Benzene
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Geometry of molecule
75.
Aromatic compound
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Aromatic molecules are very stable, and do not break apart easily to react with other substances. Organic compounds that are not aromatic are classified as aliphatic compounds—they might be cyclic, since the most common aromatic compounds are derivatives of benzene, the word “aromatic” occasionally refers informally to benzene derivatives, and so it was first defined. Nevertheless, many aromatic compounds exist. In living organisms, for example, the most common aromatic rings are the 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 compounds, many of which have odors. In terms of the 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 pi system to be delocalized around the ring, increasing the molecules stability. The molecule cannot be represented by one structure, but rather a hybrid of different structures. These molecules cannot be found in one of these representations, with the longer single bonds in one location. Rather, the molecule exhibits bond lengths in between those of single and double bonds and this commonly seen model of aromatic rings, namely the idea that benzene was formed from a six-membered carbon ring with alternating single and double bonds, was developed by August Kekulé. The model for benzene consists of two forms, which corresponds to the double and single bonds superimposing to produce six one-and-a-half bonds. Benzene is a stable molecule than would be expected without accounting for charge delocalization. As is standard for resonance diagrams, the use of an arrow indicates that two structures are not distinct entities but merely hypothetical possibilities. Neither is a representation of the actual compound, which is best represented by a hybrid of these structures. A C=C bond is shorter than a C−C bond, but benzene is perfectly hexagonal—all six carbon–carbon bonds have the same length, intermediate between that of a single and that of a double bond. In a cyclic molecule with three alternating double bonds, cyclohexatriene, the length of the single bond would be 1.54 Å. However, in a molecule of benzene, the length of each of the bonds is 1.40 Å, a better representation is that of the circular π-bond, in which the electron density is evenly distributed through a π-bond above and below the ring
Aromatic compound
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Two different resonance forms of benzene (top) combine to produce an average structure (bottom)
76.
Coronene
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Coronene is a polycyclic aromatic hydrocarbon comprising six peri-fused benzene rings. Its chemical formula is C 24H12 and it is a yellow material that dissolves in such solvents as benzene, toluene, and dichloromethane. Its solutions emit blue light fluorescence under UV light and its emission 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 and it occurs naturally as the very rare mineral carpathite, which is characterized by flakes of pure coronene embedded in sedimentary rock. This mineral may be created from ancient hydrothermal vent activity, in earlier times this mineral was also called Karpatite or Pendletonite. Coronene is produced in the process of hydrocracking, where it can dimerize to a fifteen ring PAH. The compound is of theoretical interest to chemists because of its aromaticity. Hexa-benzopericoronenes are members of the family and investigated in supramolecular electronics. They are known to self-assemble into a columnar phase, one derivative in particular forms carbon nanotubes with interesting electrical properties. In this geometry, the stacks of disks are aligned with the length of the tube. The nanotubes have sufficient length to fit between two platinum nanogap electrodes produced by scanning probe nanofabrication and are 180 nanometer apart, the nanotubes as such are insulating, but, after one-electron oxidation with nitrosonium tetrafluoroborate, they conduct electricity. Organic synthesis of a hexa-benzopericoronene starts with an Aldol condensation reaction of a derivative with a benzil derivative to substituted cyclopentadienone. This compound is reacted with an alkyne in a Diels-Alder reaction and subsequent expulsion of carbon monoxide to the hexaphenylbenzene, a related compound lacking the central core, cyclooctadecanonaene A larger polycyclic aromatic hydrocarbon with the same symmetry properties, Hexabenzocoronene Coronenes dimer, Dicoronylene
Coronene
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Coronene
77.
Basalt
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Basalt is a common extrusive igneous rock formed from the rapid cooling of basaltic lava exposed at or very near the surface of a planet or moon. Flood basalt describes the formation in a series of basalt flows. By definition, basalt is an igneous rock with generally 45-55% silica and less than 10% feldspathoid by volume. Basalt commonly features a very fine-grained or glassy matrix interspersed with visible mineral grains, the average density is 3.0 gm/cm3. Basalt is defined by its content and texture, and physical descriptions without mineralogical context may be unreliable in some circumstances. Basalt is usually grey to black in colour, but rapidly weathers to brown or rust-red due to oxidation of its mafic minerals into hematite, although usually characterized as dark, basaltic rocks exhibit a wide range of shading due to regional geochemical processes. Due to weathering or high concentrations of plagioclase, some basalts can be quite light-coloured and these phenocrysts usually are of olivine or a calcium-rich plagioclase, which have the highest melting temperatures of the typical minerals that can crystallize from the melt. Basalt with a texture is called vesicular basalt, when the bulk of the rock is mostly solid. Gabbro is often marketed commercially as black granite and these ultramafic volcanic rocks, with silica contents below 45% are usually classified as komatiites. Agricola applied basalt to the black rock of the Schloßberg at Stolpen. Tholeiitic basalt is relatively rich in silica and poor in sodium, included in this category are most basalts of the ocean floor, most large oceanic islands, and continental flood basalts such as the Columbia River Plateau. Basalt rocks are in some cases classified after their content in High-Ti and Low-Ti varieties. High-Ti and Low-Ti basalts have been distinguished in the Paraná and Etendeka traps and it has greater than 17% alumina and is intermediate in composition between tholeiite and alkali basalt, the relatively alumina-rich composition is based on rocks without phenocrysts of plagioclase. Alkali basalt is relatively poor in silica and rich in sodium and it is silica-undersaturated and may contain feldspathoids, alkali feldspar and phlogopite. Boninite is a form of basalt that is erupted generally in back-arc basins. Ocean island basalt Lunar basalt On Earth, most basalt magmas have formed by melting of the mantle. Basalt commonly erupts on Io, the third largest moon of Jupiter, and has formed on the Moon, Mars, Venus. The crustal portions of oceanic tectonic plates are composed predominantly of basalt, produced from upwelling mantle below, the mineralogy of basalt is characterized by a preponderance of calcic plagioclase feldspar and pyroxene
Basalt
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Basalt
Basalt
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Columnar basalt flows in Yellowstone National Park, USA.
Basalt
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Columnar basalt at Szent György Hill, Hungary
Basalt
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Vesicular basalt at Sunset Crater, Arizona. US quarter for scale.
78.
Northern Ireland
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Northern Ireland is a constituent unit of the United Kingdom in the north-east of Ireland. It is variously described as a country, province, region, or part of the United Kingdom, Northern Ireland shares a border to the south and west with the Republic of Ireland. In 2011, its population was 1,810,863, constituting about 30% of the total population. Northern Ireland was created in 1921, when Ireland was partitioned between Northern Ireland and Southern Ireland 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 political and social turmoil of the Troubles, its economy has grown significantly since the late 1990s. Unemployment in Northern Ireland peaked at 17. 2% in 1986, dropping to 6. 1% for June–August 2014,58. 2% of those unemployed had been unemployed for over a year. Prominent artists and sports persons from Northern Ireland include Van Morrison, Rory McIlroy, Joey Dunlop, Wayne McCullough, some people from Northern Ireland prefer to identify as Irish while others prefer to identify as British. Cultural links between Northern Ireland, the rest of Ireland, and the rest of the UK are complex, in many sports, the island of Ireland fields a single team, a notable exception being association football. Northern Ireland competes separately at the Commonwealth Games, and people from Northern Ireland may compete for either Great Britain or Ireland at the Olympic Games. The region that is now Northern Ireland was the bedrock of the Irish war of resistance against English programmes of colonialism in the late 16th century, the English-controlled Kingdom of Ireland had been declared by the English king Henry VIII in 1542, but Irish resistance made English control fragmentary. Victories by English forces in war and further Protestant victories in the Williamite War in Ireland toward the close of the 17th century solidified Anglican rule in Ireland. In Northern Ireland, the victories of the Siege of Derry and their intention was to materially disadvantage the Catholic community and, to a lesser extent, the Presbyterian community. In the context of open institutional discrimination, the 18th century saw secret, militant societies develop in communities in the region and act on sectarian tensions in violent attacks. Following this, in an attempt to quell sectarianism and force the removal of discriminatory laws, the new state, formed in 1801, the United Kingdom of Great Britain and Ireland, was governed from a single government and parliament based in London. Between 1717 and 1775 some 250,000 people from Ulster emigrated to the British North American colonies and it is estimated that there are more than 27 million Scotch-Irish Americans now living in the US. By the close of the century, autonomy for Ireland within the United Kingdom, in 1912, after decades of obstruction from the House of Lords, Home Rule became a near-certainty. A clash between the House of Commons and House of Lords over a controversial budget produced the Parliament Act 1911, which enabled the veto of the Lords to be overturned. The House of Lords veto had been the unionists main guarantee that Home Rule would not be enacted, in 1914, they smuggled thousands of rifles and rounds of ammunition from Imperial Germany for use by the Ulster Volunteers, a paramilitary organisation opposed to the implementation of Home Rule
Northern Ireland
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Scrabo Tower, County Down
Northern Ireland
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Location of Northern Ireland (dark green) – in Europe (green & dark grey) – in the United Kingdom (green)
Northern Ireland
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Cannon on Derry 's city walls
Northern Ireland
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Signing of the Ulster Covenant in 1912 in opposition to Home Rule
79.
Metropolitan France
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Metropolitan France, also known as European France, is the part of France in Europe. It comprises mainland France and nearby islands in the Atlantic Ocean, the English Channel, Overseas France is the collective name for the part of France outside Europe, French overseas regions, territories, collectivities, and 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,3. 3% of the exclusive economic zone, and 95. 9% of the population of the French Republic. The five overseas regions—Martinique, Guadeloupe, Réunion, French Guiana, in overseas France, a person from metropolitan France is often called a métro, short for métropolitain. Similar terms existed to describe other European colonial powers and this usage of the words metropolis and metropolitan itself came from Ancient Greek metropolis, which was the name for a city-state from which originated colonies across the Mediterranean. By extension metropolis and metropolitan came to mean motherland, a nation or country as opposed to its colonies overseas, today 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. As a result, since the end of the 1990s INSEE has included the five 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, and not the whole of France as INSEE does. According to the French government,64,860,000 people lived in metropolitan France as of January 2017, Metropolitan France covers a land area of 551,695 km². At sea, the economic zone of metropolitan France covers 334,604 km², while the EEZ of overseas France covers 9,821,231 km². In the second round of the 2007 French presidential election,37,342,004 French people cast a ballot. 35,907,015 of these cast their ballots in metropolitan France,1,088,679 cast their ballots in overseas France, and 346,310 cast their ballots in foreign countries. The French National Assembly is made up of 577 deputies,539 of whom are elected in metropolitan France,27 in overseas France, and 11 by French citizens living in foreign countries. Mainland France, or just the mainland, does not include the French islands in the Atlantic Ocean, English Channel or Mediterranean Sea, in Corsica, people from the mainland part of Metropolitan France are referred to as les continentaux. A casual synonym for the part of Metropolitan France is lHexagone, for its approximate shape. French colonial empire Mainland Wildlife of Metropolitan France
Metropolitan France
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Metropolitan France
80.
Martinique
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Like Guadeloupe, it is an overseas region of France, consisting of a single overseas department. One of the Windward Islands, it is north of Saint Lucia, southeast of Puerto Rico, northwest of Barbados. As with the overseas departments, Martinique is one of the eighteen regions of France. As part of France, Martinique is part of the European Union, the official language is French, and virtually the entire population also speak Antillean Creole. Martinique owes its name to Christopher Columbus, who sighted the island in 1493, 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, when Columbus returned to the island in 1502, he rechristened the island as Martinica. The name then evolved into Madinina, Madiana, and 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, Martinique was charted by Columbus in 1493, but Spain had little interest in the territory. On 15 September 1635, Pierre Belain dEsnambuc, French governor of the island of St. Kitts, dEsnambuc claimed Martinique for the French King Louis XIII and the French Compagnie des Îles de lAmérique, and established the first European settlement at Fort Saint-Pierre. DEsnambuc died in 1636, leaving the company and Martinique in the hands of his nephew, in 1637, his nephew Jacques Dyel du Parquet became governor of the island. In 1636, the indigenous Caribs rose against the settlers to drive them off the island in the first of many skirmishes. The French successfully repelled the natives and forced them to retreat to the part of the island. When the Carib revolted against French rule in 1658, the Governor Charles Houël du Petit Pré retaliated with war against them, many were killed, those who survived were taken captive and expelled from the island. Some Carib had fled to Dominica or St. Vincent, where the French agreed to them at peace. They were quite industrious and became quite prosperous, from September 1686 to early 1688, the French crown used Martinique as a threat and a dumping ground for mainland Huguenots who refused to reconvert to Catholicism. Over 1,000 Huguenots were transported to Martinique during this period, usually under miserable and those that survived the trip were distributed to the island planters as Engagés under the system of serf peonage that prevailed in the French Antilles at the time. Many of them were encouraged by their Catholic brethren who looked forward to the departure of the heretics, by 1688, nearly all of Martiniques French Protestant population had escaped to the British American colonies or Protestant countries back home
Martinique
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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
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Flag
Martinique
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The attack on the French ships at Martinique in 1667
Martinique
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The Battle of Martinique between British and French fleets in 1779
81.
French Guiana
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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 Brazil to the east and south, and Suriname to the west. Its 83,534 km2 area has a low population density of only 3 inhabitants per km2, with half of its 244,118 inhabitants in 2013 living in the metropolitan area of Cayenne. By land area, it is the second largest region of France, both the region and the department have been ruled since December 2015 by a single assembly within the framework of a new territorial collectivity, the French Guiana Territorial Collectivity. This assembly, the French Guiana Assembly, has replaced the regional council and departmental council. The French Guiana Assembly is in charge of regional and departmental government, the area was originally inhabited by Native Americans. The first French establishment is recorded in 1503 but the French presence didnt really become durable until 1643, Guiana then became a slave colony and saw its population increase until the official abolition of slavery at the time of the French revolution. During World War II, the Guianan Félix Éboué was one of the first to stand behind General de Gaulle as early as June 18,1940, Guiana officially rallied Free France in 1943. It definitively abandoned its status as a colony and became again a French department in 1946, de Gaulle, who became president, decided to establish the Guiana Space Center in 1965. It is now operated by the CNES, Arianespace and the European Space Agency, several thousand Hmong refugees from Laos migrated to French Guiana in the late 1970s and early 1980s. Nowadays fully integrated in the French central state, Guiana is a part of the European Union, the region is the most prosperous territory in South America with the highest GDP per capita. A large part of Guianas economy derives from the presence of the Guiana Space Centre, as elsewhere in France, the official language is French, but each ethnic community has its own language, of which Guianan Creole is the most widely spoken. Guiana is derived from an Amerindian language and means land of many waters, French Guiana and the two larger countries to the north and west, Guyana and Suriname, are still often collectively referred to as the Guianas and constitute one large shield landmass. French Guiana was originally inhabited by people, Kalina, Arawak, Emerillon, Galibi, Palikur, Wayampi. The French attempted to create a colony there in the 18th century in conjunction 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 Devils Island, fewer than 10% survived their sentence. In addition, in the nineteenth century, France began requiring forced residencies by prisoners who survived their hard labor. A Portuguese-British naval squadron took French Guiana for the Portuguese Empire in 1809 and it was returned to France with the signing of the Treaty of Paris in 1814
French Guiana
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Forested landscape of Remire-Montjoly.
French Guiana
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Flag
French Guiana
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View from the île Royale
French Guiana
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Liana on a palm branch near a lake in Kourou
82.
Hexagonal crystal system
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In crystallography, the hexagonal crystal family is one of the 6 crystal families. In the hexagonal family, the crystal is described by a right rhombic prism unit cell with two equal axes, an included angle of 120° and a height perpendicular to the two base axes. 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 cell with parameters a = b = c, α = β = γ ≠ 90°. In practice, the description is more commonly used because it is easier to deal with a coordinate system with two 90° angles. However, the 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 crystal system is a set of point groups in which the point groups themselves, 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 example are described by two of those 18 space groups associated with the hexagonal lattice system. The hexagonal crystal system consists of the seven point groups such that all their groups have the hexagonal lattice as underlying lattice. Graphite is an example of a crystal that crystallizes in the crystal system. Note that the atom in the center of the HCP unit cell in the hexagonal lattice system does not appear in the unit cell of the hexagonal lattice. It is part of the two atom motif associated with each point in the underlying lattice. The trigonal crystal system is the crystal system whose point groups have more than one lattice system associated with their space groups. The 5 point groups in this system are listed below, with their international number and notation, their space groups in name. The point groups in this system are listed below, followed by their representations in Hermann–Mauguin or international notation and Schoenflies notation
Hexagonal crystal system
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An example of the hexagonal crystals, beryl
Hexagonal crystal system
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Hexagonal Hanksite crystal
83.
The Hexagon
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The Hexagon is a multi-purpose theatre and arts venue in Reading, Berkshire, England. The theatre was built in 1977 by Robert Matthew Johnson Marshall, the original design featured a proscenium but no fly tower. Upon opening, the venue was comparable to Derbys Assembly Rooms—which also opened in 1977—but the Hexagon was described as architecturally and acoustically superior, as the building was designed operate as a multi-use venue, the arena-style seating was used to avoid limited visibility. This proved useful for such as snooker or boxing. The stalls, which use removable and retractable seats, have less headroom than the balcony above, the theatre 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 proscenium arch, the capacity is 946. This increases to 1,200 for a concert and 1,686 for standing with balcony seating. The theatres diameter is roughly 30 metres, originally, the Hexagon used an electronically assisted reverberation system, this has now been removed. In a review of the system, one author wrote that the system seemed inaudible in the stalls but made a contribution in the balcony. Acoustic panelling is used throughout the auditorium, the ceiling features rotatable acoustic screens to provide reflections to the balcony seats. The assisted resonance system was found to increase this to 1.5 seconds at 200 Hz, the balcony seats have been described as having an inadequate level of early reflections and speech performance was judged to be poor. It has been concluded that rectifying the rooms acoustics for the benefit of theatre, events include classical music, comedy, dance, drama, pop and rock concerts. In past years the venue has also used for snooker. Reading Arts, The Hexagon Ents24. com —Whats On & Tickets
The Hexagon
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View of The Hexagon from San Francisco Libre Walk
84.
Theatre
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The performers may communicate this experience to the audience through combinations of gesture, speech, song, music, and dance. Elements of art, such as painted scenery and stagecraft such as lighting are used to enhance the physicality, presence, the specific place of the performance is also named by the word theatre as derived from the Ancient Greek θέατρον, itself from θεάομαι. Modern theatre, broadly defined, includes performances of plays and musical theatre, there are connections between theatre and the art forms of ballet, opera and various other forms. The city-state of Athens is where western theatre originated, participation in the city-states many festivals—and mandatory attendance at the City Dionysia as an audience member in particular—was an important part of citizenship. The Greeks also developed the concepts of dramatic criticism and theatre architecture, Actors were either amateur or at best semi-professional. The theatre of ancient Greece consisted of three types of drama, tragedy, comedy, and the satyr play, the origins of theatre in ancient Greece, according to Aristotle, 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 hillsides, capable of seating 10, the stage consisted of a dancing floor, dressing room and scene-building area. Since the words were the most important part, good acoustics, the actors wore masks appropriate to the characters they represented, and each might play several parts. Athenian tragedy—the oldest surviving form of tragedy—is a type of dance-drama that formed an important part of the culture of the city-state. Having emerged sometime during the 6th century BCE, it flowered during the 5th century BCE, no tragedies from the 6th century BCE and only 32 of the more than a thousand that were performed in during the 5th century BCE have survived. We have complete texts extant by Aeschylus, Sophocles, and Euripides, the origins of tragedy remain obscure, though by the 5th century BCE it was institution alised in competitions held as part of festivities celebrating Dionysus. As contestants in the City Dionysias competition playwrights were required to present a tetralogy of plays, the performance of tragedies at the City Dionysia may have begun as early as 534 BCE, official records begin from 501 BCE, when the satyr play was introduced. More than 130 years later, the philosopher Aristotle analysed 5th-century Athenian tragedy in the oldest surviving work of dramatic theory—his Poetics, Athenian comedy is conventionally divided into three periods, Old Comedy, Middle Comedy, and New Comedy. Old Comedy survives today largely in the form of the surviving plays of Aristophanes. New Comedy is known primarily from the papyrus fragments of Menander. Aristotle defined comedy as a representation of people that involves some kind of blunder or ugliness that does not cause pain or disaster. In addition to the categories of comedy and tragedy at the City Dionysia, finding its origins in rural, agricultural rituals dedicated to Dionysus, the satyr play eventually found its way to Athens in its most well-known form. Satyrs themselves were tied to the god Dionysus as his loyal companions, often engaging in drunken revelry
Theatre
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Sarah Bernhardt as Hamlet, in 1899
Theatre
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A master (right) and his slave (left) in a Greek phlyax play, circa 350/340 BCE
Theatre
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Mosaic depicting masked actors in a play: two women consult a "witch"
Theatre
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Performer playing Sugriva in the Koodiyattam form of Sanskrit theatre.
85.
Reading, Berkshire
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Reading is a large, historically important town in Berkshire, England, of which it is the county town. The 19th century saw the coming of the Great Western Railway, Today Reading is a major commercial centre, with involvement in information technology and insurance, and, despite its proximity to London, has a net inward commuter flow. The first evidence for Reading as a settlement dates from the 8th century, by 1525, Reading was the largest town in Berkshire, and tax returns show that Reading was the 10th largest town in England when measured by taxable wealth. By 1611, it had a population of over 5000 and had grown rich on its trade in cloth, the 18th century saw the beginning of a major iron works in the town and the growth of the brewing trade for which Reading was to become famous. During the 19th century, the town rapidly as a manufacturing centre. It is ranked the UKs top economic area for economic success and wellbeing, according to such as employment, health, income. Reading is also a regional retail centre serving a large area of the Thames Valley. Every year it hosts the Reading Festival, one of Englands biggest music festivals, sporting teams based in Reading include Reading Football Club and the London Irish rugby union team, and over 15,000 runners annually compete in the Reading Half Marathon. In 2015, Reading had an population of 232,662. The town is represented in Parliament by two members, and has been continuously represented there since 1295, for ceremonial purposes the town is in the county of Berkshire and has served as its county town since 1867, previously sharing this status with Abingdon-on-Thames. It is in the Thames Valley at the confluence of the River Thames and River Kennet, and on both the Great Western Main Line railway and the M4 motorway. Reading is 75 miles east of Bristol,25 miles south of Oxford,42 miles west of London,17 miles north of Basingstoke,13 miles south-west of Maidenhead and 20 miles east of Newbury. Reading may date back to the Roman occupation of Britain, possibly as a port for Calleva Atrebatum. However the first clear evidence for Reading as a settlement dates from the 8th century, the name probably comes from the Readingas, an Anglo-Saxon tribe whose name means Readas People in Old English, or less probably the Celtic Rhydd-Inge, meaning Ford over the River. In late 870, an army of Danes invaded the kingdom of Wessex, on 4 January 871, in the first Battle of Reading, King Ethelred and his brother Alfred the Great attempted unsuccessfully to breach the Danes defences. The battle is described in the Anglo-Saxon Chronicle, and that account provides the earliest known record of the existence of Reading. The Danes remained in Reading until late in 871, when they retreated to their quarters in London. After the Battle of Hastings and the Norman conquest of England, William the Conqueror gave land in, in its 1086 Domesday Book listing, the town was explicitly described as a borough
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
86.
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 an octahedron. Its facets are simplexes of the dimension, while the cross-polytopes vertex 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 unit ball in the ℓ1-norm on Rn. In 1 dimension the cross-polytope is simply the line 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 hexadecachoron or 16-cell. It is one of six convex regular 4-polytopes and these 4-polytopes were first described by the Swiss mathematician Ludwig Schläfli in the mid-19th century. The cross polytope family is one of three regular polytope families, labeled by Coxeter as βn, the two being the hypercube family, labeled as γn, and the simplices, labeled as αn. A fourth family, the infinite tessellations of hypercubes, he labeled as δn, the n-dimensional cross-polytope has 2n vertices, and 2n facets all of which are n−1 simplices. The vertex figures are all n −1 cross-polytopes, the Schläfli symbol of the cross-polytope is. The dihedral angle of the n-dimensional cross-polytope is δ n = arccos and this gives, δ2 = arccos = 90°, δ3 = arccos =109. 47°, δ4 = arccos = 120°, δ5 = arccos =126. 87°. The volume of the n-dimensional cross-polytope is 2 n n. Petrie polygon projections map the points into a regular 2n-gon or lower order regular polygons. A second projection takes the 2-gon petrie polygon of the dimension, seen as a bipyramid, projected down the axis. The vertices of a cross polytope are all at equal distance from each other in the Manhattan distance. Kusners conjecture states that this set of 2d points is the largest possible equidistant set for this distance, Regular complex polytopes can be defined in complex Hilbert space called generalized orthoplexes, βpn =22. 2p, or. Real solutions exist with p=2, i. e. β2n = βn =22.22 =, for p>2, they exist in C n
Orthoplex
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2 dimensions square
87.
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 edition and variation of a book, for example, an e-book, a paperback and a hardcover edition of the same book would each have a different ISBN. The ISBN is 13 digits long if assigned on or after 1 January 2007, the method of assigning an ISBN is nation-based and 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 in 1967 based upon the 9-digit Standard Book Numbering created in 1966, the 10-digit ISBN format was developed by the International Organization for Standardization and was published in 1970 as international standard ISO2108. Occasionally, a book may appear without a printed ISBN if it is printed privately or the author does not follow the usual ISBN procedure, however, this can be rectified later. Another identifier, the International Standard Serial Number, identifies periodical publications such as magazines, the ISBN configuration of recognition was generated in 1967 in the United Kingdom by David Whitaker and in 1968 in the US by Emery Koltay. The 10-digit ISBN format was developed by the International Organization for Standardization and was published in 1970 as international standard ISO2108, 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 to an ISBN by prefixing the digit 0. For example, the edition of Mr. J. G. Reeder Returns, published by Hodder in 1965, has SBN340013818 -340 indicating the publisher,01381 their serial number. This can be converted to ISBN 0-340-01381-8, the check digit does not need to be re-calculated, since 1 January 2007, ISBNs have contained 13 digits, a format that is compatible with Bookland European Article Number EAN-13s. An ISBN is assigned to each edition and variation of a book, for example, an ebook, a paperback, and a hardcover edition of the same book would each have a different ISBN. The ISBN is 13 digits long if assigned on or after 1 January 2007, a 13-digit ISBN can be separated into its parts, and when this is done it is customary to separate the parts with hyphens or spaces. Separating the parts of a 10-digit ISBN is also done with either hyphens or spaces, figuring out how to correctly separate a given ISBN number is complicated, because most of the parts do not use a fixed number of digits. ISBN issuance is country-specific, in that ISBNs are issued by the ISBN registration agency that is responsible for country or territory regardless of the publication language. Some ISBN registration agencies are based in national libraries or within ministries of culture, in other cases, the ISBN registration service is provided by organisations such as bibliographic data providers that are not government funded. In Canada, ISBNs are issued at no cost with the purpose of encouraging Canadian culture. In the United Kingdom, United States, and some countries, where the service is provided by non-government-funded organisations. Australia, ISBNs are issued by the library services agency Thorpe-Bowker
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
88.
List of polygons
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In geometry, a polygon /ˈpɒlɪɡɒn/ is traditionally a plane figure that is bounded by a finite chain of straight line segments closing in a loop to form a closed chain or circuit. These segments are called its edges or sides, and the points where two edges meet are the vertices or corners. The word polygon comes from Late Latin polygōnum, from Greek πολύγωνον, noun use of neuter of πολύγωνος, individual polygons are named according to the number of sides, combining a Greek-derived numerical prefix with the suffix -gon, e. g. pentagon, dodecagon. The triangle, quadrilateral and nonagon are exceptions, although the regular forms trigon, tetragon, polygons are primarily named by prefixes from Greek numbers. To construct the name of a polygon with more than 20 and less than 100 edges, the kai connector is not included by some authors. Extending the system up to 999 is expressed with these prefixes, platonic solid Dice NAMING POLYGONS Benjamin Franklin Finkel, A Mathematical Solution Book Containing Systematic Solutions to Many of the Most Difficult Problems,1888
List of polygons
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A pentagon is a five sided polygon. A regular pentagon has 5 equal edges and 5 equal angles.
89.
Scalene triangle
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A triangle is a polygon with three edges and three vertices. It is one of the shapes in geometry. A triangle with vertices A, B, and C is denoted △ A B C, in Euclidean geometry any three points, when non-collinear, determine a unique triangle and 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 all sides the same length. An equilateral triangle is also a 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 triangles isosceles triangles, the 45–45–90 right triangle, which appears in the tetrakis 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, a side can be marked with a pattern of ticks, short line segments in the form of tally marks, two sides have equal lengths if they are both marked with the same pattern. In a triangle, the pattern is no more than 3 ticks. Similarly, patterns of 1,2, or 3 concentric arcs inside the angles are used to indicate equal angles, triangles can also be classified according to their internal angles, measured here in degrees. A right triangle has one of its interior angles measuring 90°, the side opposite to the right angle is the hypotenuse, the longest side of the triangle. The other two sides are called the legs or catheti of the triangle, special right triangles are right triangles with additional properties that make calculations involving them easier. One of the two most famous is the 3–4–5 right triangle, where 32 +42 =52, in this situation,3,4, and 5 are a Pythagorean triple. The other one is a triangle that has 2 angles that each measure 45 degrees. Triangles that do not have an angle measuring 90° are called oblique triangles, a triangle with all interior angles measuring less than 90° is an acute triangle or acute-angled triangle. If c is the length of the longest side, then a2 + b2 > c2, a triangle with one interior angle measuring more than 90° is an obtuse triangle or obtuse-angled triangle. If c is the length of the longest side, then a2 + b2 < c2, a triangle with an interior angle of 180° is degenerate
Scalene triangle
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The Flatiron Building in New York is shaped like a triangular prism
Scalene triangle
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A triangle
90.
Quadrilateral
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In Euclidean plane geometry, a quadrilateral is a polygon with four edges and four vertices or corners. Sometimes, the quadrangle is used, by analogy with triangle. The origin of the quadrilateral is the two Latin words quadri, a variant of four, and latus, meaning side. Quadrilaterals are simple or complex, also called crossed, simple quadrilaterals are either convex or concave. The interior angles of a simple quadrilateral ABCD add up to 360 degrees of arc and this is a special case of the n-gon interior angle sum formula × 180°. All non-self-crossing quadrilaterals tile the plane by repeated rotation around the midpoints of their edges, any quadrilateral that is not self-intersecting is a simple quadrilateral. In a convex quadrilateral, all angles are less than 180°. Irregular quadrilateral or trapezium, no sides are parallel, trapezium or trapezoid, at least one pair of opposite sides are parallel. Isosceles trapezium or isosceles trapezoid, one pair of sides are parallel. Alternative definitions are a quadrilateral with an axis of symmetry bisecting one pair of opposite sides, parallelogram, a quadrilateral with two pairs of parallel sides. Equivalent conditions are that opposite sides are of length, that opposite angles are equal. In other words, parallelograms include all rhombi and all rhomboids, rhombus or rhomb, all four sides are of equal length. An equivalent condition is that the diagonals bisect each other. Rhomboid, a parallelogram in which adjacent sides are of unequal lengths, not all references agree, some define a rhomboid as a parallelogram which is not a rhombus. Rectangle, all four angles are right angles, an equivalent condition is that the diagonals bisect each other and are equal in length. Square, all four sides are of length, and all four angles are right angles. An equivalent condition is that opposite sides are parallel, that the diagonals bisect each other. A quadrilateral is a if and only if it is both a rhombus and a rectangle
Quadrilateral
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Six different types of quadrilaterals
91.
Trapezoid
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The parallel sides are called the bases of the trapezoid and the other two sides are called the legs or the lateral sides. A scalene trapezoid is a trapezoid with no sides of equal measure, the first recorded use of the Greek word translated trapezoid was by Marinus Proclus in his Commentary on the first book of Euclids Elements. This article uses the term trapezoid in the sense that is current in the United States, in many other languages using a word derived from the Greek for this figure, the form closest to trapezium is used. A right trapezoid has two adjacent right angles, right trapezoids are used in the trapezoidal rule for estimating areas under a curve. An acute trapezoid has two adjacent acute angles on its longer base edge, while an obtuse trapezoid has one acute, an acute trapezoid is also an isosceles trapezoid, if its sides have the same length, and the base angles have the same measure. An obtuse trapezoid with two pairs of sides is a parallelogram. A parallelogram has central 2-fold rotational symmetry, a Saccheri quadrilateral is similar to a trapezoid in the hyperbolic plane, with two adjacent right angles, while it is a rectangle in the Euclidean plane. A Lambert quadrilateral in the plane has 3 right angles. A tangential trapezoid is a trapezoid that has an incircle, there is some disagreement whether parallelograms, which have two pairs of parallel sides, should be regarded as trapezoids. Some define a trapezoid as a quadrilateral having one pair of parallel sides. Others define a trapezoid as a quadrilateral with at least one pair of parallel sides, the latter definition is consistent with its uses in higher mathematics such as calculus. The former definition would make such concepts as the trapezoidal approximation to a definite integral ill-defined and this article uses the inclusive definition and considers parallelograms as special cases of a trapezoid. This is also advocated in the taxonomy of quadrilaterals, under the inclusive definition, all parallelograms are trapezoids. Rectangles have mirror symmetry on mid-edges, rhombuses have mirror symmetry on vertices, while squares have mirror symmetry on both mid-edges and vertices. Four lengths a, c, b, d can constitute the sides of a non-parallelogram trapezoid with a and b parallel only when | d − c | < | b − a | < d + c. The quadrilateral is a parallelogram when d − c = b − a =0, the angle between a side and a diagonal is equal to the angle between the opposite side and the same diagonal. The diagonals cut each other in mutually the same ratio, the diagonals cut the quadrilateral into four triangles of which one opposite pair are similar. The diagonals cut the quadrilateral into four triangles of which one pair have equal areas
Trapezoid
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The Temple of Dendur in the Metropolitan Museum of Art in New York City
Trapezoid
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Trapezoid
Trapezoid
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Example of a trapeziform pronotum outlined on a spurge bug
92.
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. The area of a regular hendecagon with side length a is given by A =114 a 2 cot π11 ≃9.36564 a 2, as 11 is not a Fermat prime, the regular hendecagon is not constructible with compass and straightedge. Because 11 is not a Pierpont prime, construction of a regular hendecagon is still impossible even with the usage of an angle trisector and it can, however, be constructed via neusis construction. Close approximations to the regular hendecagon can be constructed, however, for instance, the ancient Greek mathematicians approximated the side length of a hendecagon inscribed in a unit circle as being 14/25 units long. The following construction description is given by T, the regular hendecagon has Dih11 symmetry, order 22. Since 11 is a number there is one subgroup with dihedral symmetry, Dih1, and 2 cyclic group symmetries, Z11. These 4 symmetries can be seen in 4 distinct symmetries on the hendecagon, john Conway labels these by a letter and group order. Full symmetry of the form is r22 and no symmetry is labeled a1. The dihedral symmetries are divided depending on whether they pass through vertices or edges, 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 g11 subgroup has no degrees of freedom but can seen as directed edges. The Canadian dollar coin, the loonie, is similar to, but not exactly, the cross-section of a loonie is actually a Reuleaux hendecagon. Anthony dollar has a hendecagonal outline along the inside of its edges
Hendecagon
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A regular hendecagon
93.
Hexadecagon
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In mathematics, a hexadecagon or 16-gon is a sixteen-sided polygon. A regular hexadecagon is a hexadecagon in which all angles are equal and its Schläfli symbol is and can be constructed as a truncated octagon, t, and a twice-truncated square tt. A truncated hexadecagon, t, is a triacontadigon, as 16 =24, a regular hexadecagon is constructible using compass and straightedge, this was already known to ancient Greek mathematicians. Each angle of a regular hexadecagon is 157.5 degrees, the area of a regular hexadecagon with edge length t is A =4 t 2 cot π16 =4 t 2. Since the area of the circumcircle is π R2, the regular hexadecagon fills approximately 97. 45% of its circumcircle, the regular hexadecagon has Dih16 symmetry, order 32. There are 4 dihedral subgroups, Dih8, Dih4, Dih2, and Dih1, and 5 cyclic subgroups, Z16, Z8, Z4, Z2, and Z1, on the regular hexadecagon, there are 14 distinct symmetries. John Conway labels full symmetry as r32 and no symmetry is labeled a1, the dihedral symmetries are divided depending on whether they pass through vertices or edges Cyclic symmetries in the middle column are labeled as g for their central gyration orders. These two forms are duals of each other and have half the order of the regular hexadecagon. Each subgroup 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. A skew hexadecagon is a polygon with 24 vertices and edges. The interior of such an hexadecagon is not generally defined, a skew zig-zag hexadecagon has vertices alternating between two parallel planes. A regular skew hexadecagon is vertex-transitive with equal edge lengths, in 3-dimensions it will be a zig-zag skew hexadecagon and can be seen in the vertices and side edges of a octagonal antiprism with the same D8d, symmetry, order 32. The octagrammic antiprism, s and octagrammic crossed-antiprism, s also have regular skew octagons, there are three regular star polygons, using the same vertices, but connecting every third, fifth or seventh points. There are also three compounds, is reduced to 2 as two octagons, is reduced to 4 as four squares and reduces to 2 as two octagrams, and finally is reduced to 8 as eight digons. Deeper truncations of the octagon and octagram can produce isogonal intermediate hexadecagram forms with equally spaced vertices. A truncated octagon is a hexadecagon, t=, a quasitruncated octagon, inverted as, is a hexadecagram, t=. A truncated octagram is a hexadecagram, t= and a quasitruncated octagram, inverted as, is a hexadecagram, hexadecagrams are included in the Girih patterns in the Alhambra. An octagonal star can be seen as a concave hexadecagon, Weisstein, Eric W. Hexadecagon
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
94.
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, as 17 is a Fermat prime, the regular heptadecagon is a constructible polygon, this was shown by Carl Friedrich Gauss in 1796 at the age of 19. This proof represented the first progress in regular polygon construction in over 2000 years, constructing a regular heptadecagon thus involves finding the cosine of 2 π /17 in terms of square roots, which involves an equation of degree 17—a Fermat prime. Gauss book Disquisitiones Arithmeticae gives this as,16 cos 2 π17 = −1 +17 +34 −217 +217 +317 −34 −217 −234 +217. 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. Make OI one-fourth of OB, and the angle OIE one-fourth of OIA, another more recent construction is given by Callagy. The regular heptadecagon has Dih17 symmetry, order 34, since 17 is a prime number there is one subgroup with dihedral symmetry, Dih1, and 2 cyclic group symmetries, Z17, and Z1. These 4 symmetries can be seen in 4 distinct symmetries on the heptadecagon, john Conway labels these by a letter and group order. Full symmetry of the form is r34 and no symmetry is labeled a1. The dihedral symmetries are divided depending on whether they pass through vertices or edges, 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 g17 subgroup has no degrees of freedom but can seen as directed edges. A heptadecagram is a 17-sided star polygon, there are seven regular forms given by Schläfli symbols, and. The regular heptadecagon is the Petrie polygon for one higher-dimensional regular convex polytope, projected in an orthogonal projection. – Describes the algebraic aspect, by Gauss, contains a description of the construction. Heptadecagon trigonometric functions heptadecagon building New R&D center for SolarUK BBC video of New R&D center for SolarUK Eisenbud, Heptadecagon Heptadecagon, a construction with only one point N, a variation of the design according to H. W. Richmond
Heptadecagon
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A regular heptadecagon
95.
Octadecagon
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An octadecagon or 18-gon is an eighteen-sided polygon. A regular octadecagon has a Schläfli symbol and can be constructed as a truncated enneagon, t. As 18 =2 ×32, a regular octadecagon cannot be constructed using a compass, however, it is constructible using neusis, or an angle trisection with a tomahawk. The following approximate construction is similar to that of the enneagon. It is also feasible with exclusive use of compass and straightedge, the regular octadecagon has Dih18 symmetry, order 36. There are 5 subgroup dihedral symmetries, Dih9, and, and 6 cyclic group symmetries and these 15 symmetries can be seen in 12 distinct symmetries on the octadecagon. John Conway labels these by a letter and group order, full symmetry of the regular form is r36 and no symmetry is labeled a1. The dihedral symmetries are divided depending on whether they pass through vertices or edges, 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 g18 subgroup has no degrees of freedom but can seen as directed edges. A regular triangle, nonagon, and octadecagon can completely surround a point in the plane, one of 17 different combinations of polygons with this property. The regular octadecagon can tessellate the plane with concave hexagonal gaps, and another tiling mixes in nonagons and octagonal gaps. The first tiling is related to a hexagonal tiling. An octadecagram is an 18-sided star polygon, represented by symbol, there are two regular star polygons, and, using the same points, but connecting every fifth or seventh points. Deeper truncations of the regular enneagon and enneagrams can produce isogonal intermediate octadecagram forms with equally spaced vertices, other truncations form double coverings, t==2, t==2, t==2. The regular octadecagon is the Petrie polygon for a number of higher-dimensional polytopes, shown in these orthogonal projections from Coxeter planes, octadecagon Weisstein
Octadecagon
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A regular octadecagon
96.
Enneadecagon
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In geometry an enneadecagon or 19-gon is a nineteen-sided polygon. It is also known as an enneakaidecagon or a nonadecagon, a regular enneadecagon is represented by Schläfli symbol. The radius of the circumcircle of the regular enneadecagon with side length t is R = t 2 csc 18019, the area, where t is the edge length, is 194 t 2 cot π19 ≃28.4652 t 2. As 19 is a Pierpont prime but not a Fermat prime, however, it is constructible using neusis, or an angle trisector. Another animation of an approximate construction, based on the unit circle r =1 Constructed side length of the enneadecagon in GeoGebra a =0.329189180561468. Side length of the enneadecagon a s h o u l d =2 ⋅ sin =0.329189180561467788, the regular enneadecagon has Dih19 symmetry, order 38. Since 19 is a number there is one subgroup with dihedral symmetry, Dih1, and 2 cyclic group symmetries, Z19. These 4 symmetries can be seen in 4 distinct symmetries on the enneadecagon, john Conway labels these by a letter and group order. Full symmetry of the form is r38 and no symmetry is labeled a1. The dihedral symmetries are divided depending on whether they pass through vertices or edges, 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 g19 subgroup has no degrees of freedom but can seen as directed edges. A enneadecagram is a 19-sided star polygon, there are eight regular forms given by Schläfli symbols, and. The regular enneadecagon is the Petrie polygon for one higher-dimensional polytope, projected in an orthogonal projection, enneadecagon Weisstein
Enneadecagon
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A regular enneadecagon
97.
Pentacontagon
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In geometry, a pentacontagon or pentecontagon or 50-gon is a fifty-sided polygon. The sum of any pentacontagons interior angles is 8640 degrees, a regular pentacontagon is represented by Schläfli symbol and can be constructed as a quasiregular truncated icosipentagon, t, which alternates two types of edges. One interior angle in a regular pentacontagon is 172 4⁄5°, meaning that one exterior angle would be 7 1⁄5°, the regular pentacontagon has Dih50 dihedral symmetry, order 100, represented by 50 lines of reflection. Dih50 has 5 dihedral subgroups, Dih25, and and it also has 6 more cyclic symmetries as subgroups, and, with Zn representing π/n radian rotational symmetry. John Conway labels these symmetries with a letter and order of the symmetry follows the letter. He gives d with mirror lines through vertices, p with mirror lines through edges and 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 star polygon. There are 9 regular forms given by Schläfli symbols, and, as well as 16 compound star figures with the same vertex configuration
Pentacontagon
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A regular pentacontagon
98.
Octacontagon
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In geometry, an octacontagon is an eighty-sided polygon. The sum of any octacontagons interior angles is 14040 degrees, one interior angle in a regular octacontagon is 175 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, and, john Conway labels these lower symmetries with a letter and order of the symmetry follows the letter. R160 represents full symmetry and a1 labels no symmetry and he gives d with mirror lines through vertices, p with mirror lines through edges, i with mirror lines through both vertices and edges, and g for rotational 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 star polygon, there are 15 regular forms given by Schläfli symbols, and, as well as 24 regular star figures with the same vertex configuration
Octacontagon
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A regular octacontagon
99.
Hectogon
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In geometry, a hectogon or hecatontagon or 100-gon is a hundred-sided polygon. The sum of any hectogons interior angles is 17640 degrees, a regular hectogon is represented by Schläfli symbol and can be constructed as a truncated pentacontagon, t, or a twice-truncated icosipentagon, tt. One interior angle in a regular hectogon is 176 2⁄5°, meaning that one exterior angle would be 3 3⁄5°, thus the regular hectogon is not a constructible polygon. Indeed, it is not even constructible with the use of neusis or an angle trisector, as the number of sides is neither a product of distinct Pierpont primes, the regular hectogon has Dih100 dihedral symmetry, order 200, represented by 100 lines of reflection. It also has 9 more cyclic symmetries as subgroups, and, john Conway labels these lower symmetries with a letter and order of the symmetry follows the letter. R200 represents full symmetry and a1 labels no symmetry and he gives d with mirror lines through vertices, p with mirror lines through edges, i with mirror lines through both vertices and edges, and g for rotational 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 star polygon, there are 19 regular forms given by Schläfli symbols, and, as well as 30 regular star figures with the same vertex configuration
Hectogon
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A regular hectogon
100.
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°. The area of a regular myriagon with sides of length a is given by A =2500 a 2 cot π10000 The result differs from the area of its circumscribed circle by up to 40 parts per billion. Because 10000 =24 ×54, the number of sides is neither a product of distinct Fermat primes nor a power of two, thus the regular myriagon is not a constructible polygon. Indeed, it is not even constructible with the use of neusis or an angle trisector, as the number of sides is neither a product of distinct Pierpont primes, the regular myriagon has Dih10000 dihedral symmetry, order 20000, represented by 10000 lines of reflection. Dih100 has 24 dihedral subgroups, and and it also has 25 more cyclic symmetries as subgroups, and, with Zn representing π/n radian rotational symmetry. John Conway labels these symmetries with a letter and order of the symmetry follows the letter. R20000 represents full symmetry, and a1 labels no symmetry and he gives d with mirror lines through vertices, p with mirror lines through edges, i with mirror lines through both vertices and edges, and g for rotational 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 star polygon, there are 1999 regular forms given by Schläfli symbols of the form, where n is an integer between 2 and 5000 that is coprime to 10000. There are also 3000 regular star figures in the remaining cases
Myriagon
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A regular myriagon
101.
65537-gon
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In geometry, a 65537-gon is a polygon with 65537 sides. The sum of the angles of any non-self-intersecting 65537-gon is 11796300°. The regular 65537-gon is of interest for being a constructible polygon, that is, it can be constructed using a compass and this is because 65537 is a Fermat prime, being of the form 22n +1. Thus, the values cos π65537 and cos 2 π65537 are of a 32768-degree algebraic numbers, although it was known to Gauss by 1801 that the regular 65537-gon was constructible, the first explicit construction of a regular 65537-gon was given by Johann Gustav Hermes. The construction is complex, Hermes spent 10 years completing the 200-page manuscript. Another method involves the use of at most 1332 Carlyle circles, and this method faces practical problems, as one of these Carlyle circles solves the quadratic equation x2 + x −16384 =0. The regular 65537-gon has Dih65537 symmetry, order 131074, since 65537 is a prime number there is one subgroup with dihedral symmetry, Dih1, and 2 cyclic group symmetries, Z65537, and Z1. A 65537-gram is a 65537-sided star polygon, as 65537 is prime, there are 32767 regular forms generated by Schläfli symbols for all integers 2 ≤ n ≤32768 as ⌊655372 ⌋ =32768. New York, Dover, p.53,1991, benjamin Bold, Famous Problems of Geometry and How to Solve Them New York, Dover, p.70,1982. ISBN 978-0486242972 H. S. M. Coxeter Introduction to Geometry, chapter 2, Regular polygons Leonard Eugene Dickson Constructions with Ruler and Compasses, Regular Polygons Ch.8 in Monographs on Topics of Modern Mathematics Relevant to the Elementary Field. New York, Dover, pp. 352–386,1955, 65537-gon mathematik-olympiaden. de, with images of the documentation HERMES, retrieved on April 25,2016 Wikibooks 65573-Eck Approximate construction of the first side in two main steps
65537-gon
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A regular 65537-gon
102.
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 difficult to distinguish from a circle. A regular megagon has an angle of 179. 99964°. The area of a regular megagon with sides of length a is given by A =250000 a 2 cot π1000000, the perimeter of a regular megagon inscribed in the unit circle is,2000000 sin π1000000, which is very close to 2π. In fact, for a circle the size of the Earths equator, with a circumference of 40,075 kilometres, the difference between the perimeter of the inscribed megagon and the circumference of this circle comes to less than 1/16 millimeters. Because 1000000 =26 ×56, the number of sides is not a product of distinct Fermat primes, thus the regular megagon is not a constructible polygon. Indeed, it is not even constructible with the use of neusis or an angle trisector, as the number of sides is neither a product of distinct Pierpont primes, nor a product of powers of two and three. Like René Descartes example of the chiliagon, the polygon has been used as an illustration of a well-defined concept that cannot be visualised. The megagon is also used as an illustration of the convergence of regular polygons to a circle, the regular megagon has Dih1000000 dihedral symmetry, order 2000000, represented by 1000000 lines of reflection. Dih100 has 48 dihedral subgroups, and and it also has 49 more cyclic symmetries as subgroups, and, with Zn representing π/n radian rotational symmetry. John Conway labels these symmetries with a letter and order of the symmetry follows the letter. R2000000 represents full symmetry and a1 labels no symmetry and he gives d with mirror lines through vertices, p with mirror lines through edges, i with mirror lines through both vertices and edges, and g for rotational symmetry. These lower symmetries allows degrees of freedom in defining irregular megagons, only the g1000000 subgroup has no degrees of freedom but can seen as directed edges. A megagram is a star polygon. There are 199,999 regular forms given by Schläfli symbols of the form, there are also 300,000 regular star figures in the remaining cases
Megagon
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A regular megagon
103.
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 a polygon as n approaches infinity. The interior of an apeirogon can be defined by a direction order of vertices. This article describes an apeirogon in its form as a tessellation or partition of a line. A regular apeirogon has equal edge lengths, just like any regular polygon and its Schläfli symbol is, and its Coxeter-Dynkin diagram is. It is the first in the family of regular hypercubic honeycombs. This line may be considered as a circle of radius, by analogy with regular polygons with great number of edges. In two dimensions, a regular apeirogon divides the plane into two half-planes as a regular apeirogonal dihedron, the interior of an apeirogon can be defined by its orientation, filling one half plane. Dually the apeirogonal hosohedron has digon faces and a vertex figure. A truncated apeirogonal hosohedron becomes a apeirogonal prism, with each vertex bounded by two squares and an apeirogon, an alternated apeirogonal prism is a apeirogonal antiprism, with each vertex bounded by three triangles and an apeirogon. The regular apeirogon can also be seen as linear sets within 4 of the regular, uniform tilings, an isogonal apeirogon has a single type of vertex and alternates two types of edges. A quasiregular apeirogon is an isogonal apeirogon with equal edge lengths, an isotoxal apeirogon, being the dual of an isogonal one, has one type of edge, and two types of vertices, and is therefore geometrically identical to the regular apeirogon. It can be seen by drawing vertices in alternate colors. All of these will have half the symmetry of the regular apeirogon, Regular apeirogons that are scaled to converge at infinity have the symbol and exist on horocycles, while more generally they can exist on hypercycles. The regular tiling has regular apeirogon faces, hypercyclic apeirogons can also be isogonal or quasiregular, with truncated apeirogon faces, t, like the tiling tr, with two types of edges, alternately connecting to triangles or other apeirogons. Apeirogonal tiling Apeirogonal prism Apeirogonal antiprism Apeirohedron Circle Coxeter, H. S. M. Regular Polytopes, Regular polyhedra - old and new, Aequationes Math. 16 p. 1-20 Coxeter, H. S. M. & Moser, W. O. J. Generators, archived from the original on 4 February 2007
Apeirogon
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Regular apeirogon
104.
Pentagram
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A pentagram is the shape of a five-pointed star drawn with five straight strokes. The word pentagram comes from the Greek word πεντάγραμμον, from πέντε, five + γραμμή, the word pentacle is sometimes used synonymously with pentagram The word pentalpha is a learned modern revival of a post-classical Greek name of the shape. The pentagram is the simplest regular star polygon, the pentagram contains ten points and fifteen line segments. It is represented by the Schläfli symbol, like a regular pentagon, and a regular pentagon with a pentagram constructed inside it, the regular pentagram has as its symmetry group the dihedral group of order 10. The pentagram can be constructed by connecting alternate vertices of a pentagon and it can also be constructed as a stellation of a pentagon, by extending the edges of a pentagon until the lines intersect. Each intersection of edges sections the edges in the golden ratio, also, the ratio of the length of the shorter segment to the segment bounded by the two intersecting edges is φ. As the four-color illustration shows, r e d g r e e n = g r e e n b l u e = b l u e m a g e n t a = φ. The pentagram includes ten isosceles triangles, five acute and 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 obtuse 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, the tips of the five loops at the center of the figure have the same geometric relationship to one another as the five vertices, or points, of a pentagram. Groups of five intersections of curves, equidistant from the center, have the same geometric relationship. In early monumental Sumerian script, or cuneiform, a pentagram glyph served as a logogram for the word ub, meaning corner, angle, nook, the word Pentemychos was the title of the cosmogony of Pherecydes of Syros. Here, the five corners are where the seeds of Chronos are placed within the Earth in order for the cosmos to appear. The pentangle plays an important symbolic role in the 14th-century English poem Sir Gawain, heinrich Cornelius Agrippa and others perpetuated the popularity of the pentagram as a magic symbol, attributing the five neoplatonic elements to the five points, in typical Renaissance fashion. By the mid-19th century a distinction had developed amongst occultists regarding the pentagrams orientation. With a single point upwards it depicted spirit presiding over the four elements of matter, however, the influential writer Eliphas Levi called it evil whenever the symbol appeared the other way up. It is the goat of lust attacking the heavens with its horns and it is the sign of antagonism and fatality. It is the goat of lust attacking the heavens with its horns, faust, The pentagram thy peace doth mar
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).
105.
Heptagram
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A heptagram, septagram, or septogram 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 star polygon that can be drawn in two forms, as irreducible fractions, the two heptagrams are sometimes called the heptagram and the great heptagram. The previous one, the 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, 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 symbol in various modern pagan. Blue Star Wicca also uses the symbol, where it is referred to as a septegram, the second heptagram is a symbol of magical power in some pagan spiritualities. The heptagram is used by members of the otherkin subculture as an identifier. In alchemy, a star can refer to the seven planets which were known to ancient alchemists. The seven-pointed star is incorporated into the flags of the bands of the Cherokee Nation. The Bennington flag, a historical American Flag, has thirteen seven-pointed stars along with the numerals 76 in the canton, the Flag of Jordan contains a seven-pointed star. The Flag of Australia employs five heptagrams and one pentagram to depict the Southern Cross constellation, some old versions of the coat of arms of Georgia including the Georgian Soviet Socialist Republic used the heptagram as an element. A seven-pointed star is used as the badge in many sheriffs departments, the seven-pointed star is used as the logo for the international Danish shipping company A. P. Moller–Maersk Group, sometimes known simply as Maersk. In George R. R. Martins novel series A Song of Ice and Fire, Star polygon Stellated polygons Two-dimensional regular polytopes Bibliography Grünbaum, B. and G. C. Shephard, Tilings and Patterns, New York, W. H. Freeman & Co, polyhedra with Hollow Faces, Proc of NATO-ASI Conference on Polytopes
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
106.
Octagram
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In geometry, an octagram is an eight-angled star polygon. The name octagram combine a Greek numeral prefix, octa-, with the Greek suffix -gram, the -gram 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 a dihedral, Dih4, symmetry, The symbol Rub el Hizb is a Unicode glyph ۞ at U+06DE. Deeper truncations of the square can produce intermediate star polygon forms with equal spaced vertices. A truncated square is an octagon, t=, a quasitruncated square, inverted as, is an octagram, t=. The uniform star polyhedron stellated truncated hexahedron, t=t has octagram faces constructed from the cube in this way, there are two regular octagrammic star figures of the form, the first constructed as two squares =2, and second as four degenerate digons, =4. There are other isogonal and isotoxal compounds including rectangular and rhombic forms, an octagonal star can be seen as a concave hexadecagon, with internal intersecting geometry erased. It can also be dissected by radial lines, usage Rub el Hizb - Islamic character Star of Lakshmi - Indian character Auseklis – usage of regular octagram by Latvians Guñelve – representation of Venus in Mapuche iconography. Selburose – usage of regular octagram in Norwegian design Stars generally Star Stellated polygons Two-dimensional regular polytopes Grünbaum, shephard, Tilings and Patterns, New York, W. H. Freeman & Co. Polyhedra with Hollow Faces, Proc of NATO-ASI Conference on Polytopes, etc. ed T. Bisztriczky et al. John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 Weisstein, Eric W. Octagram
Octagram
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A regular octagram
107.
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, the name decagram combine a numeral prefix, deca-, with the Greek suffix -gram. The -gram suffix derives from γραμμῆς meaning a line, for a regular decagram with unit edge lengths, the proportions of the crossing points on each edge are as shown below. Decagrams have been used as one of the motifs in girih tiles. A regular decagram is a 10-sided polygram, represented by symbol, deeper truncations of the regular pentagon and 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
108.
Dodecagram
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A dodecagram is a star polygon that has 12 vertices. A regular dodecagram has the vertex arrangement as a regular dodecagon. The name dodecagram combine a numeral prefix, dodeca-, with the Greek suffix -gram, the -gram suffix derives from γραμμῆς meaning a line. A regular dodecagram can be seen as a hexagon, t=. Other isogonal variations with equal spaced vertices can be constructed with two edge lengths, there are four regular dodecagram star figures, =2, =3, =4, and =6. The first is a compound of two hexagons, the second is a compound of three squares, the third is a compound of four triangles, and the fourth is a compound of six straight-sided digons. Superimposing all the dodecagons and dodecagrams on each other – including the degenerate compound of six digons, 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. Shephard, Tilings and Patterns, New York, W. H. Freeman & Co, polyhedra with Hollow Faces, Proc of NATO-ASI Conference on Polytopes. Etc. ed T. Bisztriczky et al, john H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, ISBN 978-1-56881-220-5
Dodecagram
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A regular dodecagram
109.
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 representations, with dimension 7 and 14. The Lie algebra g 2, being the smallest exceptional simple Lie algebra, was the first of these to be discovered in the attempt to classify simple Lie algebras. On May 23,1887, Wilhelm Killing wrote a letter to Friedrich Engel saying that he had found a 14-dimensional simple Lie algebra, in the same year, in the same journal, Engel noticed the same thing. Later it was discovered that the 2-dimensional distribution is related to a ball rolling on another ball. The space of configurations of the ball is 5-dimensional, with a 2-dimensional distribution that describes motions of the ball where it rolls without slipping or twisting. In 1900, Engel discovered that a generic antisymmetric trilinear form on a 7-dimensional complex vector space is preserved by a group isomorphic to the form of G2. In 1908 Cartan mentioned that the group of the octonions is a 14-dimensional simple Lie group. In 1914 he stated that this is the real form of G2. In older books and papers, G2 is sometimes denoted by E2, there are 3 simple real Lie algebras associated with this root system, The underlying real Lie algebra of the complex Lie algebra G2 has dimension 28. It has complex conjugation as an automorphism and is simply connected. The maximal compact subgroup of its associated group is the form of G2. The Lie algebra of the form is 14-dimensional. The associated Lie group has no outer automorphisms, no center, the Lie algebra of the non-compact form has dimension 14. The associated simple Lie group has fundamental group of order 2 and its maximal compact subgroup is SU × SU/. It has a double cover that is simply connected. The Dynkin diagram for G2 is given by and its Cartan matrix is, Although they span a 2-dimensional space, as drawn, it is much more symmetric to consider them as vectors in a 2-dimensional subspace of a three-dimensional space
G2 (mathematics)
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Algebraic structure → Group theory Group theory
110.
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 and it is a 4-simplex, the simplest possible convex regular 4-polytope, and is analogous to the tetrahedron in three dimensions and the triangle in two dimensions. The pentachoron is a four dimensional pyramid with a tetrahedral base, the regular 5-cell is bounded by regular tetrahedra, and is one of the six regular convex 4-polytopes, represented by Schläfli symbol. Pentachoron 4-simplex Pentatope Pentahedroid Pen Hyperpyramid, tetrahedral pyramid The 5-cell is self-dual and its maximal intersection with 3-dimensional space is the triangular prism. Its dihedral angle is cos−1, or approximately 75. 52°, the 5-cell can be constructed from a tetrahedron 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 edge length 2√2, a 5-cell can be constructed as a Boerdijk–Coxeter helix of five chained tetrahedra, folded into a 4-dimensional ring. The 10 triangle faces can be seen in a 2D net within a triangular tiling, with 6 triangles around every vertex, the purple edges represent the Petrie polygon of the 5-cell. The A4 Coxeter plane projects the 5-cell into a regular pentagon, the four sides of the pyramid are made of tetrahedron cells. Many uniform 5-polytopes have tetrahedral pyramid vertex figures, Other uniform 5-polytopes have irregular 5-cell vertex figures, the symmetry of a vertex figure of a uniform polytope is represented by removing the ringed nodes of the Coxeter diagram. The compound of two 5-cells in dual configurations can be seen in this A5 Coxeter plane projection, with a red and this compound has symmetry, order 240. The intersection of these two 5-cells is a uniform birectified 5-cell, the pentachoron is the simplest of 9 uniform polychora constructed from the Coxeter group. It is in the sequence of regular polychora, the tesseract, 120-cell, of Euclidean 4-space, all of these have a tetrahedral vertex figure. It is similar to three regular polychora, the tesseract, 600-cell of Euclidean 4-space, and the order-6 tetrahedral honeycomb of hyperbolic space, all of these have a tetrahedral cell. 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, 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
5-cell
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Vertex figure: tetrahedron
5-cell
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Schlegel diagram (vertices and edges)
111.
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 consists of six square faces. The tesseract is one of the six convex regular 4-polytopes, the tesseract is also called an 8-cell, C8, octachoron, octahedroid, cubic prism, and tetracube. It is the four-dimensional hypercube, or 4-cube as a part of the family of hypercubes or measure polytopes. In this publication, as well as some of Hintons later work, 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 as a composite Schläfli symbol ×, with symmetry order 96. As a 4-4 duoprism, 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 by composite Schläfli symbol × × × or 4, with symmetry order 16. Since each vertex of a tesseract is adjacent to four edges, the dual polytope of the tesseract is called the hexadecachoron, or 16-cell, with Schläfli symbol. The standard tesseract in Euclidean 4-space is given as the hull of the points. That is, it consists of the points, A tesseract is bounded by eight hyperplanes, each pair of non-parallel hyperplanes intersects to form 24 square faces in a tesseract. Three cubes and three squares intersect at each edge, there are four cubes, six squares, and four edges meeting at every vertex. All in all, it consists of 8 cubes,24 squares,32 edges, the construction of a hypercube can be imagined the following way, 1-dimensional, Two points A and B can be connected to a line, giving a new line segment AB. 2-dimensional, Two parallel line segments AB and CD can be connected to become a square, 3-dimensional, Two parallel squares ABCD and EFGH can be connected to become a cube, with the corners marked as ABCDEFGH. 4-dimensional, Two parallel cubes ABCDEFGH and IJKLMNOP can be connected to become a hypercube and it is possible to project tesseracts into three- or two-dimensional spaces, as projecting a cube is possible on a two-dimensional space. Projections on the 2D-plane become more instructive by rearranging the positions of the projected vertices, the scheme is similar to the construction of a cube from two squares, juxtapose two copies of the lower-dimensional cube and connect the corresponding vertices. Each edge of a tesseract is of the same length, the regular complex polytope 42, in C2 has a real representation as a tesseract or 4-4 duoprism in 4-dimensional space. 42 has 16 vertices, and 8 4-edges and its symmetry is 42, order 32. It also has a lower construction, or 4×4, with symmetry 44
Tesseract
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Schlegel diagram
112.
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, dodecacontachoron and hecatonicosahedroid, the boundary of the 120-cell is composed of 120 dodecahedral cells with 4 meeting at each vertex. It can be thought of as the 4-dimensional analog of the dodecahedron and has called a dodecaplex. Just as a dodecahedron can be built up as a model with 12 pentagons,3 around each vertex, there are 120 cells,720 pentagonal faces,1200 edges, and 600 vertices. There are 4 dodecahedra,6 pentagons, and 4 edges meeting at every vertex, there are 3 dodecahedra and 3 pentagons meeting every edge. The dual polytope of the 120-cell is the 600-cell, the vertex figure of the 120-cell is a tetrahedron. The dihedral angle of the 120-cell is 144° The 600 vertices of the 120-cell include all permutations of, 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 in a straight line bent in the 4th direction 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 stereographic projection, and a structure of intertwining rings. The cell locations lend themselves to a hyperspherical description, pick an arbitrary cell and label it the North Pole. Twelve great circle meridians radiate out in 3 dimensions, converging at the 5th South Pole cell and this skeleton accounts for 50 of the 120 cells. Starting at the North Pole, we can build up the 120-cell in 9 latitudinal layers, with the exception of the poles, each layer represents a separate 2-sphere, with the equator being a great 2-sphere. The centroids of the 30 equatorial cells form the vertices of an icosidodecahedron, the cells labeled interstitial in the following table do not fall on meridian great circles. Layers 2,4,6 and 8 cells are located over the cells faces. Layers 3 and 7s cells are located directly over the pole cells vertices, layer 5s cells are located over the pole cells edges. The 120-cell can be partitioned into 12 disjoint 10-cell great circle rings, starting with one 10-cell ring, one can place another ring alongside it that spirals around the original ring one complete revolution in ten cells. Five such 10-cell rings can be placed adjacent to the original 10-cell ring, although the outer rings spiral around the inner ring, they actually have no helical torsion. The spiraling is a result of the 3-sphere curvature, the inner ring and the five outer rings now form a six ring, 60-cell solid torus
120-cell
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Two intertwining rings of the 120-cell.
120-cell
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Schlegel diagram (vertices and edges)
113.
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, hexacosichoron and hexacosidedroid, the 600-cell is regarded as the 4-dimensional analog of the icosahedron, since it has five tetrahedra meeting at every edge, just as the icosahedron has five triangles meeting at every vertex. It is also called a tetraplex and polytetrahedron, being bounded by tetrahedral cells and its boundary is composed of 600 tetrahedral cells with 20 meeting at each vertex. Together they form 1200 triangular faces,720 edges, and 120 vertices, the edges form 72 flat 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 vertex figure is an icosahedron, and its dual polytope is the 120-cell and it has a dihedral angle of cos−1 = ~164. 48°. Each cell touches, in manner,56 other cells. One cell contacts each of the four faces, two cells contact each of the six edges, but not a face, and ten cells contact each of the four vertices, but not a face or edge. The vertices of a 600-cell centered at the origin of 4-space, with edges of length 1/φ, can be given as follows,16 vertices of the form, the remaining 96 vertices are obtained by taking even permutations of ½. Note that the first 16 vertices are the vertices of a tesseract, the eight are the vertices of a 16-cell. The final 96 vertices are the vertices of a snub 24-cell, when interpreted as quaternions, the 120 vertices of the 600-cell form a group under quaternionic multiplication. This group is 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 elements of 2IL and 2IR. The centre of RSG consists of the non-rotation Id and the central inversion -Id and we have the isomorphism RSG ≅ /. The order of RSG equals 120 ×120 /2 =7200, the binary icosahedral group is isomorphic to SL. The full symmetry group of the 600-cell is the Weyl group of H4 and this is a group of order 14400. It consists of 7200 rotations and 7200 rotation-reflections, the rotations form an invariant subgroup of the full symmetry group. The rotational symmetry group was described by S. L. van Oss, one can start by realizing the 600-cell is the dual of the 120-cell
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)
114.
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 first being regular with Schläfli symbol, and the second with alternately labeled facets and it is a part of an infinite family of polytopes, called cross-polytopes or orthoplexes. The dual polytope is the 5-hypercube or 5-cube, pentacross, derived from combining the family name cross polytope with pente for five in Greek. Triacontaditeron - as a 32-facetted 5-polytope and this polytope is one of 31 uniform 5-polytopes generated from the B5 Coxeter plane, including the regular 5-cube and 5-orthoplex. Coxeter, Regular Polytopes, 3rd Edition, Dover New York,1973 Kaleidoscopes, 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, 5D uniform polytopes x3o3o3o4o - tac. Archived from the original on 4 February 2007, Polytopes of Various Dimensions Multi-dimensional Glossary
5-orthoplex
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Orthogonal projection inside Petrie polygon
115.
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 semiregular 5-polytope, he called it a 5-ic semi-regular. E. L. Elte identified it in 1912 as a semiregular polytope, Coxeter named this polytope as 121 from its Coxeter diagram, which has branches of length 2,1 and 1 with a ringed node on one of the short branches, and Schläfli symbol or. It exists in the k21 polytope family as 121 with the Gosset polytopes,221,321, the graph formed by the vertices and edges of the demipenteract is sometimes called the Clebsch graph, though that name sometimes refers to the folded cube graph of order five instead. Cartesian coordinates for the vertices of a demipenteract centered at the origin and edge length 2√2 are alternate halves of the penteract and it is a part of a dimensional family of uniform polytopes called demihypercubes for being alternation of the hypercube family. There are 23 Uniform 5-polytopes that can be constructed from the D5 symmetry of the demipenteract,8 of which are unique to this family, the 5-demicube is third in a dimensional series of semiregular polytopes. Each progressive uniform polytope is constructed vertex figure of the previous polytope, Thorold Gosset identified this series in 1900 as containing all regular polytope facets, containing all simplexes and orthoplexes. In Coxeters notation the 5-demicube is given the symbol 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, 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. 5D uniform polytopes x3o3o *b3o3o - hin, archived from the original on 4 February 2007
5-demicube
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Petrie polygon projection
116.
6-cube
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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, and 12 5-cube 5-faces. It has Schläfli symbol, being composed of 3 5-cubes around each 4-face and it can be called a hexeract, a portmanteau of tesseract with hex for six in Greek. It can also be called a regular dodeca-6-tope or dodecapeton, being a 6-dimensional polytope constructed from 12 regular facets and it is a part of an infinite family of polytopes, called hypercubes. The dual of a 6-cube can be called a 6-orthoplex, and is a part of the family of cross-polytopes. Applying an alternation operation, 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. Cartesian coordinates for the vertices of a 6-cube centered at the origin and this polytope is one of 63 Uniform 6-polytopes generated from the B6 Coxeter plane, including the regular 6-cube or 6-orthoplex. Regular Polytopes, Dover edition, ISBN 0-486-61480-8 p.296, Table I, Regular Polytopes, 6D uniform polytopes o3o3o3o3o4x - ax. Archived from the original on 4 February 2007
6-cube
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Hexeract Quasicrystal structure orthographically projected to 3D using the Golden Ratio.
6-cube
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Orthogonal projection inside Petrie polygon Orange vertices are doubled, and the center yellow has 4 vertices
117.
1 22 polytope
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In 6-dimensional geometry, the 122 polytope is a uniform polytope, constructed from the E6 group. It was first published in E. L. Eltes 1912 listing of semiregular polytopes and its Coxeter symbol is 122, describing its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of the 1-node sequence. There are two rectifications of the 122, construcated by positions 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 face centers of the 122. The 1_22 polytope contains 72 vertices, and 54 5-demicubic facets and it has a birectified 5-simplex vertex figure. Its 72 vertices represent the vectors of the simple Lie group 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 facet information can be extracted from its Coxeter-Dynkin diagram. Removing the node on either of 2-length branches leaves the 5-demicube,131, the vertex figure is determined by removing the ringed node and ringing the neighboring node. This makes the birectified 5-simplex,022, the regular complex polyhedron 332, in C2 has a real representation as the 122 polytope in 4-dimensional space. It has 72 vertices,216 3-edges, and 5433 faces and its complex reflection group is 332, order 1296. It has a half-symmetry quasiregular construction as, as a rectification of the Hessian polyhedron, the 122 is related to the 24-cell by a geometric folding E6 → F4 of Coxeter-Dynkin diagrams, E6 corresponding to 122 in 6 dimensions, F4 to the 24-cell in 4 dimensions. This can be seen in the Coxeter plane projections, the 24 vertices of the 24-cell are projected in the same two rings as seen in the 122. This polytope is the figure for a uniform tessellation of 6-dimensional space,222. The rectified 122 polytope can tessellate 6-dimensional space as the Voronoi cell of the E6* honeycomb lattice, removing the ring on the short branch leaves the birectified 5-simplex. Removing the ring on the either 2-length branch leaves the birectified 5-orthoplex in its alternated form, the vertex figure is determined by removing the ringed node and ringing the neighboring ring. This makes 3-3 duoprism prism, ××, Vertices are colored by their multiplicity in this projection, in progressive order, red, orange, yellow. Truncated 122 polytope Its construction is based on the E6 group, Vertices are colored by their multiplicity in this projection, in progressive order, red, orange, yellow. Bicantellated 221 Birectified pentacontitetrapeton Vertices are colored by their multiplicity in this projection, in order, red, orange
1 22 polytope
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1 22
118.
Uniform 7-polytope
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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 which is vertex-transitive, and constructed from uniform 6-polytope facets. Regular 7-polytopes are represented by the Schläfli symbol with u 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, 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 distinguish between different topologies in higher dimensions led to the discovery of the more sophisticated Betti numbers. Similarly, the notion of orientability of a polyhedron is insufficient to characterise the surface twistings of toroidal polytopes, There are 71 forms based on all permutations of the Coxeter-Dynkin diagrams with one or more rings. Norman Johnsons 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, see also a list of B7 polytopes for symmetric Coxeter plane graphs of these polytopes. The D7 family has symmetry of order 322560 and this family has 3×32−1=95 Wythoffian uniform polytopes, generated by marking one or more nodes of the D7 Coxeter-Dynkin diagram. Of these,63 are repeated from the B7 family and 32 are unique to this family, bowers names and acronym are given for cross-referencing. See also list of D7 polytopes for Coxeter plane graphs of these polytopes, the E7 Coxeter group has order 2,903,040. There are 127 forms based on all permutations of the Coxeter-Dynkin diagrams with one or more rings, see also a list of E7 polytopes for symmetric Coxeter plane graphs of these polytopes. Coxeter calls the first one a quarter 6-cubic honeycomb, however, there are 3 noncompact hyperbolic Coxeter groups of rank 7, each generating uniform honeycombs in 6-space as permutations of rings of the Coxeter diagrams. The reflective 7-dimensional uniform polytopes are constructed through a Wythoff construction process, and represented by a Coxeter-Dynkin diagram, an active mirror is represented by a ringed node. Each combination of active mirrors generates a unique uniform polytope, Uniform polytopes are named in relation to the regular polytopes in each family. Some families have two regular constructors and thus may be named in two equally valid ways, here are the primary operators available for constructing and naming the uniform 7-polytopes. The prismatic forms and bifurcating graphs can use the same truncation indexing notation, T. Gosset, On the Regular and Semi-Regular Figures in Space of n Dimensions, Messenger of Mathematics, Macmillan,1900 A. S. M
Uniform 7-polytope
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7-simplex
119.
7-cube
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In geometry, a 7-cube is a seven-dimensional hypercube with 128 vertices,448 edges,672 square faces,560 cubic cells,280 tesseract 4-faces,84 penteract 5-faces, and 14 hexeract 6-faces. It can be named by its Schläfli symbol, being composed of 3 6-cubes around each 5-face and it can be called a hepteract, a portmanteau of tesseract and hepta for seven in Greek. It can also be called a regular tetradeca-7-tope or tetradecaexon, being a 7 dimensional polytope constructed from 14 regular facets and it is a part of an infinite family of polytopes, called hypercubes. The dual of a 7-cube is called a 7-orthoplex, and is a part of the family of cross-polytopes. Applying an alternation operation, deleting alternating vertices of the hepteract, creates another uniform polytope, called a demihepteract, which has 14 demihexeractic and 64 6-simplex 6-faces. Cartesian coordinates for the vertices of a hepteract centered at the origin, hepteract 7D simple rotation through 2Pi with 7D perspective projection to 3D. 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, 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, 7D uniform polytopes o3o3o3o3o3o4x - hept. Archived from the original on 4 February 2007, multi-dimensional Glossary, hypercube Garrett Jones Rotation of 7D-Cube www. 4d-screen. de
7-cube
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Orthogonal projection inside Petrie polygon The central orange vertex is doubled
120.
3 21 polytope
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In 7-dimensional geometry, the 321 polytope is a uniform 7-polytope, constructed within the symmetry of the E7 group. It was discovered by Thorold Gosset, published in his 1900 paper and he called it an 7-ic semi-regular figure. Its Coxeter symbol is 321, describing its bifurcating Coxeter-Dynkin diagram, the rectified 321 is constructed by points at the mid-edges of the 321. The birectified 321 is constructed by points at the face centers of the 321. The trirectified 321 is constructed by points at the centers of the 321. In 7-dimensional geometry, the 321 is a uniform polytope and it has 56 vertices, and 702 facets,126311 and 576 6-simplexes. For visualization this 7-dimensional polytope is often displayed in a special skewed orthographic projection direction that fits its 56 vertices within an 18-gonal regular polygon and its 756 edges are drawn between 3 rings of 18 vertices, and 2 vertices in the center. Specific higher elements can also be extracted and 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 331 and it is also called the Hess polytope for Edmund Hess who first discovered it. It was enumerated by Thorold Gosset in his 1900 paper and he called it an 7-ic semi-regular figure. E. L. Elte named it V56 in his 1912 listing of semiregular polytopes. Coxeter called it 321 due to its bifurcating Coxeter-Dynkin diagram, having 3 branches of length 3,2, and 1, Coxeter named it as 321 by its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of the 3-node sequence. The facet information can be extracted from its Coxeter-Dynkin diagram, removing the node on the short branch leaves the 6-simplex. Removing the node on the end of the 2-length branch leaves the 6-orthoplex in its form,311. Every simplex facet touches an 6-orthoplex facet, while alternate facets of the orthoplex touch either a simplex or another orthoplex, the vertex figure is determined by removing the ringed node and ringing the neighboring node. The 321 is fifth in a series of semiregular polytopes. Each progressive uniform polytope is constructed vertex figure of the previous polytope, Thorold Gosset identified this series in 1900 as containing all regular polytope facets, containing all simplexes and orthoplexes. It is in a series of uniform polytopes and honeycombs
3 21 polytope
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3 21
121.
Uniform 8-polytope
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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 which is vertex-transitive, and constructed from uniform 7-polytope facets. Regular 8-polytopes can be represented by the Schläfli symbol, with v 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, the value of the Euler characteristic used to characterise polyhedra does not generalize usefully to higher dimensions, and is zero for all 8-polytopes, whatever their underlying topology. This inadequacy of the Euler characteristic to distinguish between different topologies in higher dimensions led to the discovery of the more sophisticated Betti numbers. Similarly, the notion of orientability of a polyhedron is insufficient to characterise the surface twistings of toroidal polytopes, There are 135 forms based on all permutations of the Coxeter-Dynkin diagrams with one or more rings. 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. This family has 191 Wythoffian uniform polytopes, from 3x64-1 permutations of the D8 Coxeter-Dynkin diagram with one or more rings,127 are repeated from the B8 family and 64 are unique to this family, all listed below. See list of D8 polytopes for Coxeter plane graphs of these polytopes, the E8 family has symmetry order 696,729,600. There are 255 forms based on all permutations of the Coxeter-Dynkin diagrams with one or more rings, eight forms are shown below,4 single-ringed,3 truncations, and the final omnitruncation are given below. Bowers-style acronym names are given for cross-referencing, see also list of E8 polytopes for Coxeter plane graphs of this family. However, there are 4 noncompact hyperbolic Coxeter groups of rank 8, T. Gosset, On the Regular and Semi-Regular Figures in Space of n Dimensions, Messenger of Mathematics, Macmillan,1900 A. S. M. Miller, Uniform Polyhedra, Philosophical Transactions of the Royal Society of London, Coxeter, Regular Polytopes, 3rd Edition, Dover New York,1973 Kaleidoscopes, Selected Writings of H. S. M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, thompson, Asia Ivic Weiss, Wiley-Interscience Publication,1995, ISBN 978-0-471-01003-6 Wiley, Kaleidoscopes, Selected Writings of 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, N. W, johnson, The Theory of Uniform Polytopes and Honeycombs, Ph. D
Uniform 8-polytope
122.
8-simplex
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In geometry, an 8-simplex is a self-dual regular 8-polytope. It has 9 vertices,36 edges,84 triangle faces,126 tetrahedral cells,126 5-cell 4-faces,84 5-simplex 5-faces,36 6-simplex 6-faces and its dihedral angle is cos−1, or approximately 82. 82°. It can also be called an enneazetton, or ennea-8-tope, as a 9-facetted polytope in eight-dimensions, the name enneazetton is derived from ennea for nine facets in Greek and -zetta for having seven-dimensional facets, and -on. This construction is based on facets of the 9-orthoplex and this polytope is a facet in the uniform tessellations,251, and 521 with respective Coxeter-Dynkin diagrams, This polytope is one of 135 uniform 8-polytopes with A8 symmetry. 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, 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. 8D uniform polytopes x3o3o3o3o3o3o3o - ene, Polytopes of Various Dimensions Multi-dimensional Glossary
8-simplex
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Orthogonal projection inside Petrie polygon
123.
8-cube
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In geometry, an 8-cube is an eight-dimensional hypercube. It has 256 vertices,1024 edges,1792 square faces,1792 cubic cells,1120 tesseract 4-faces,448 5-cube 5-faces,112 6-cube 6-faces and it is represented by Schläfli symbol, being composed of 3 7-cubes around each 6-face. It is called an octeract, a portmanteau of tesseract and oct for eight in Greek and it can also be called a regular hexdeca-8-tope or hexadecazetton, being an 8-dimensional polytope constructed from 16 regular facets. It is a part of an family of polytopes, called hypercubes. The dual of an 8-cube can be called a 8-orthoplex, and is a part of the family of cross-polytopes. Cartesian coordinates for the vertices of an 8-cube centered at the origin, applying an alternation operation, deleting alternating vertices of the octeract, creates another uniform polytope, called a 8-demicube, which has 16 demihepteractic and 128 8-simplex facets. Coxeter, Coxeter, Regular Polytopes, Dover edition, ISBN 0-486-61480-8, p.296, Table I, Regular Polytopes, 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, 8D uniform polytopes o3o3o3o3o3o3o4x - octo. Archived from the original on 4 February 2007
8-cube
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Orthogonal projection inside Petrie polygon
124.
2 41 polytope
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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 bifurcating Coxeter-Dynkin diagram, the rectified 241 is constructed by points at the mid-edges of the 241. The birectified 241 is constructed by points at the face centers of the 241. 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,69,120 edges and its vertex figure is a 7-demicube. This polytope is a facet in the uniform tessellation,251 with Coxeter-Dynkin diagram and it is named 241 by Coxeter for its bifurcating Coxeter-Dynkin diagram, 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 at the positions of the 240 vertices in the 421 polytope, the vertex figure is determined by removing the ringed node and ringing the neighboring node. Petrie polygon projections can be 12,18, or 30-sided based on the E6, E7, the 2160 vertices are all displayed, but lower symmetry forms have projected positions overlapping, shown as different colored vertices. For comparison, a B6 coxeter group is also shown, the rectified 241 is a rectification of the 241 polytope, with vertices positioned at the mid-edges of the 241. The facet information can be extracted from its Coxeter-Dynkin diagram, Removing the node on the short branch leaves the rectified 7-simplex. Removing the node on the end of the 4-length branch leaves the rectified 231, Removing the node on the end of the 2-length branch leaves the 7-demicube,141. The vertex figure is determined by removing the ringed node and ringing the neighboring node and this makes the rectified 6-simplex prism. Petrie polygon projections can be 12,18, or 30-sided based on the E6, E7, the 2160 vertices are all displayed, but lower symmetry forms have projected positions overlapping, shown as different colored vertices. For comparison, a B6 coxeter group is also shown, list of E8 polytopes Elte, E. L. The Semiregular Polytopes of the Hyperspaces, Groningen, University of Groningen H. S. M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York,1973 Kaleidoscopes, 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 III, Klitzing, Richard. X3o3o3o *c3o3o3o3o - bay, o3x3o3o *c3o3o3o3o - robay
2 41 polytope
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421
125.
9-simplex
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In geometry, a 9-simplex is a self-dual regular 9-polytope. It has 10 vertices,45 edges,120 triangle faces,210 tetrahedral cells,252 5-cell 4-faces,210 5-simplex 5-faces,120 6-simplex 6-faces,45 7-simplex 7-faces and its dihedral angle is cos−1, or approximately 83. 62°. It can also be called a decayotton, or deca-9-tope, as a 10-facetted polytope in 9-dimensions, the name decayotton is derived from deca for ten facets in Greek and -yott, having 8-dimensional facets, and -on. This construction is based on facets of the 10-orthoplex, 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, 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. 9D uniform polytopes x3o3o3o3o3o3o3o3o - day, Polytopes of Various Dimensions Multi-dimensional Glossary
9-simplex
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Orthogonal projection inside Petrie polygon
126.
9-orthoplex
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It has two constructed forms, the first being regular with Schläfli symbol, and the second with alternately labeled facets, with Schläfli symbol or Coxeter symbol 611. It is one of an family of polytopes, called cross-polytopes or orthoplexes. The dual polytope is the 9-hypercube or enneract, cartesian coordinates for the vertices of a 9-orthoplex, centered at the origin, are, Every vertex pair is connected by an edge, except opposites. Coxeter, Regular Polytopes, 3rd Edition, Dover New York,1973 Kaleidoscopes, 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, 9D uniform polytopes x3o3o3o3o3o3o3o4o - vee. Archived from the original on 4 February 2007, Polytopes of Various Dimensions Multi-dimensional Glossary
9-orthoplex
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Orthogonal projection inside Petrie polygon
127.
9-cube
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It can be named by its Schläfli symbol, being composed of three 8-cubes around each 7-face. It is also called an enneract, a portmanteau of tesseract and it can also be called a regular octadeca-9-tope or octadecayotton, as a nine-dimensional polytope constructed with 18 regular facets. It is a part of an family of polytopes, called hypercubes. The dual of a 9-cube can be called a 9-orthoplex, and is a part of the family of cross-polytopes. Cartesian coordinates for the vertices of a 9-cube centered at the origin, applying an alternation operation, 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. 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, 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, 9D uniform polytopes o3o3o3o3o3o3o3o4x - enne. Archived from the original on 4 February 2007
9-cube
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Orthogonal projection inside Petrie polygon Orange vertices are doubled, yellow have 4, and the green center has 8
128.
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 which is vertex-transitive, and constructed from uniform facets, Regular 10-polytopes can be represented by the Schläfli symbol, with x 9-polytope facets around each peak. There are exactly three convex regular 10-polytopes, - 10-simplex - 10-cube - 10-orthoplex There are no nonconvex regular 10-polytopes. The topology of any given 10-polytope is defined by its Betti numbers, the value of the Euler characteristic used to characterise polyhedra does not generalize usefully to higher dimensions, and is zero for all 10-polytopes, whatever their underlying topology. This inadequacy of the Euler characteristic to distinguish between different topologies in higher dimensions led to the discovery of the more sophisticated Betti numbers. Similarly, the notion of orientability of a polyhedron is insufficient to characterise the surface twistings of toroidal polytopes, 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, and 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, ten single-ring forms, and 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 3×256−1=767 Wythoffian uniform polytopes, generated by marking one or more nodes of the D10 Coxeter-Dynkin diagram, of these,511 are repeated from the B10 family and 256 are unique to this family, with 2 listed below. Bowers-style acronym names are given in parentheses for cross-referencing, however, there are 3 noncompact hyperbolic Coxeter groups of rank 9, each generating uniform honeycombs in 9-space as permutations of rings of the Coxeter diagrams. Miller, Uniform Polyhedra, Philosophical Transactions of the Royal Society of London, Coxeter, Regular Polytopes, 3rd Edition, Dover New York,1973 Kaleidoscopes, Selected Writings of H. S. M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, 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, N. W, johnson, The Theory of Uniform Polytopes and Honeycombs, Ph. D. Dissertation, University of Toronto,1966 Klitzing, Richard, polytope names Polytopes of Various Dimensions, Jonathan Bowers Multi-dimensional Glossary Glossary for hyperspace, George Olshevsky
Uniform 10-polytope
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10-simplex
129.
10-simplex
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In geometry, a 10-simplex is a self-dual regular 10-polytope. Its dihedral angle is cos−1, or approximately 84. 26° and it can also be called a hendecaxennon, or hendeca-10-tope, as an 11-facetted polytope in 10-dimensions. The name hendecaxennon is derived from hendeca for 11 facets in Greek and -xenn, having 9-dimensional facets and this construction is based on facets of the 11-orthoplex. The 2-skeleton of the 10-simplex is topologically related to the 11-cell abstract regular polychoron which has the same 11 vertices,55 edges, but only 1/3 the faces. 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, 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. 10D uniform polytopes x3o3o3o3o3o3o3o3o3o - ux, Polytopes of Various Dimensions Multi-dimensional Glossary
10-simplex
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Orthogonal projection inside Petrie polygon
130.
10-orthoplex
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It has two constructed forms, the first being regular with Schläfli symbol, and the second with alternately labeled facets, with Schläfli symbol or Coxeter symbol 711. It is one of an family of polytopes, called cross-polytopes or orthoplexes. The dual polytope is the 10-hypercube or 10-cube, decacross is derived from combining the family name cross polytope with deca for ten in Greek Chilliaicositetraxennon as a 1024-facetted 10-polytope. Cartesian coordinates for the vertices of a 10-orthoplex, centred at the origin are, Every vertex pair is connected by an edge, Coxeter, Regular Polytopes, 3rd Edition, Dover New York,1973 Kaleidoscopes, Selected Writings of H. S. M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, 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, Johnson, The Theory of Uniform Polytopes and Honeycombs, Ph. D. 10D uniform polytopes x3o3o3o3o3o3o3o3o4o - ka, archived from the original on 4 February 2007. Polytopes of Various Dimensions Multi-dimensional Glossary
10-orthoplex
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Orthogonal projection inside Petrie polygon
131.
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 polytope which is the convex hull of its k +1 vertices. More formally, suppose the k +1 points u 0, …, u k ∈ R k are affinely independent, then, the simplex determined by them is the set of points C =. For example, a 2-simplex is a triangle, a 3-simplex is a tetrahedron, a single point may be considered a 0-simplex, and 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 that is also a regular polytope. A regular n-simplex may be constructed from a regular -simplex by connecting a new vertex to all original vertices by the 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 abstract simplicial complex, in which context the word “simplex” simply means any finite set of vertices. A 1-simplex is a line segment, the convex hull of any nonempty subset of the n+1 points that define an n-simplex is called a face of the simplex. In particular, the hull of a subset of size m+1 is an m-simplex. The 0-faces are called the vertices, the 1-faces are called the edges, the -faces are called the facets, in general, the number of m-faces is equal to the binomial coefficient. Consequently, the number of m-faces of an n-simplex may be found in column of row of Pascals triangle, a simplex A is a coface of a simplex B if B is a face of A. Face and facet can have different meanings when describing types of simplices in a simplicial complex, see simplical complex for more detail. The regular simplex family is the first of three regular polytope families, labeled by Coxeter as αn, the two being the cross-polytope family, labeled as βn, and the hypercubes, labeled as γn. A fourth family, the infinite tessellation of hypercubes, he labeled as δn, an -simplex can be constructed as a join of an n-simplex and a point. An -simplex can be constructed as a join of an m-simplex, the two simplices are oriented to be completely normal from each other, with translation in a direction orthogonal to both of them. A 1-simplex is a joint of two points, ∨ =2, a general 2-simplex is the join of 3 points, ∨∨. An isosceles triangle is the join of a 1-simplex and a point, a general 3-simplex is the join of 4 points, ∨∨∨. A 3-simplex with mirror symmetry can be expressed as the join of an edge and 2 points, a 3-simplex with triangular symmetry can be expressed as the join of an equilateral triangle and 1 point,3. ∨ or ∨
Simplex
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A regular 3-simplex or tetrahedron
132.
Hypercube
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In geometry, a hypercube is an n-dimensional analogue of a square and a cube. A unit hypercubes longest diagonal in n-dimensions is equal to n, an n-dimensional hypercube is also called an n-cube or an n-dimensional cube. The term measure polytope is also used, notably in the work of H. S. M. Coxeter, the hypercube is the special case of a hyperrectangle. A unit hypercube is a hypercube whose side has one unit. Often, the hypercube whose corners are the 2n points in Rn with coordinates equal to 0 or 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,2 – If one moves this line segment 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 perpendicular to the plane it lies on. 4 – If one moves the cube one unit length into the fourth dimension and this can be generalized to any number of dimensions. The 1-skeleton of a hypercube is a hypercube graph, a unit hypercube of n dimensions is the convex hull of the points given by all sign permutations of the Cartesian coordinates. It has a length of 1 and an n-dimensional volume of 1. An n-dimensional hypercube is also regarded as the convex hull of all sign permutations of the coordinates. This form is chosen due to ease of writing out the coordinates. Its edge length is 2, and its volume is 2n. Every n-cube of n >0 is composed of elements, or n-cubes of a dimension, on the -dimensional surface on the parent hypercube. A side is any element of -dimension of the parent hypercube, a hypercube of dimension n has 2n sides. The number of vertices of a hypercube is 2 n, the number of m-dimensional hypercubes on the boundary of an n-cube is E m, n =2 n − m, where = n. m. and n. denotes the factorial of n. For example, the boundary of a 4-cube contains 8 cubes,24 squares,32 lines and 16 vertices and this identity can be proved by combinatorial arguments, each of the 2 n vertices defines a vertex in a m-dimensional boundary. There are ways of choosing which lines that defines the subspace that the boundary is in, but, each side is counted 2 m times since it has that many vertices, we need to divide with this number
Hypercube
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Perspective projections
133.
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 by its bifurcating Coxeter-Dynkin diagram and it can be named by an extended Schläfli symbol. The family starts uniquely as 6-polytopes, but can be extended backwards to include the 5-orthoplex in 5-dimensions, each polytope is constructed from -simplex and 2k-1,1 -polytope facets, each has a vertex figure as an -demicube. The sequence ends with k=6, as an infinite tessellation of 9-space. Geometrical Deduction of Semiregular from Regular Polytopes and Space Fillings, 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, Geometrical Deduction of Semiregular from Regular Polytopes and Space Fillings. Analytical treatment of the polytopes regularly derived from the regular polytopes, Ver. der Koninklijke Akad. van Wetenschappen te Amsterdam, H. S. M. Coxeter, Regular and Semi-Regular Polytopes, Part I, Mathematische Zeitschrift, Springer, Berlin,1940 N. W. Johnson, The Theory of Uniform Polytopes and Honeycombs, Ph. D, dissertation, University of Toronto,1966 H. S. M. Coxeter, Regular and Semi-Regular Polytopes, Part II, Mathematische Zeitschrift, Springer, Coxeter, Regular and Semi-Regular Polytopes, Part III, Mathematische Zeitschrift, Springer, Berlin,1988 PolyGloss v0.05, Gosset figures
Uniform 2 k1 polytope
134.
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 and it can be named by its Schläfli symbol as or. The family starts as 1-polytopes and ends with n =5 as infinite tessellations of 4-dimensional hyperbolic space, there are two types of pentagonal polytopes, they may be termed the dodecahedral and icosahedral types, by their three-dimensional members. The two types are duals of each other and their vertex figures are the simplices of one less dimension. Their vertex figures are icosahedral pentagonal polytopes of one less dimension, the pentagonal polytopes can be stellated to form new star regular polytopes, In three dimensions, this forms the four Kepler–Poinsot polyhedra, and. In four dimensions, this forms the ten Schläfli–Hess polychora, in four-dimensional hyperbolic space there are four regular star-honeycombs, and. Coxeter, edited by F. Arthur Sherk, Peter McMullen, 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
135.
List of regular polytopes and compounds
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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, a Schläfli symbol describing an n-polytope equivalently describes a tessellation of a -sphere. Another related symbol is the Coxeter-Dynkin diagram which represents a group with no rings. For example, the cube has Schläfli symbol, and with its octahedral symmetry, the regular polytopes are grouped by dimension and subgrouped by convex, nonconvex and infinite forms. Nonconvex forms use the same vertices as the 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 scale. This allows vertex figures to have negative angle defects, like making a vertex with seven equilateral triangles and it cannot be done in a regular plane, but can be at the right scale of a hyperbolic plane. A more general definition of regular polytopes which do not have simple Schläfli symbols includes regular skew polytopes and this table shows a summary of regular polytope counts by dimension. *1 if the number of dimensions is of the form 2k −1,2 if the number of dimensions is a power of two,0 otherwise, There are no Euclidean regular star tessellations in any number of dimensions. A one-dimensional polytope or 1-polytope is a line segment, bounded by its two endpoints. A 1-polytope is regular by definition and is represented by Schläfli symbol, norman Johnson calls it a ditel and gives it the Schläfli symbol. Although trivial as a polytope, it appears as the edges of polygons and it is used in the definition of uniform prisms like Schläfli symbol ×, or Coxeter diagram as a Cartesian product of a line segment and a regular polygon. The two-dimensional polytopes are called polygons, Regular polygons are equilateral and cyclic. A p-gonal regular polygon is represented by Schläfli symbol, usually only convex polygons are considered regular, but star polygons, like the pentagram, can also be considered regular. They use the same vertices as the forms, but connect in an alternate connectivity which passes around the circle more than once to complete. Star polygons should be called nonconvex rather than concave because the edges do not generate new vertices. The Schläfli symbol represents a regular p-gon, the regular digon can be considered to be a degenerate regular polygon
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
136.
Hexagon
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In geometry, a hexagon is a six sided polygon or 6-gon. The total of the angles of any hexagon is 720°. A regular hexagon has Schläfli symbol and can also be constructed as an equilateral triangle, t. A regular hexagon is defined as a hexagon that is both equilateral and equiangular and it is bicentric, meaning that it is both cyclic and tangential. The common length of the sides equals the radius of the circumscribed circle, all internal angles are 120 degrees. A regular hexagon has 6 rotational symmetries and 6 reflection symmetries, the longest diagonals of a regular hexagon, connecting diametrically opposite vertices, are twice the length of one side. Like squares and equilateral triangles, regular hexagons fit together without any gaps to tile the plane, the cells of a beehive honeycomb are hexagonal for this reason and because the shape makes efficient use of space and building materials. The Voronoi diagram of a triangular lattice is the honeycomb tessellation of hexagons. It is not usually considered a triambus, although it is equilateral, the maximal diameter, D is twice the maximal radius or circumradius, R, which equals the side length, t. The minimal diameter or the diameter of the circle, d, is twice the minimal radius or inradius. If a regular hexagon has successive vertices A, B, C, D, E, F, the regular hexagon has Dih6 symmetry, order 12. There are 3 dihedral subgroups, Dih3, Dih2, and Dih1, and 4 cyclic subgroups, Z6, Z3, Z2 and these symmetries express 9 distinct symmetries of a regular hexagon. John Conway labels these by a letter and group order, r12 is full symmetry, and a1 is no symmetry. These two forms are duals of each other and have half the order of the regular hexagon. The i4 forms are regular hexagons flattened or stretched along one symmetry direction and it can be seen as an elongated rhombus, while d2 and p2 can be seen as horizontally and vertically elongated kites. G2 hexagons, with sides parallel are also called hexagonal parallelogons. Each subgroup symmetry allows one or more degrees of freedom for irregular forms, only the g6 subgroup has no degrees of freedom but can seen as directed edges. Hexagons of symmetry g2, i4, and r12, as parallelogons can tessellate the Euclidean plane by translation, other hexagon shapes can tile the plane with different orientations
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