1.
Regular polygon
–
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
–
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
–
Regular convex and star polygons with 3 to 12 vertices labelled with their Schläfli symbols
2.
Edge (geometry)
–
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)
–
Three edges AB, BC, and CA, each between two
vertices of a
triangle.
3.
Dihedral symmetry
–
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
–
The
symmetry group of a
snowflake is Dih 6, a dihedral symmetry, the same as for a regular
hexagon.
4.
Dual polygon
–
In geometry, polygons are associated into pairs called duals, where the vertices of one correspond to the edges of the other. The dual of a polygon is an isotoxal polygon. For example, the rectangle and rhombus are duals, in a cyclic polygon, longer sides correspond to larger exterior angles in the dual, and shorter sides to smaller angles. Further, congruent sides in the original polygon yields congruent angles in the dual, for example, the dual of a highly acute isosceles triangle is an obtuse isosceles triangle. In the Dorman Luke construction, each face of a polyhedron is the dual polygon of the corresponding vertex figure. As an example of the duality of polygons we compare properties of the cyclic. This duality is perhaps more clear when comparing an isosceles trapezoid to a kite. The simplest qualitative construction of a polygon is a rectification operation. New edges are formed between these new vertices and that is, the polygon generated by applying it twice is in general not similar to the original polygon. As with dual polyhedra, one can take a circle and perform polar reciprocation in it. Combinatorially, one can define a polygon as a set of vertices, a set of edges, then the dual polygon is obtained by simply switching the vertices and edges. Thus for the triangle with vertices and edges, the triangle has vertices, and edges, where B connects AB & BC. This is not a particularly fruitful avenue, as combinatorially, there is a family of polygons, geometric duality of polygons is more varied. Dual curve Dual polyhedron Self-dual polygon Dual Polygon Applet by Don Hatch
Dual polygon
–
Dorman Luke construction, showing a
rhombus face being dual to a
rectangle vertex figure.
5.
Convex polygon
–
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
–
An example of a convex polygon: a
regular pentagon
6.
Isotoxal figure
–
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
–
A
rhombic dodecahedron is an isohedral and isotoxal polyhedron
Isotoxal figure
7.
Truncation (geometry)
–
In geometry, a truncation is an operation in any dimension that cuts polytope vertices, creating a new facet in place of each vertex. The term originates from Keplers names for the Archimedean solids, in general any polyhedron can also be truncated with a degree of freedom as to how deep the cut is, as shown in Conway polyhedron notation truncation operation. A special kind of truncation, usually implied, is a uniform truncation, there are no degrees of freedom, and it represents a fixed geometric, just like the regular polyhedra. In general all single ringed uniform polytopes have a uniform truncation, for example, the icosidodecahedron, represented as Schläfli symbols r or, and Coxeter-Dynkin diagram or has a uniform truncation, the truncated icosidodecahedron, represented as tr or t. In the Coxeter-Dynkin diagram, the effect of a truncation is to ring all the adjacent to the ringed node. A truncated n-sided polygon will have 2n sides, a regular polygon uniformly truncated will become another regular polygon, t is. A complete truncation, r, is another regular polygon in its dual position, a regular polygon can also be represented by its Coxeter-Dynkin diagram, and its uniform truncation, and its complete truncation. Star polygons can also be truncated, a truncated pentagram will look like a pentagon, but is actually a double-covered decagon with two sets of overlapping vertices and edges. A truncated great heptagram gives a tetradecagram and this sequence shows an example of the truncation of a cube, using four steps of a continuous truncating process between a full cube and a rectified cube. The final polyhedron is a cuboctahedron, the middle image is the uniform truncated cube. It is represented by a Schläfli symbol t, a bitruncation is a deeper truncation, removing all the original edges, but leaving an interior part of the original faces. The truncated octahedron is a cube, 2t is an example. A complete bitruncation is called a birectification that reduces original faces to points, for polyhedra, this becomes the dual polyhedron. An octahedron is a birectification of the cube, = 2r is an example, another type of truncation is called cantellation, cuts edge and vertices, removing original edges and replacing them with rectangles. Higher dimensional polytopes have higher truncations, runcination cuts faces, edges, in 5-dimensions sterication cuts cells, faces, and edges. Edge-truncation is a beveling or chamfer for polyhedra, similar to cantellation but retains original vertices, in 4-polytopes edge-truncation replaces edges with elongated bipyramid cells. Alternation or partial truncation only removes some of the original vertices, a partial truncation or alternation - Half of the vertices and connecting edges are completely removed. The operation only applies to polytopes with even-sided faces, faces are reduced to half as many sides, and square faces degenerate into edges
Truncation (geometry)
–
Truncated cubic honeycomb t{4,3,4} or
Truncation (geometry)
–
Truncated square is a regular octagon: t{4} = {8} =
8.
Intersection
–
In mathematics, the intersection of two or more objects is another, usually smaller object. All objects are presumed to lie in a common space except in set theory. The intersection is one of basic concepts of geometry, intuitively, the intersection of two or more objects is a new object that lies in each of original objects. An intersection can have various shapes, but a point is the most common in a plane geometry. It is always defined, but may be empty, incidence geometry defines an intersection as an object of lower dimension that is incident to each of original objects. In this approach an intersection can be sometimes undefined, such as for parallel lines, in both cases the concept of intersection relies on logical conjunction. Algebraic geometry defines intersections in its own way with the intersection theory, there can be more than one primitive object, such as points that form an intersection. It can be understood ambiguously, either the intersection is all of them, constructive solid geometry, Boolean Intersection is one of the ways of combining 2D/3D shapes Meet Weisstein, Eric W. Intersection
Intersection
–
This
circle (black) intersects this
line (purple) in two points
9.
Circumscribed circle
–
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
–
Circumscribed circle, C, and circumcenter, O, of a cyclic polygon, P
10.
Bicentric polygon
–
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
–
An
equilateral triangle
11.
Tangential polygon
–
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
–
A tangental trapezoid
12.
Apothem
–
The apothem of a regular polygon is a line segment from the center to the midpoint of one of its sides. Equivalently, it is the line drawn from the center of the polygon that is perpendicular to one of its sides, the word apothem can also refer to the length of that line segment. Regular polygons are the polygons that have apothems. Because of this, all the apothems in a polygon will be congruent. For a regular pyramid, which is a pyramid base is a regular polygon, the apothem is the slant height of a lateral face, that is. For a truncated pyramid, the apothem is the height of a trapezoidal lateral face. An apothem of a polygon will always be a radius of the inscribed circle. It is also the distance between any side of the polygon and its center. A = p a 2 = r 2 = π r 2 The apothem of a regular polygon can be multiple ways. The apothem a of a regular n-sided polygon with side length s, or circumradius R, can be using the following formula. The apothem can also be found by a = s 2 tan and these formulae can still be used even if only the perimeter p and the number of sides n are known because s = p n. Circumradius of a regular polygon Sagitta Chord Apothem of a regular polygon With interactive animation Apothem of pyramid or truncated pyramid Pegg, Jr. Ed
Apothem
–
Apothem of a
hexagon
13.
Angle
–
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
–
An angle enclosed by rays emanating from a vertex.
14.
Rotational symmetries
–
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
–
The starting position in
shogi
Rotational symmetries
–
The
triskelion appearing on the
Isle of Man flag.
15.
Reflection symmetries
–
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 symmetries
–
Many animals, such as this
spider crab Maja crispata, are bilaterally symmetric.
Reflection symmetries
–
Figures with the axes of
symmetry drawn in. The figure with no axes is
asymmetric.
Reflection symmetries
–
Mirror symmetry is often used in
architecture, as in the facade of
Santa Maria Novella,
Venice, 1470.
16.
Square (geometry)
–
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 (geometry)
–
A regular
quadrilateral (tetragon)
17.
Equilateral
–
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
–
A regular tetrahedron is made of four equilateral triangles.
Equilateral
–
Equilateral triangle
18.
Beehive (beekeeping)
–
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)
–
Painted wooden beehives with active honey bees
Beehive (beekeeping)
–
Natural beehive in the hollow of a tree
Beehive (beekeeping)
–
Hives from the collection of
Radomysl Castle,
Ukraine, 19th century
Beehive (beekeeping)
–
Beehives –
watercolour painted by
Stanisław Masłowski in
Wola Rafałowska village,
Poland in 1924,
Silesian Museum in
Katowice,
Poland
19.
Voronoi diagram
–
In mathematics, a Voronoi diagram is a partitioning of a plane into regions based on distance to points in a specific subset of the plane. That set of points is specified beforehand, and for each seed there is a region consisting of all points closer to that seed than to any other. These regions are called Voronoi cells, the Voronoi diagram of a set of points is dual to its Delaunay triangulation. It is named after Georgy Voronoi, and is called a Voronoi tessellation, a Voronoi decomposition. Voronoi diagrams have practical and theoretical applications to a number of fields, mainly in science and technology. They are also known as Thiessen polygons, in the simplest case, shown in the first picture, we are given a finite set of points in the Euclidean plane. Each such cell is obtained from the intersection of half-spaces, the line segments of the Voronoi diagram are all the points in the plane that are equidistant to the two nearest sites. The Voronoi vertices are the points equidistant to three sites, let X be a metric space with distance function d. Let K be a set of indices and let k ∈ K be a tuple of nonempty subsets in the space X. In other words, if d = inf denotes the distance between the point x and the subset A, then R k = The Voronoi diagram is simply the tuple of cells k ∈ K. In principle some of the sites can intersect and even coincide, in addition, infinitely many sites are allowed in the definition, but again, in many cases only finitely many sites are considered. Sometimes the induced combinatorial structure is referred to as the Voronoi diagram, however, in general the Voronoi cells may not be convex or even connected. In the usual Euclidean space, we can rewrite the definition in usual terms. Each Voronoi polygon R k is associated with a generator point P k, let X be the set of all points in the Euclidean space. Let P1 be a point that generates its Voronoi region R1, P2 that generates R2, and P3 that generates R3, and so on. Then, as expressed by Tran et al all locations in the Voronoi polygon are closer to the point of that polygon than any other generator point in the Voronoi diagram in Euclidian plane. As a simple illustration, consider a group of shops in a city, suppose we want to estimate the number of customers of a given shop. With all else being equal, it is reasonable to assume that customers choose their preferred shop simply by distance considerations, they will go to the shop located nearest to them
Voronoi diagram
–
John Snow's original diagram
Voronoi diagram
–
20 points and their Voronoi cells (larger version below).
Voronoi diagram
–
Periodic
Voronoi diagram
20.
Diagonal
–
In geometry, a diagonal is a line segment joining two vertices of a polygon or polyhedron, when those vertices are not on the same edge. Informally, any sloping line is called diagonal, in matrix algebra, a diagonal of a square matrix is a set of entries extending from one corner to the farthest corner. There are also other, non-mathematical uses, diagonal pliers are wire-cutting pliers defined by the cutting edges of the jaws intersects the joint rivet at an angle or on a diagonal, hence the name. A diagonal lashing is a type of lashing used to bind spars or poles together applied so that the cross over the poles at an angle. In association football, the system of control is the method referees. As applied to a polygon, a diagonal is a line segment joining any two non-consecutive vertices, therefore, a quadrilateral has two diagonals, joining opposite pairs of vertices. For any convex polygon, all the diagonals are inside the polygon, in a convex polygon, if no three diagonals are concurrent at a single point, the number of regions that the diagonals divide the interior into is given by + =24. The number of regions is 1,4,11,25,50,91,154,246, in a polygon with n angles the number of diagonals is given by n ∗2. The number of intersections between the diagonals is given by, in the case of a square matrix, the main or principal diagonal is the diagonal line of entries running from the top-left corner to the bottom-right corner. For a matrix A with row index specified by i and column index specified by j, the off-diagonal entries are those not on the main diagonal. A diagonal matrix is one whose off-diagonal entries are all zero, a superdiagonal entry is one that is directly above and to the right of the main diagonal. Just as diagonal entries are those A i j with j = i and this plays an important part in geometry, for example, the fixed points of a mapping F from X to itself may be obtained by intersecting the graph of F with the diagonal. In geometric studies, the idea of intersecting the diagonal with itself is common, not directly and this is related at a deep level with the Euler characteristic and the zeros of vector fields. For example, the circle S1 has Betti numbers 1,1,0,0,0, a geometric way of expressing this is to look at the diagonal on the two-torus S1xS1 and observe that it can move off itself by the small motion to. Topics In Algebra, Waltham, Blaisdell Publishing Company, ISBN 978-1114541016 Nering, linear Algebra and Matrix Theory, New York, Wiley, LCCN76091646 Diagonals of a polygon with interactive animation Polygon diagonal from MathWorld. Diagonal of a matrix from MathWorld
Diagonal
–
A stand of basic scaffolding on a house construction site, with diagonal braces to maintain its structure
Diagonal
–
The diagonals of a
cube with side length 1. AC' (shown in blue) is a
space diagonal with length, while AC (shown in red) is a
face diagonal and has length.
21.
Circumradius
–
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
–
Circumscribed circle, C, and circumcenter, O, of a cyclic polygon, P
22.
Inradius
–
In geometry, the incircle or inscribed circle of a triangle is the largest circle contained in the triangle, it touches the three sides. The center of the incircle is a center called the triangles incenter. An excircle or escribed circle of the triangle is a circle lying outside the triangle, tangent to one of its sides, every triangle has three distinct excircles, each tangent to one of the triangles sides. The center of the incircle, called the incenter, can be found as the intersection of the three angle bisectors. The center of an excircle is the intersection of the internal bisector of one angle, the center of this excircle is called the excenter relative to the vertex A, or the excenter of A. Because the internal bisector of an angle is perpendicular to its external bisector, polygons with more than three sides do not all have an incircle tangent to all sides, those that do are called tangential polygons. See also Tangent lines to circles, suppose △ A B C has an incircle with radius r and center I. The distance from vertex A to the incenter I is, d = c sin cos = b sin cos The trilinear coordinates for a point in the triangle is the ratio of distances to the triangle sides. Because the Incenter is the distance of all sides the trilinear coordinates for the incenter are 1,1,1. The barycentric coordinates for a point in a triangle give weights such that the point is the average of the triangle vertex positions. The Cartesian coordinates of the incenter are an average of the coordinates of the three vertices using the side lengths of the triangle relative to the perimeter—i. e. Using the barycentric coordinates given above, normalized to sum to unity—as weights. If the three vertices are located at, and, and the sides opposite these vertices have corresponding lengths a, b, additionally, I A ⋅ I B ⋅ I C =4 R r 2, where R and r are the triangles circumradius and inradius respectively. The collection of triangle centers may be given the structure of a group under multiplication of trilinear coordinates, in this group. Then the incircle has the radius r = x y z x + y + z, the product of the incircle radius r and the circumcircle radius R of a triangle with sides a, b, and c is r R = a b c 2. Some relations among the sides, incircle radius, and circumcircle radius are, a b + b c + c a = s 2 + r, any line through a triangle that splits both the triangles area and its perimeter in half goes through the triangles incenter. There are either one, two, or three of these for any given triangle, the distance from any vertex to the incircle tangency on either adjacent side is half the sum of the vertexs adjacent sides minus half the opposite side. Thus for example for vertex B and adjacent tangencies TA and TC, the incircle radius is no greater than one-ninth the sum of the altitudes
Inradius
–
A triangle (black) with incircle (blue),
incenter (I), excircles (orange), excenters (J A,J B,J C), internal
angle bisectors (red) and external angle bisectors (green)
23.
Reflection symmetry
–
Reflection symmetry, line symmetry, mirror symmetry, mirror-image symmetry, is symmetry with respect to reflection. That is, a figure which does not change upon undergoing a reflection has reflectional symmetry, in 2D there is a line/axis of symmetry, in 3D a plane of symmetry. An object or figure 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
–
Many animals, such as this
spider crab Maja crispata, are bilaterally symmetric.
Reflection symmetry
–
Figures with the axes of
symmetry drawn in. The figure with no axes is
asymmetric.
Reflection symmetry
–
Mirror symmetry is often used in
architecture, as in the facade of
Santa Maria Novella,
Venice, 1470.
24.
Rhombus
–
In Euclidean geometry, a rhombus is a simple quadrilateral whose four sides all have the same length. Another name is equilateral quadrilateral, since equilateral means that all of its sides are equal in length, every rhombus is a 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
–
Some polyhedra with all rhombic faces
Rhombus
–
Two rhombi.
Rhombus
25.
Kite (geometry)
–
In Euclidean geometry, a kite is a quadrilateral whose four sides can be grouped into two pairs of equal-length sides that are adjacent to each other. In contrast, a parallelogram also has two pairs of sides, but they are opposite to each other rather than adjacent. Kite quadrilaterals are named for the wind-blown, flying kites, which often have this shape, kites are also known as deltoids, but the word deltoid may also refer to a deltoid curve, an unrelated geometric object. A kite, as defined above, may be convex or concave. A concave kite is called a dart or arrowhead, and is a type of pseudotriangle. If all four sides of a kite have the same length, if a kite is equiangular, meaning that all four of its angles are equal, then it must also be equilateral and thus a square. A kite with three equal 108° angles and one 36° angle forms the hull of the lute of Pythagoras. The kites that are cyclic quadrilaterals are exactly the ones formed from two congruent right triangles. That is, for these kites the two angles on opposite sides of the symmetry axis are each 90 degrees. These shapes are called right kites and they are in fact bicentric quadrilaterals, among all the bicentric quadrilaterals with a given two circle radii, the one with maximum area is a right kite. The tiling that it produces by its reflections is the deltoidal trihexagonal tiling, among all quadrilaterals, the shape that has the greatest ratio of its perimeter to its diameter is an equidiagonal kite with angles π/3, 5π/12, 5π/6, 5π/12. Its four vertices lie at the three corners and one of the midpoints of the Reuleaux triangle. In non-Euclidean geometry, a Lambert quadrilateral is a kite with three right angles. A quadrilateral is a if and only if any one of the following conditions is true. One diagonal is the bisector of the other diagonal. One diagonal is a line of symmetry, one diagonal bisects a pair of opposite angles. The kites are the quadrilaterals that have an axis of symmetry along one of their diagonals, if crossings are allowed, the list of quadrilaterals with axes of symmetry must be expanded to also include the antiparallelograms. Every kite is orthodiagonal, meaning that its two diagonals are at angles to each other
Kite (geometry)
–
V4.3.4.3
Kite (geometry)
–
A kite showing its sides equal in length and its inscribed circle.
Kite (geometry)
–
V4.3.4.4
Kite (geometry)
–
V4.3.4.5
26.
Parallelogon
–
A parallelogon is a polygon such that images of the polygon under translations only tile the plane when fitted together along entire sides. A parallelogon must have a number of sides and opposite sides must be equal in length. A less obvious restriction is that a parallelogon can only have four or six sides, in general a parallelogon has 180-degree rotational symmetry around its center. Quadrilateral and hexagonal parallelogons each have varied geometric symmetric forms, in general they all have central inversion symmetry, order 2. Hexagonal parallelogons enable the possibility of nonconvex polygons, parallelogram can tile the plane as a distorted square tiling while hexagonal parallelogon can tiling the plane as a distorted regular hexagonal tiling. Parallelohedron - Dimensional extension of parallelogons into 3D The facts on file, Geometry handbook, Catherine A. Gorini,2003, ISBN 0-8160-4875-4, p.117 Grünbaum, Branko, and Shephard, G. C. CS1 maint, Multiple names, authors list list of 107 isohedral tilings, p. 473-481 Fedorovs Five Parallelohedra
Parallelogon
–
A parallelogon is constructed by two or three pairs of parallel line segments. The vertices and edges on the interior of the hexagon are suppressed.
27.
Directed edge
–
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
–
A directed graph.
28.
Dodecagon
–
In geometry, a dodecagon or 12-gon is any twelve-sided polygon. A regular dodecagon is a figure with sides of the same length. It has twelve lines of symmetry and rotational symmetry of order 12. A regular dodecagon is represented by the Schläfli symbol and can be constructed as a hexagon, t, or a twice-truncated triangle. The internal angle at each vertex of a regular dodecagon is 150°, as 12 =22 ×3, regular dodecagon is constructible using compass and straightedge, Coxeter states that every parallel-sided 2m-gon can be divided into m/2 rhombs. For the dodecagon, m=6, and it can be divided into 15 rhombs and this decomposition is based on a Petrie polygon projection of a 6-cube, with 15 of 240 faces. One of the ways the mathematical manipulative pattern blocks are used is in creating a number of different dodecagons, the regular dodecagon has Dih12 symmetry, order 24. There are 15 distinct subgroup dihedral and cyclic symmetries, each subgroup symmetry allows one or more degrees of freedom for irregular forms. Only the g12 subgroup has no degrees of freedom but can seen as directed edges, the interior of such an dodecagon is not generally defined. A skew zig-zag dodecagon has vertices alternating between two parallel planes, a regular skew dodecagon is vertex-transitive with equal edge lengths. In 3-dimensions it will be a zig-zag skew dodecagon and can be seen in the vertices and side edges of a antiprism with the same D5d, symmetry. The dodecagrammic antiprism, s and dodecagrammic crossed-antiprism, s also have regular skew dodecagons, the regular dodecagon is the Petrie polygon for many higher-dimensional polytopes, seen as orthogonal projections in Coxeter planes. Examples in 4 dimensionare the 24-cell, snub 24-cell, 6-6 duoprism, in 6 dimensions 6-cube, 6-orthoplex,221,122. It is also the Petrie polygon for the grand 120-cell and great stellated 120-cell, a dodecagram is a 12-sided star polygon, represented by symbol. There is one regular star polygon, using the same vertices, but connecting every fifth point. There are also three compounds, is reduced to 2 as two hexagons, and is reduced to 3 as three squares, is reduced to 4 as four triangles, and is reduced to 6 as six degenerate digons. Deeper truncations of the regular dodecagon and dodecagrams can produce intermediate star polygon forms with equal spaced vertices. A truncated hexagon is a dodecagon, t=, a quasitruncated hexagon, inverted as, is a dodecagram, t=
Dodecagon
–
pattern blocks
Dodecagon
–
A regular dodecagon
Dodecagon
–
The Vera Cruz church in
Segovia
Dodecagon
–
A 1942 British threepence, reverse
29.
Alternation (geometry)
–
In geometry, an alternation or partial truncation, is an operation on a polygon, polyhedron, tiling, or higher dimensional polytope that removes alternate vertices. Coxeter labels an alternation by a prefixed by an h, standing for hemi or half, because alternation reduce all polygon faces to half as many sides, it can only be applied for polytopes with all even-sided faces. An alternated square face becomes a digon, and being degenerate, is reduced to a single edge. More generally any vertex-uniform polyhedron or tiling with a configuration consisting of all even-numbered elements can be alternated. For example, the alternation a vertex figure with 2a. 2b. 2c is a.3. b.3. c.3 where the three is the number of elements in this vertex figure. A special case is square faces whose order divide in half into degenerate digons, a snub can be seen as an alternation of a truncated regular or truncated quasiregular polyhedron. In general a polyhedron can be snubbed if its truncation has only even-sided faces, all truncated rectified polyhedra can be snubbed, not just from regular polyhedra. The snub square antiprism is an example of a general snub and this alternation operation applies to higher-dimensional polytopes and honeycombs as well, but in general most of the results of this operation will not be uniform. The voids created by the vertices will not in general create uniform facets. Examples, Honeycombs An alternated cubic honeycomb is the tetrahedral-octahedral honeycomb, an alternated hexagonal prismatic honeycomb is the gyrated alternated cubic honeycomb. 4-polytope An alternated truncated 24-cell is the snub 24-cell, 4-honeycombs, An alternated truncated 24-cell honeycomb is the snub 24-cell honeycomb. A hypercube can always be alternated into a uniform demihypercube, cube → Tetrahedron → Tesseract → 16-cell → Penteract → demipenteract Hexeract → demihexeract. Coxeter also used the operator a, which contains both halves, so retains the original symmetry, for even-sided regular polyhedra, a represents a compound polyhedron with two opposite copies of h. For odd-sided, greater than 3, regular polyhedra a, becomes a star polyhedron, Norman Johnson extended the use of the altered operator a, b for blended, and c for converted, as, and respectively. The compound polyhedron, stellated octahedron can be represented by a, the star-polyhedron, small ditrigonal icosidodecahedron, can be represented by a, and. Here all the pentagons have been alternated into pentagrams, and triangles have been inserted to take up the free edges. A similar operation can truncate alternate vertices, rather than just removing them, below is a set of polyhedra that can be generated from the Catalan solids. These have two types of vertices which can be alternately truncated, truncating the higher order vertices and both vertex types produce these forms, Conway polyhedral notation Wythoff construction Coxeter, H. S. M
Alternation (geometry)
–
Two
snub cubes from
truncated cuboctahedron See that red and green dots are placed at alternate vertices. A
snub cube is generated from deleting either set of vertices, one resulting in clockwise gyrated squares, and other counterclockwise.
30.
Hexagram
–
A hexagram or sexagram is a six-pointed geometric star figure with the Schläfli symbol,2, or. It is the compound of two equilateral triangles, the intersection is a regular hexagon. It is used 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
–
Diagram showing the two mystic syllables Om and Hrim
Hexagram
–
A regular hexagram
Hexagram
–
The Star of David in the oldest surviving complete copy of the
Masoretic text, the
Leningrad Codex, dated 1008.
Hexagram
–
Star of David on the
Salt Lake Assembly Hall
31.
Triangular tiling
–
In geometry, the triangular tiling or triangular tessellation is one of the three regular tilings of the Euclidean plane. Because the internal angle of the triangle is 60 degrees. The triangular tiling has Schläfli symbol of, Conway calls it a deltille, named from the triangular shape of the Greek letter delta. The triangular tiling can also be called a kishextille by a kis operation that adds a center point and it is one of three regular tilings of the plane. The other two are the square tiling and the hexagonal tiling, there are 9 distinct uniform colorings of a triangular tiling. Three of them can be derived from others by repeating colors,111212 and 111112 from 121213 by combining 1 and 3, there is one class of Archimedean colorings,111112, which is not 1-uniform, containing alternate rows of triangles where every third is colored. The example shown is 2-uniform, but there are many such Archimedean colorings that can be created by arbitrary horizontal shifts of the rows. The vertex arrangement of the tiling is called an A2 lattice. It is the 2-dimensional case of a simplectic honeycomb, the A*2 lattice can be constructed by the union of all three A2 lattices, and equivalent to the A2 lattice. + + = dual of = The vertices of the tiling are the centers of the densest possible circle packing. Every circle is in contact with 6 other circles in the packing, the packing density is π⁄√12 or 90. 69%. Since the union of 3 A2 lattices is also an A2 lattice, the voronoi cell of a triangular tiling is a hexagon, and so the voronoi tessellation, the hexagonal tiling has a direct correspondence to the circle packings. Triangular tilings can be made with the equivalent topology as the regular tiling, with identical faces and vertex-transitivity, there are 5 variations. Symmetry given assumes all faces are the same color, the planar tilings are related to polyhedra. Putting fewer triangles on a vertex leaves a gap and allows it to be folded into a pyramid and these can be expanded to Platonic solids, five, four and three triangles on a vertex define an icosahedron, octahedron, and tetrahedron respectively. This tiling is related as a part of sequence of regular polyhedra with Schläfli symbols. It is also related as a part of sequence of Catalan solids with face configuration Vn.6.6. Like the uniform there are eight uniform tilings that can be based from the regular hexagonal tiling
Triangular tiling
–
Periodic
Triangular tiling
–
Triangular tiling
Triangular tiling
32.
Star figure
–
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
–
Regular polygrams {n/d}, with red lines showing constant d, and blue lines showing compound sequences k{n/d}
33.
Cube
–
Beryllium copper, also known as copper beryllium, beryllium bronze and spring copper, is a copper alloy with 0. 5—3% beryllium and sometimes other elements. Beryllium copper combines high strength with non-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
–
Example of a non-sparking tool made of beryllium copper
Cube
34.
Hexagonal grid
–
In geometry, the hexagonal tiling or hexagonal tessellation is a regular tiling of the Euclidean plane, in which three hexagons meet at each vertex. It has Schläfli symbol of or t, 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 grid
–
Chicken wire fencing
Hexagonal grid
–
Hexagonal tiling
Hexagonal grid
–
Graphene
Hexagonal grid
–
Periodic
35.
Wax
–
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
–
Commercial
honeycomb foundation, made by pressing beeswax between patterned metal rollers.
Wax
–
Ceroline brand wax for floors and furniture, first half of 20th century. From the
Museo del Objeto del Objeto collection
Wax
–
Wax candle.
Wax
–
A typical modern
wax sculpture of
Cecilia Cheung at
Madame Tussauds Hong Kong.
36.
Hexagonal prism
–
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
–
Uniform Hexagonal prism
37.
Parallelohedron
–
In geometry a parallelohedron is a polyhedron that can tessellate 3-dimensional spaces with face-to-face contacts via translations. This requires all opposite faces be congruent, Parallelohedra can only have parallelogonal faces, either parallelograms or hexagons with parallel opposite edges. There are 5 types, first identified by Evgraf Fedorov in his studies of crystallographic systems, the vertices of parallelohedra can be computed by linear combinations of 3 to 6 vectors. Each vector can have any greater than zero, with zero length becoming degenerate. A belt mn means n directional vectors, each containing m coparallel congruent edges, every type has order 2 Ci central inversion symmetry in general, but each has higher symmetry geometries as well. There are 5 types of parallelohedra, although each has forms of varied symmetry, in higher dimensions a parallelohedron is called a parallelotope. There are 52 variations for 4-dimensional parallelotopes, gorini,2003, ISBN 0-8160-4875-4, p.117 Coxeter, H. S. M. Regular polytopes, 3rd ed. New York, Dover, pp. 29–30, p.257,1973, tutton, A. E. H. Crystallography and Practical Crystal Measurement, 2nd ed. London, Lubrecht & Cramer,1964. E. S. Fedorov, Nachala Ucheniya o Figurah, Fedorovs five parallelohedra in R³ Fedorovs Five Parallelohedra
Parallelohedron
–
Images
38.
Hexagonal prismatic honeycomb
–
The triangular prismatic honeycomb or triangular prismatic cellulation is a space-filling tessellation in Euclidean 3-space. It is composed entirely of triangular prisms and it is constructed from a triangular tiling extruded into prisms. It is one of 28 convex uniform honeycombs, the hexagonal prismatic honeycomb or hexagonal prismatic cellulation is a space-filling tessellation in Euclidean 3-space made up of hexagonal prisms. It is constructed from a hexagonal tiling extruded into prisms and it is one of 28 convex uniform honeycombs. This honeycomb can be alternated into the gyrated tetrahedral-octahedral honeycomb, with pairs of tetrahedra existing in the alternated gaps, the trihexagonal prismatic honeycomb or trihexagonal prismatic cellulation is a space-filling tessellation in Euclidean 3-space. It is composed of hexagonal prisms and triangular prisms in a ratio of 1,2 and it is constructed from a trihexagonal tiling extruded into prisms. It is one of 28 convex uniform honeycombs, the truncated hexagonal prismatic honeycomb or tomo-trihexagonal prismatic cellulation is a space-filling tessellation in Euclidean 3-space. It is composed of dodecagonal prisms, and triangular prisms in a ratio of 1,2 and it is constructed from a truncated hexagonal tiling extruded into prisms. It is one of 28 convex uniform honeycombs, the rhombitrihexagonal prismatic honeycomb or rhombitrihexagonal prismatic cellulation is a space-filling tessellation in Euclidean 3-space. It is composed of hexagonal prisms, cubes, and triangular prisms in a ratio of 1,3,2 and it is constructed from a rhombitrihexagonal tiling extruded into prisms. It is one of 28 convex uniform honeycombs, the snub hexagonal prismatic honeycomb or simo-trihexagonal prismatic cellulation is a space-filling tessellation in Euclidean 3-space. It is composed of hexagonal prisms and triangular prisms in a ratio of 1,8 and it is constructed from a snub hexagonal tiling extruded into prisms. It is one of 28 convex uniform honeycombs, the truncated trihexagonal prismatic honeycomb or tomo-trihexagonal prismatic cellulation is a space-filling tessellation in Euclidean 3-space. It is composed of dodecagonal prisms, hexagonal prisms, and cubes in a ratio of 1,2,3 and it is constructed from a truncated trihexagonal tiling extruded into prisms. It is one of 28 convex uniform honeycombs, the elongated triangular prismatic honeycomb or elongated antiprismatic prismatic cellulation is a space-filling tessellation in Euclidean 3-space. It is composed of cubes and triangular prisms in a ratio of 1,2 and it is constructed from an elongated triangular tiling extruded into prisms. It is one of 28 convex uniform honeycombs, the gyrated triangular prismatic honeycomb or parasquare fastigial cellulation is a space-filling tessellation in Euclidean 3-space made up of triangular prisms. It is vertex-uniform with 12 triangular prisms per vertex and it can be seen as parallel planes of square tiling with alternating offsets caused by layers of paired triangular prisms
Hexagonal prismatic honeycomb
–
Triangular prismatic honeycomb
39.
Extended side
–
In plane geometry, an extended side or sideline of a polygon is the line that contains one side of the polygon. The extension of a side arises in various contexts, in an obtuse triangle, the altitudes from the acute angled vertices intersect the corresponding extended base sides but not the base sides themselves. The excircles of a triangle, as well as the triangles inconics that are not inellipses, are tangent to one side. Trilinear coordinates locate a point in the plane by its relative distances from the sides of a reference triangle. If the point is outside the triangle, the perpendicular from the point to the sideline may meet the sideline outside the triangle—that is, in a triangle, three intersection points, each of an external angle bisector with the opposite extended side, are collinear. An ex-tangential quadrilateral is a quadrilateral for which exists a circle that is tangent to all four extended sides. The excenter lies at the intersection of six angle bisectors
Extended side
–
Each of a triangle's excircles (orange) is tangent to one of the triangle's sides and to the other two extended sides.
40.
Symmedian point
–
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
–
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.
41.
Concurrent lines
–
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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verification. Please help improve this article by
adding citations to reliable sources. Unsourced material may be challenged and removed. (March 2011)
42.
Circumcircle
–
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
Circumcircle
–
Circumscribed circle, C, and circumcenter, O, of a cyclic polygon, P
43.
Tangent line
–
In geometry, the tangent line to a plane curve at a given point is the straight line that just touches the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve, a similar definition applies to space curves and curves in n-dimensional Euclidean space. Similarly, the tangent plane to a surface at a point is the plane that just touches the surface at that point. The concept of a tangent is one of the most fundamental notions in geometry and has been extensively generalized. The word tangent comes from the Latin tangere, to touch, euclid makes several references to the tangent to a circle in book III of the Elements. In Apollonius work Conics he defines a tangent as being a line such that no other straight line could fall between it and the curve, archimedes found the tangent to an Archimedean spiral by considering the path of a point moving along the curve. Independently Descartes used his method of normals based on the observation that the radius of a circle is always normal to the circle itself and these methods led to the development of differential calculus in the 17th century. Many people contributed, Roberval discovered a method of drawing tangents. René-François de Sluse and Johannes Hudde found algebraic algorithms for finding tangents, further developments included those of John Wallis and Isaac Barrow, leading to the theory of Isaac Newton and Gottfried Leibniz. An 1828 definition of a tangent was a line which touches a curve. This old definition prevents inflection points from having any tangent and it has been dismissed and the modern definitions are equivalent to those of Leibniz who defined the tangent line as the line through a pair of infinitely close points on the curve. The tangent at A is the limit when point B approximates or tends to A, the existence and uniqueness of the tangent line depends on a certain type of mathematical smoothness, known as differentiability. At most points, the tangent touches the curve without crossing it, a point where the tangent crosses the curve is called an inflection point. Circles, parabolas, hyperbolas and ellipses do not have any point, but more complicated curves do have, like the graph of a cubic function. Conversely, it may happen that the curve lies entirely on one side of a line passing through a point on it. This is the case, for example, for a passing through the vertex of a triangle. In convex geometry, such lines are called supporting lines, the geometrical idea of the tangent line as the limit of secant lines serves as the motivation for analytical methods that are used to find tangent lines explicitly. The question of finding the tangent line to a graph, or the tangent line problem, was one of the central questions leading to the development of calculus in the 17th century, suppose that a curve is given as the graph of a function, y = f
Tangent line
–
Tangent to a curve. The red line is tangential to the curve at the point marked by a red dot.
44.
Skew polygon
–
In geometry, a skew polygon is a polygon whose vertices are not all coplanar. Skew polygons must have at least 4 vertices, the interior surface of such a polygon is not uniquely defined. Skew infinite polygons have vertices which are not all collinear, a zig-zag skew polygon or antiprismatic polygon has vertices which alternate on two parallel planes, and thus must be even-sided. Regular skew polygon in 3 dimensions are always zig-zag, a regular skew polygon is isogonal with equal edge lengths. In 3 dimensions a regular polygon is a zig-zag skew. The sides of an n-antiprism can define a regular skew 2n-gons, a regular skew n-gonal can be given a symbol # as a blend of a regular polygon, and an orthogonal line segment. The symmetry operation between sequential vertices is glide reflection, examples are shown on the uniform square and pentagon antiprisms. The star antiprisms also generate regular skew polygons with different connection order of the top, the filled top and bottom polygons are drawn for structural clarity, and are not part of the skew polygons. A regular compound skew 2n-gon can be constructed by adding a second skew polygon by a rotation. These shares the same vertices as the compound of antiprisms. Petrie polygons are regular skew polygons defined within regular polyhedra and polytopes, for example, the 5 Platonic solids have 4,6, and 10-sided regular skew polygons, as seen in these orthogonal projections with red edges around the projective envelope. The tetrahedron and octahedron include all the vertices in the zig-zag skew polygon and can be seen as a digonal, the regular skew polyhedron have regular faces, and regular skew polygon vertex figures. Three are infinite space-filling in 3-space and others exist in 4-space, an isogonal skew polygon is a skew polygon with one type of vertex, connected by two types of edges. Isogonal skew polygons with equal edge lengths can also be considered quasiregular and it is similar to a zig-zag skew polygon, existing on two planes, except allowing one edge to cross to the opposite plane, and the other edge to stay on the same plane. Isogonal skew polygons can be defined on even-sided n-gonal prisms, alternatingly following an edge of one side polygon, for example, on the vertices of a cube. Vertices alternate between top and bottom squares with red edges between sides, and blue edges along each side, in 4 dimensions a regular skew polygon can have vertices on a Clifford torus and related by a Clifford displacement. Unlike zig-zag skew polygons, skew polygons on double rotations can include an odd-number of sides, the petrie polygons of the regular 4-polytope define regular skew polygons. The Coxeter number for each coxeter group symmetry expresses how many sides a petrie polygon has and this is 5 sides for a 5-cell,8 sides for a tesseract and 16-cell,12 sides for a 24-cell, and 30 sides for a 120-cell and 600-cell
Skew polygon
–
The (red) side edges of
tetragonal disphenoid represent a regular zig-zag skew quadrilateral.
45.
Regular polytope
–
In mathematics, a regular polytope is a polytope whose symmetry group acts transitively on its flags, thus giving it the highest degree of symmetry. All its elements or j-faces — cells, faces and so on — are also transitive on the symmetries of the polytope, Regular polytopes are the generalized analog in any number of dimensions of regular polygons and regular polyhedra. The strong symmetry of the regular polytopes gives them an aesthetic quality that interests both non-mathematicians and mathematicians, classically, a regular polytope in n dimensions may be defined as having regular facets and regular vertex figures. These two conditions are sufficient to ensure that all faces are alike and all vertices are alike, note, however, that this definition does not work for abstract polytopes. A regular polytope can be represented by a Schläfli symbol of the form, with regular facets as, Regular polytopes are classified primarily according to their dimensionality. They can be classified according to symmetry. For example, the cube and the regular octahedron share the same symmetry, indeed, symmetry groups are sometimes named after regular polytopes, for example the tetrahedral and icosahedral symmetries. Three special classes of regular polytope exist in every dimensionality, Regular simplex Measure polytope Cross polytope In two dimensions there are many regular polygons. In three and four dimensions there are more regular polyhedra and 4-polytopes besides these three. In five dimensions and above, these are the only ones, see also the list of regular polytopes. The idea of a polytope is sometimes generalised to include related kinds of geometrical object, some of these have regular examples, as discussed in the section on historical discovery below. A concise symbolic representation for regular polytopes was developed by Ludwig Schläfli in the 19th Century, the notation is best explained by adding one dimension at a time. A convex regular polygon having n sides is denoted by, so an equilateral triangle is, a square, and so on indefinitely. A regular star polygon which winds m times around its centre is denoted by the fractional value, a regular polyhedron having faces with p faces joining around a vertex is denoted by. The nine regular polyhedra are and. is the figure of the polyhedron. A regular 4-polytope having cells with q cells joining around an edge is denoted by, the vertex figure of the 4-polytope is a. A five-dimensional regular polytope is an, the dual of a regular polytope is also a regular polytope. The Schläfli symbol for the dual polytope is just the original written backwards, is self-dual, is dual to, to
Regular polytope
–
Platonic solids
Regular polytope
–
A regular
pentagon is a
polygon, a two-dimensional polytope with 5
edges, represented by
Schläfli symbol {5}.
Regular polytope
Regular polytope
46.
Orthogonal projection
–
In linear algebra and functional analysis, a projection is a linear transformation P from a vector space to itself such that P2 = P. That is, whenever P is applied twice to any value, though abstract, this definition of projection formalizes and generalizes the idea of graphical projection. One can also consider the effect of a projection on an object by examining the effect of the projection on points in the object. For example, the function maps the point in three-dimensional space R3 to the point is an orthogonal projection onto the x–y plane. This function is represented by the matrix P =, the action of this matrix on an arbitrary vector is P =. To see that P is indeed a projection, i. e. P = P2, a simple example of a non-orthogonal projection is P =. Via matrix multiplication, one sees that P2 = = = P. proving that P is indeed a projection, the projection P is orthogonal if and only if α =0. Let W be a finite dimensional space and P be a projection on W. Suppose the subspaces U and V are the range and kernel of P respectively, then P has the following properties, By definition, P is idempotent. P is the identity operator I on U ∀ x ∈ U, P x = x and we have a direct sum W = U ⊕ V. Every vector x ∈ W may be decomposed uniquely as x = u + v with u = P x and v = x − P x = x, the range and kernel of a projection are complementary, as are P and Q = I − P. The operator Q is also a projection and the range and kernel of P become the kernel and range of Q and we say P is a projection along V onto U and Q is a projection along U onto V. In infinite dimensional spaces, the spectrum of a projection is contained in as −1 =1 λ I +1 λ P. Only 0 or 1 can be an eigenvalue of a projection, the corresponding eigenspaces are the kernel and range of the projection. Decomposition of a space into direct sums is not unique in general. Therefore, given a subspace V, there may be many projections whose range is V, if a projection is nontrivial it has minimal polynomial x 2 − x = x, which factors into distinct roots, and thus P is diagonalizable. The product of projections is not, in general, a projection, if projections commute, then their product is a projection. When the vector space W has a product and is complete the concept of orthogonality can be used
Orthogonal projection
–
The transformation P is the orthogonal projection onto the line m.
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
–
3-3 duoprisms
48.
3-3 duopyramid
–
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
–
3-3 duoprisms
49.
Truncated octahedron
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In geometry, the truncated octahedron is an Archimedean solid. It has 14 faces,36 edges, and 24 vertices, since each of its faces has point symmetry the truncated octahedron is a zonohedron. It is also the Goldberg polyhedron GIV, containing square and hexagonal faces, like the cube, it can tessellate 3-dimensional space, as a permutohedron. Its dual polyhedron is the tetrakis hexahedron, if the original truncated octahedron has unit edge length, its dual tetrakis cube has edge lengths 9/8√2 and 3/2√2. A truncated octahedron is constructed from an octahedron with side length 3a by the removal of six right square pyramids. These pyramids have both base side length and lateral side length of a, to form equilateral triangles, the base area is then a2. Note that this shape is similar to half an octahedron or Johnson solid J1. The truncated octahedron has five special orthogonal projections, centered, on a vertex, the last two correspond to the B2 and A2 Coxeter planes. The truncated octahedron 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, all permutations of are Cartesian coordinates of the vertices of a truncated octahedron of edge length a = √2 centered at the origin. The vertices are also the corners of 12 rectangles whose long edges are parallel to the coordinate axes. The edge vectors have Cartesian coordinates and permutations of these, the face normals of the 6 square faces are, and. The face normals of the 8 hexagonal faces are, the dot product between pairs of two face normals is the cosine of the dihedral angle between adjacent faces, either −1/3 or −1/√3. The dihedral angle is approximately 1.910633 radians at edges shared by two hexagons or 2.186276 radians at edges shared by a hexagon and a square. The truncated octahedron can be dissected into an octahedron, surrounded by 8 triangular cupola on each face. Therefore, the octahedron is the permutohedron of order 4, each vertex corresponds to a permutation of. The area A and the volume V of an octahedron of edge length a are. There are two uniform colorings, with symmetry and octahedral symmetry, and two 2-uniform coloring with dihedral symmetry as a truncated triangular antiprism
Truncated octahedron
–
(Click here for rotating model)
Truncated octahedron
–
Tetrakis hexahedron (
dual polyhedron)
50.
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
–
(Click here for rotating model)
Truncated icosahedron
–
Pentakis dodecahedron (
dual polyhedron)
Truncated icosahedron
–
Fullerene C 60 molecule
Truncated icosahedron
–
Truncated icosahedral
radome on a
weather station
51.
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
–
(Click here for rotating model)
Truncated cuboctahedron
–
Disdyakis dodecahedron (
dual polyhedron)
52.
Tetrahedral symmetry
–
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
–
A regular
tetrahedron, an example of a solid with full tetrahedral symmetry
Tetrahedral symmetry
Tetrahedral symmetry
–
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.
53.
Octahedral symmetry
–
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
–
Each face of the
disdyakis dodecahedron is a fundamental domain
Octahedral symmetry
54.
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
55.
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
–
Example chamfered pentagon (v,e) --> (v+2e,3e)
56.
Chamfered cube
–
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
–
Example chamfered pentagon (v,e) --> (v+2e,3e)
57.
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
–
The
elongated square gyrobicupola (J 37), a Johnson solid
58.
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
–
Triangular cupola
59.
Gyroelongated triangular cupola
–
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
60.
Parabiaugmented hexagonal prism
–
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
61.
Metabiaugmented hexagonal prism
–
In geometry, the metabiaugmented 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, nonparallel equatorial faces. Attaching the pyramids to opposite equatorial faces yields a parabiaugmented 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, hexagonal prism Eric W. Weisstein, Metabiaugmented hexagonal prism at MathWorld
Metabiaugmented hexagonal prism
–
Metabiaugmented hexagonal prism
62.
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
63.
Prismoid
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In geometry, a prismatoid is a polyhedron whose vertices all lie in two parallel planes. Its lateral faces can be trapezoids or triangles, if both planes have the same number of vertices, and the lateral faces are either parallelograms or trapezoids, it is called a prismoid. For example, in four dimensions, two polyhedra can be placed in two parallel 3-spaces, and connected with polyhedral sides
Prismoid
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Pyramids
64.
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
65.
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)
66.
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
67.
Cassini-Huygens
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Cassini–Huygens is an unmanned spacecraft sent to the planet Saturn. It is a Flagship-class NASA–ESA–ASI robotic spacecraft, Cassini is the fourth space probe to visit Saturn and the first to enter orbit, and its mission is ongoing as of April 2017. It has studied the planet and its natural satellites since arriving there in 2004. Its design includes a Saturn orbiter and a lander for the moon Titan, the two spacecraft are named after astronomers Giovanni Cassini and Christiaan Huygens. On December 25,2004, Huygens separated from the orbiter and it successfully returned data to Earth, using the orbiter as a relay. This was the first landing ever accomplished in the outer Solar System, on November 30,2016, Cassini entered the final phase of the project. Cassini will dive through the ring of Saturn 20 times. The spacecraft will enter areas that have been untouched up until this point, the first pass of the rings took place on December 4,2016. Cassini continued to study the Saturn system in the following years, however, due to the spacecrafts dwindling fuel resources for further orbital corrections, it is currently planned to be destroyed by diving into the planets atmosphere in September 2017. This method of disposal was chosen to avoid potential biological contamination of Saturns moons, Cassinis death spiral is set to officially begin on April 22,2017 and will take its final plunge on September 15,2017 before it beams its last batch of images. Sixteen European countries and the United States make up the team responsible for designing, building, flying and collecting data from the Cassini orbiter, the mission is managed by NASAs Jet Propulsion Laboratory in the United States, where the orbiter was assembled. Huygens was developed by the European Space Research and Technology Centre, the Centres prime contractor, Aérospatiale of France, assembled the probe with equipment and instruments supplied by many European countries. The VIMS infrared counterpart was provided by NASA, as well as Main Electronic Assembly, on April 16,2008, NASA announced a two-year extension of the funding for ground operations of this mission, at which point it was renamed the Cassini Equinox Mission. This was again extended in February 2010 with the Cassini Solstice Mission, the mission was commonly called Saturn Orbiter Titan Probe during gestation, both as a Mariner Mark II mission and generically. Cassini-Huygens is a mission to the outer planets. The other planetary flagships include Galileo, Voyager, and Viking, Cassini has several objectives, including, Determine the three-dimensional structure and dynamic behavior of the rings of Saturn. Determine the composition of the surfaces and the geological history of each object. Determine the nature and origin of the material on Iapetuss leading hemisphere
Cassini-Huygens
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Artist's concept of Cassini 's
orbit insertion around Saturn
Cassini-Huygens
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Huygens' explanation for the aspects of Saturn, Systema Saturnium, 1659.
Cassini-Huygens
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Saturn's north side (2014)
Cassini-Huygens
68.
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
–
Two different
resonance forms of benzene (top) combine to produce an average structure (bottom)
69.
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
–
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.
70.
Northern Ireland
–
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
–
Scrabo Tower,
County Down
Northern Ireland
–
Location of Northern Ireland (dark green) – in
Europe (green & dark grey) – in the
United Kingdom (green)
Northern Ireland
–
Cannon on
Derry 's
city walls
Northern Ireland
–
Signing of the
Ulster Covenant in 1912 in opposition to Home Rule
71.
James Webb Space Telescope
–
The James Webb Space Telescope, previously known as Next Generation Space Telescope, is a part of NASAs ongoing Flagship program. It is under construction and scheduled to launch in October 2018, the JWST will offer unprecedented resolution and sensitivity from long-wavelength visible light, through near-infrared to the mid-infrared. While the Hubble Space Telescope has a 2. 4-meter mirror, a large sunshield will keep its mirror and four science instruments below 50 K. JWSTs capabilities will enable a broad range of investigations across the fields of astronomy and cosmology. One particular goal involves observing some of the most distant events and objects in the Universe and these types of targets are beyond the reach of current ground and space-based instruments. Another goal is understanding the formation of stars and planets and this will include direct imaging of exoplanets. In gestation since 1996, the project represents a collaboration of the European Space Agency, Canadian Space Agency. It is named after James E. Webb, the administrator of NASA. NASA has described JWST as the successor of the Hubble Space Telescope. JWST has the objective to see objects, typically both older and farther away than previous instruments could assess. The result was to extend the life of Hubble until JWST as the next generation telescope could go online. This led to an altered design for JWST to obtain images deeper into the infrared than Hubble. The JWST originated in 1996 as the Next Generation Space Telescope, in 2002 it was renamed after NASAs second administrator James E. Webb, noted for playing a key role in the Apollo program and establishing scientific research as a core NASA activity. The telescope has a mass about half of Hubble Space Telescopes. The JWST is oriented toward near-infrared astronomy, but can also see orange and red light, as well as the mid-infrared region. The JWST will operate near the Earth-Sun L2 point, approximately 930,000 mi beyond the Earth, by way of comparison, Hubble orbits 340 miles above Earths surface, and the Moon is roughly 250,000 miles from Earth. This distance makes post-launch repair or upgrade of the JWST hardware virtually impossible and this will keep the temperature of the spacecraft below 50 K, necessary for infrared observations. The prime contractor is Northrop Grumman and its nominal mission time is five years, with a goal of ten years. JWST needs to use propellant to maintain its orbit around L2, which provides an upper limit to its designed lifetime
James Webb Space Telescope
–
Full-scale James Webb Space Telescope model at South by Southwest in Austin
James Webb Space Telescope
–
Two alternate
Hubble Space Telescope views of the
Carina Nebula, comparing visible (top) and infrared (bottom) astronomy
James Webb Space Telescope
–
Infrared observations can see objects hidden in visible light,
HUDF-JD2 shown
James Webb Space Telescope
–
JWST will not be exactly at the L2 point, but circle around it in a
halo orbit.
72.
Metropolitan France
–
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
–
Metropolitan France
73.
Martinique
–
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
–
Saint-Pierre. Before the total destruction of Saint-Pierre in 1902 by a volcanic eruption, it was the most important city of Martinique culturally and economically, being known as "the Paris of the Caribbean".
Martinique
–
Flag
Martinique
–
The
attack on the French ships at Martinique in 1667
Martinique
–
The
Battle of Martinique between British and French fleets in 1779
74.
French Guiana
–
French Guiana, officially called Guiana, is an overseas department and region of France, located on the north Atlantic coast of South America in the Guyanas. It borders 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
–
Forested landscape of
Remire-Montjoly.
French Guiana
–
Flag
French Guiana
–
View from the
île Royale
French Guiana
–
Liana on a
palm branch near a lake in
Kourou
75.
Hanksite
–
Hanksite is a sulfate mineral, distinguished as one of only a handful that contain both carbonate and sulfate ion groups. It has the formula, Na22K92Cl. It was first described in 1888 for an occurrence in Searles Lake, California, Hanksite is normally found in crystal form as evaporite deposits. Hanksite crystals are large but not complex in structure and it is often found in Searles Lake, Soda Lake, Mono Lake, and in Death Valley. It is associated with halite, borax, trona and aphthitalite in the Searles Lake area, Hanksite can be colorless, white, gray, green or yellow and is transparent or translucent. The minerals hardness is approximately 3 to 3.5, the specific gravity is approximately 2.5. It is salty to the taste and sometimes pale yellow in ultra-violet light. Typical growth habits are hexagonal prisms or tabular with pyramidal terminations, the streak of Hanksite is white. It can contain inclusions of clay that the crystal formed around while developing
Hanksite
–
Hanksite crystal from
Searles Lake
76.
Hexagonal crystal system
–
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
–
An example of the hexagonal crystals,
beryl
Hexagonal crystal system
–
Hexagonal
Hanksite crystal
77.
Theatre
–
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
–
Sarah Bernhardt as
Hamlet, in 1899
Theatre
–
A master (right) and his slave (left) in a
Greek phlyax play, circa 350/340 BCE
Theatre
–
Mosaic depicting masked actors in a play: two women consult a "witch"
Theatre
–
Performer playing
Sugriva in the
Koodiyattam form of
Sanskrit theatre.
78.
Reading, Berkshire
–
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
–
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
–
The earliest map of Reading, published in 1611 by
John Speed
Reading, Berkshire
–
View of Reading from
Caversham by
Joseph Farington in 1793
Reading, Berkshire
–
Reading
Crown Court
79.
24-cell
–
In geometry, the 24-cell is the convex regular 4-polytope with Schläfli symbol. It is also called C24, icositetrachoron, octaplex, icosatetrahedroid, octacube, hyper-diamond or polyoctahedron, the boundary of the 24-cell is composed of 24 octahedral cells with six meeting at each vertex, and three at each edge. Together they have 96 triangular faces,96 edges, and 24 vertices, the vertex figure is a cube. In fact, the 24-cell is the unique convex self-dual regular Euclidean polytope which is neither a polygon nor a simplex, due to this singular property, it does not have a good analogue in 3 dimensions. A 24-cell is given as the hull of its vertices. The vertices of a 24-cell centered at the origin of 4-space, with edges of length 1, the first 8 vertices are the vertices of a regular 16-cell and the other 16 are the vertices of the dual tesseract. This gives an equivalent to cutting a tesseract into 8 cubical pyramids. This is equivalent to the dual of a rectified 16-cell, the analogous construction in 3-space gives the rhombic dodecahedron which, however, is not regular. We can further divide the last 16 vertices into two groups, those with an number of minus signs and those with an odd number. Each of groups of 8 vertices also define a regular 16-cell, the vertices of the 24-cell can then be grouped into three sets of eight with each set defining a regular 16-cell, and with the complement defining the dual tesseract. The vertices of the dual 24-cell are given by all permutations of, the dual 24-cell has edges of length √2 and is inscribed in a 3-sphere of radius √2. Another method of constructing the 24-cell is by the rectification of the 16-cell, the vertex figure of the 16-cell is the octahedron, thus, cutting the vertices of the 16-cell at the midpoint of its incident edges produce 8 octahedral cells. This process also rectifies the tetrahedral cells of the 16-cell which also become octahedra, a regular tessellation of 4-dimensional Euclidean space exists with 24-cells, called an icositetrachoric honeycomb, with Schläfli symbol. Hence, the angle of a 24-cell is 120°. The regular dual tessellation, has 16-cells, the 24 vertices of the 24-cell represent the root vectors of the simple Lie group D4. The vertices can be seen in 3 hyperplanes, with the 6 vertices of a cell on each of the outer hyperplanes and 12 vertices of a cuboctahedron on a central hyperplane. These vertices, combined with the 8 vertices of the 16-cell, represent the 32 root vectors of the B4, the 48 vertices of the union of the 24-cell and its dual form the root system of type F4. The 24 vertices of the original 24-cell form a system of type D4
24-cell
–
Schlegel diagram (vertices and edges)
80.
Four-dimensional space
–
For example, the volume of a rectangular box is found by measuring its length, width, and depth. More than two millennia ago Greek philosophers explored in detail the implications of this uniformity, culminating in Euclids Elements. However, it was not until recent times that a handful of insightful mathematical innovators generalized the concept of dimensions to more than three. The idea of adding a fourth dimension began with Joseph-Louis Lagrange in the mid 1700s, in 1880 Charles Howard Hinton popularized these insights in an essay titled What is the Fourth Dimension. Which was notable for explaining the concept of a cube by going through a step-by-step generalization of the properties of lines, squares. The simplest form of Hintons method is to draw two ordinary cubes separated by a distance, and then draw lines between their equivalent vertices. This form can be seen in the accompanying animation whenever it shows a smaller inner cube inside a larger outer cube, the eight lines connecting the vertices of the two cubes in that case represent a single direction in the unseen fourth dimension. Higher dimensional spaces have become one of the foundations for formally expressing modern mathematics and physics. Large parts of these topics could not exist in their current forms without the use of such spaces, calendar entries for example are usually 4D locations, such as a meeting at time t at the intersection of two streets on some building floor. In list form such a meeting place at the 4D location. Einsteins concept of spacetime uses such a 4D space, though it has a Minkowski structure that is a bit more complicated than Euclidean 4D space, when dimensional locations are given as ordered lists of numbers such as they are called vectors or n-tuples. It is only when such locations are linked together into more complicated shapes that the richness and geometric complexity of 4D. A hint of that complexity can be seen in the animation of one of simplest possible 4D objects. Lagrange wrote in his Mécanique analytique that mechanics can be viewed as operating in a four-dimensional space — three dimensions of space, and one of time, the possibility of geometry in higher dimensions, including four dimensions in particular, was thus established. An arithmetic of four dimensions called quaternions was defined by William Rowan Hamilton in 1843 and this associative algebra was the source of the science of vector analysis in three dimensions as recounted in A History of Vector Analysis. Soon after tessarines and coquaternions were introduced as other four-dimensional algebras over R, one of the first major expositors of the fourth dimension was Charles Howard Hinton, starting in 1880 with his essay What is the Fourth Dimension. Published in the Dublin University magazine and he coined the terms tesseract, ana and kata in his book A New Era of Thought, and introduced a method for visualising the fourth dimension using cubes in the book Fourth Dimension. Hintons ideas inspired a fantasy about a Church of the Fourth Dimension featured by Martin Gardner in his January 1962 Mathematical Games column in Scientific American, in 1886 Victor Schlegel described his method of visualizing four-dimensional objects with Schlegel diagrams
Four-dimensional space
–
5-cell
Four-dimensional space
–
3D projection of a
tesseract undergoing a
simple rotation in four dimensional space.
81.
Orthoplex
–
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
–
2 dimensions
square
82.
Regular tiling
–
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
Regular tiling
–
Periodic
Regular tiling
–
A regular tiling has one type of regular face.
Regular tiling
83.
Honeycomb conjecture
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The honeycomb conjecture states that a regular hexagonal grid or honeycomb is the best way to divide a surface into regions of equal area with the least total perimeter. The conjecture was proven in 1999 by mathematician Thomas C, let Γ be a locally finite graph in R2, consisting of smooth curves, and such that R2 \Γ has infinitely many bounded connected components, all of unit area. Let C be the union of these bounded components, then lim sup r → ∞ perim area ≥124. Equality is attained for the regular hexagonal tile, the first record of the conjecture dates back to 36BC, from Marcus Terentius Varro, but is often attributed to Pappus of Alexandria. The conjecture was proven in 1999 by mathematician Thomas C, hales, who mentions in his work that there is reason to believe that the conjecture may have been present in the minds of mathematicians before Varro. It is also related to the densest circle packing of the plane, weaire–Phelan structure, a counter-example to the Kelvin conjecture on the solution of the similar problem in 3D
Honeycomb conjecture
–
A regular
hexagonal grid
84.
International Standard Book Number
–
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
–
A 13-digit ISBN, 978-3-16-148410-0, as represented by an
EAN-13 bar code
85.
Isosceles triangle
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In geometry, an isosceles triangle is a triangle that has two sides of equal length. By the isosceles triangle theorem, the two angles opposite the sides are themselves equal, while if the third side is different then the third angle is different. By the Steiner–Lehmus theorem, every triangle with two angle bisectors of equal length is isosceles, in an isosceles triangle that has exactly two equal sides, the equal sides are called legs and the third side is called the base. The angle included by the legs is called the vertex angle, the vertex opposite the base is called the apex. In the equilateral triangle case, since all sides are equal, any side can be called the base, if needed, and the term leg is not generally used. A triangle with two equal sides has exactly one axis of symmetry, which goes through the vertex angle. Thus the axis of symmetry coincides with the bisector of the vertex angle, the median drawn to the base, the altitude drawn from the vertex angle. Whether the isosceles triangle is acute, right or obtuse depends on the vertex angle, in Euclidean geometry, the base angles cannot be obtuse or right because their measures would sum to at least 180°, the total of all angles in any Euclidean triangle. The Euler line of any triangle goes through the orthocenter, its centroid. In an isosceles triangle with two equal sides, the Euler line coincides with the axis of symmetry. This can be seen as follows, if the vertex angle is acute, then the orthocenter, the centroid, and the circumcenter all fall inside the triangle. In an isosceles triangle the incenter lies on the Euler line, the Steiner inellipse of any triangle is the unique ellipse that is internally tangent to the triangles three sides at their midpoints. For any isosceles triangle with area T and perimeter p, we have 2 p b 3 − p 2 b 2 +16 T2 =0. By substituting the height, the formula for the area of a triangle can be derived from the general formula one-half the base times the height. This is what Herons formula reduces to in the isosceles case, if the apex angle and leg lengths of an isosceles triangle are known, then the area of that triangle is, T =2 = a 2 sin cos . This is derived by drawing a line from the base of the triangle. The bases of two right triangles are both equal to the hypotenuse times the sine of the bisected angle by definition of the term sine. For the same reason, the heights of these triangles are equal to the times the cosine of the bisected angle
Isosceles triangle
–
The
Flatiron Building in New York is shaped like a
triangular prism
Isosceles triangle
86.
Rectangle
–
In Euclidean plane geometry, a rectangle is a quadrilateral with four right angles. It can also be defined as a quadrilateral, since equiangular means that all of its angles are equal. It can also be defined as a parallelogram containing a right angle, a rectangle with four sides of equal length is a square. The term oblong is occasionally used to refer to a non-square rectangle, a rectangle with vertices ABCD would be denoted as ABCD. The word rectangle comes from the Latin rectangulus, which is a combination of rectus and angulus, a crossed rectangle is a crossed quadrilateral which consists of two opposite sides of a rectangle along with the two diagonals. It is a case of an antiparallelogram, and its angles are not right angles. Other geometries, such as spherical, elliptic, and hyperbolic, have so-called rectangles with sides equal in length. Rectangles are involved in many tiling problems, such as tiling the plane by rectangles or tiling a rectangle by polygons, a convex quadrilateral with successive sides a, b, c, d whose area is 12. A rectangle is a case of a parallelogram in which each pair of adjacent sides is perpendicular. A parallelogram is a case of a trapezium in which both pairs of opposite sides are parallel and equal in length. A trapezium is a quadrilateral which has at least one pair of parallel opposite sides. A convex quadrilateral is Simple, The boundary does not cross itself, star-shaped, The whole interior is visible from a single point, without crossing any edge. De Villiers defines a more generally as any quadrilateral with axes of symmetry through each pair of opposite sides. This definition includes both right-angled rectangles and crossed rectangles, quadrilaterals with two axes of symmetry, each through a pair of opposite sides, belong to the larger class of quadrilaterals with at least one axis of symmetry through a pair of opposite sides. These quadrilaterals comprise isosceles trapezia and crossed isosceles trapezia, a rectangle is cyclic, all corners lie on a single circle. It is equiangular, all its corner angles are equal and it is isogonal or vertex-transitive, all corners lie within the same symmetry orbit. It has two lines of symmetry and rotational symmetry of order 2. The dual polygon of a rectangle is a rhombus, as shown in the table below, the figure formed by joining, in order, the midpoints of the sides of a rectangle is a rhombus and vice versa
Rectangle
–
Running bond
Rectangle
–
Rectangle
Rectangle
–
Basket weave
87.
Heptagon
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In geometry, a heptagon is a seven-sided polygon or 7-gon. The heptagon is also referred to as the septagon, using sept- together with the Greek suffix -agon meaning angle. A regular heptagon, in all sides and all angles are equal, has internal angles of 5π/7 radians. The area of a regular heptagon of side length a is given by, the apothem is half the cotangent of π /7, and the area of each of the 14 small triangles is one-fourth of the apothem. This expression cannot be rewritten without complex components, since the indicated cubic function is casus irreducibilis. As 7 is a Pierpont prime but not a Fermat prime and this type of construction is called a neusis construction. It is also constructible with compass, straightedge and angle trisector, the impossibility of straightedge and compass construction follows from the observation that 2 cos 2 π7 ≈1.247 is a zero of the irreducible cubic x3 + x2 − 2x −1. Consequently, this polynomial is the polynomial of 2cos, whereas the degree of the minimal polynomial for a constructible number must be a power of 2. An approximation for practical use with an error of about 0. 2% is shown in the drawing and it is attributed to Albrecht Dürer. Let A lie on the circumference of the circumcircle, then B D =12 B C gives an approximation for the edge of the heptagon. Example to illustrate the error, At a circumscribed circle radius r =1 m, since 7 is a prime number there is one subgroup with dihedral symmetry, Dih1, and 2 cyclic group symmetries, Z7, and Z1. These 4 symmetries can be seen in 4 distinct symmetries on the heptagon, john Conway labels these by a letter and group order. Full symmetry of the form is r14 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 g7 subgroup has no degrees of freedom but can seen as directed edges. However, no algebraic expressions with purely real terms exist for the solutions of this equation, because it is an example of casus irreducibilis. A heptagonal triangle has vertices coinciding with the first, second, and fourth vertices of a regular heptagon and angles π /7,2 π /7, thus its sides coincide with one side and two particular diagonals of the regular heptagon. Two kinds of star heptagons can be constructed from regular heptagons, labeled by Schläfli symbols, blue, and green star heptagons inside a red heptagon
Heptagon
–
Cactus
Heptagon
–
A regular heptagon
88.
Octadecagon
–
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
–
A regular octadecagon
89.
Icosagon
–
In geometry, an icosagon or 20-gon is a twenty-sided polygon. The sum of any icosagons interior angles is 3240 degrees, the regular icosagon has Schläfli symbol, and can also be constructed as a truncated decagon, t, or a twice-truncated pentagon, tt. One interior angle in a regular icosagon is 162°, meaning that one exterior angle would be 18°, the area of a regular icosagon with edge length t is A =5 t 2 ≃31.5687 t 2. In terms of the radius R of its circumcircle, the area is A =5 R22, since the area of the circle is π R2, the Big Wheel on the popular US game show The Price Is Right has an icosagonal cross-section. The Globe, the outdoor theater used by William Shakespeares acting company, was discovered to have built on an icosagonal foundation when a partial excavation was done in 1989. As a golygonal path, the swastika is considered to be an irregular icosagon, a regular square, pentagon, and icosagon can completely fill a plane vertex. E20 E1 ¯ E1 F ¯ = E20 F ¯ E20 E1 ¯ =1 +52 = φ ≈1.618 The regular icosagon has Dih20 symmetry, order 40. There are 5 subgroup dihedral symmetries, and, and 6 cyclic group symmetries and these 10 symmetries can be seen in 16 distinct symmetries on the icosagon, a larger number because the lines of reflections can either pass through vertices or edges. John Conway labels these by a letter and group order, full symmetry of the regular form is r40 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 g20 subgroup has no degrees of freedom but can seen as directed edges. These two forms are duals of each other and have half the order of the regular icosagon. An icosagram is a 20-sided star polygon, represented by symbol, there are three regular forms given by Schläfli symbols, and. There are also five regular star figures using the vertex arrangement,2,4,5,2,4. Deeper truncations of the regular decagon and decagram can produce isogonal intermediate icosagram forms with equally spaced vertices, a regular icosagram, can be seen as a quasitruncated decagon, t=. Similarly a decagram, has a quasitruncation t=, and finally a simple truncation of a decagram gives t=
Icosagon
–
A regular icosagon
90.
Icositetragon
–
In geometry, an icositetragon or 24-gon is a twenty-four-sided polygon. The sum of any icositetragons interior angles is 3960 degrees, the regular icositetragon is represented by Schläfli symbol and can also be constructed as a truncated dodecagon, t, or a twice-truncated hexagon, tt, or thrice-truncated triangle, ttt. One interior angle in a regular icositetragon is 165°, meaning that one exterior angle would be 15°, the area of a regular icositetragon is, A =6 t 2 cot π24 =6 t 2. The icositetragon appeared in Archimedes polygon approximation of pi, along with the hexagon, dodecagon, tetracontaoctagon, as 24 =23 ×3, a regular icositetragon is constructible using a compass and straightedge. As a truncated dodecagon, it can be constructed by an edge-bisection of a regular dodecagon, the regular icositetragon has Dih24 symmetry, order 48. There are 7 subgroup dihedral symmetries, and, and 8 cyclic group symmetries and these 16 symmetries can be seen in 22 distinct symmetries on the icositetragon. John Conway labels these by a letter and group order, the full symmetry of the regular form is r48 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 g24 subgroup has no degrees of freedom but can seen as directed edges. A regular triangle, octagon, and icositetragon can completely fill a plane vertex, an icositetragram is a 24-sided star polygon. There are 3 regular forms given by Schläfli symbols, and, there are also 7 regular star figures using the same vertex arrangement,2,3,4,6,8,3, and 2. There are also isogonal icositetragrams constructed as deeper truncations of the regular dodecagon and these also generate two quasitruncations, t=, and t=. A skew icositetragon is a polygon with 24 vertices and edges. The interior of such an icositetragon is not generally defined, a skew zig-zag icositetragon has vertices alternating between two parallel planes. A regular skew icositetragon is vertex-transitive with equal edge lengths, in 3-dimensions it will be a zig-zag skew icositetragon and can be seen in the vertices and side edges of a dodecagonal antiprism with the same D12d, symmetry, order 48. The dodecagrammic antiprism, s and dodecagrammic crossed-antiprism, s also have regular skew dodecagons, the regular icositetragon is the Petrie polygon for many higher-dimensional polytopes, seen as orthogonal projections in Coxeter planes, including, Weisstein, Eric W. Icositetragon. Naming Polygons and Polyhedra polygon icosatetragon
Icositetragon
–
A regular icositetragon
91.
Pentacontagon
–
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
–
A regular pentacontagon
92.
Octacontagon
–
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
–
A regular octacontagon
93.
Enneacontagon
–
In geometry, an enneacontagon or enenecontagon or 90-gon is a ninety-sided polygon. The sum of any enneacontagons interior angles is 15840 degrees, a regular enneacontagon is represented by Schläfli symbol and can be constructed as a truncated tetracontapentagon, t, which alternates two types of edges. One interior angle in a regular enneacontagon is 176°, meaning that one exterior angle would be 4°, the regular enneacontagon has Dih90 dihedral symmetry, order 180, represented by 90 lines of reflection. Dih90 has 11 dihedral subgroups, Dih45, and, and 12 more cyclic symmetries, and, with Zn representing π/n radian rotational symmetry. These 24 symmmetries are related to 30 distinct symmetries on the enneacontagon, john Conway labels these lower symmetries with a 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 enneacontagons. Only the g90 symmetry has no degrees of freedom but can seen as directed edges, an enneacontagram is a 90-sided star polygon. There are 11 regular forms given by Schläfli symbols, and, as well as 33 regular star figures with the same vertex configuration
Enneacontagon
–
A regular enneacontagon
94.
257-gon
–
In geometry, a 257-gon is a polygon with 257 sides. The sum of the angles of any non-self-intersecting 257-gon is 91800°. The area of a regular 257-gon is A =2574 t 2 cot π257 ≈5255.751 t 2. A whole regular 257-gon is not visually discernible from a circle, the regular 257-gon is of interest for being a constructible polygon, that is, it can be constructed using a compass and an unmarked straightedge. This is because 257 is a Fermat prime, being of the form 22n +1. Thus, the values cos π257 and cos 2 π257 are 128-degree algebraic numbers, another method involves the use of 150 circles,24 being Carlyle circles, this method is pictured below. One of these Carlyle circles solves the quadratic equation x2 + x −64 =0, the regular 257-gon has Dih257 symmetry, order 514. Since 257 is a number there is one subgroup with dihedral symmetry, Dih1, and 2 cyclic group symmetries, Z257. A 257-gram is a 257-sided star polygon, as 257 is prime, there are 127 regular forms generated by Schläfli symbols for all integers 2 ≤ n ≤128 as ⌊2572 ⌋ =128. Below is a view of, with 257 nearly radial edges, 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. 257-gon, approximate construction of the first side, with construction instruction
257-gon
–
A regular 257-gon
95.
Chiliagon
–
In geometry, a chiliagon or 1000-gon is a polygon with 1000 sides. Philosophers commonly refer to chiliagons to illustrate ideas about the nature and workings of thought, meaning, a regular chiliagon is represented by Schläfli symbol and can be constructed as a truncated 500-gon, t, or a twice-truncated 250-gon, tt, or a thrice-truncated 125-gon, ttt. The measure of each internal angle in a regular chiliagon is 179. 64°, because 1000 =23 ×53, the number of sides is neither a product of distinct Fermat primes nor a power of two. Thus the regular chiliagon 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. René Descartes uses the chiliagon as an example in his Sixth Meditation to demonstrate the difference between pure intellection and imagination. He says that, when one thinks of a chiliagon, he does not imagine the thousand sides or see them as if they were present before him – as he does when one imagines a triangle, for example. The imagination constructs a confused representation, which is no different from that which it constructs of a myriagon, however, he does clearly understand what a chiliagon is, just as he understands what a triangle is, and he is able to distinguish it from a myriagon. Therefore, the intellect is not dependent on imagination, Descartes claims, as it is able to entertain clear, the example of a chiliagon is also referenced by other philosophers, such as Immanuel Kant. David Hume points out that it is impossible for the eye to determine the angles of a chiliagon to be equal to 1996 right angles, or make any conjecture, that approaches this proportion. Gottfried Leibniz comments on a use of the chiliagon by John Locke, noting that one can have an idea of the polygon without having an image of it, inspired by Descartess chiliagon example, Roderick Chisholm and other 20th-century philosophers have used similar examples to make similar points. Chisholms speckled hen, which need not have a number of speckles to be successfully imagined, is perhaps the most famous of these. The regular chiliagon has Dih1000 dihedral symmetry, order 2000, represented by 1000 lines of reflection. Dih100 has 15 dihedral subgroups, Dih500, Dih250, Dih125, Dih200, Dih100, Dih50, Dih25, Dih40, Dih20, Dih10, Dih5, Dih8, Dih4, Dih2, and Dih1. It also has 16 more cyclic symmetries as subgroups, Z1000, Z500, Z250, Z125, Z200, Z100, Z50, Z25, Z40, Z20, Z10, Z5, Z8, Z4, Z2, and Z1, 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 chiliagons. Only the g1000 subgroup has no degrees of freedom but can seen as directed edges, a chiliagram is a 1000-sided star polygon. There are 199 regular forms given by Schläfli symbols of the form, there are also 300 regular star figures in the remaining cases
Chiliagon
–
A regular chiliagon
96.
Myriagon
–
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
–
A regular myriagon
97.
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
98.
Enneagram (geometry)
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In geometry, an enneagram is a nine-pointed plane figure. It is sometimes called a nonagram, the name enneagram combines the numeral prefix, ennea-, with the Greek suffix -gram. The -gram suffix derives from γραμμῆς meaning a line, a regular enneagram is constructed using the same points as the regular enneagon but connected in fixed steps. It has two forms, represented by a Schläfli symbol as and, connecting every second and every fourth points respectively, there is also a star figure, or 3, made from the regular enneagon points but connected as a compound of three equilateral triangles. This star figure is known as the star of Goliath, after or 2. The nine-pointed star or enneagram can also symbolize the nine gifts or fruits of the Holy Spirit, the heavy metal band Slipknot uses the star figure enneagram as a symbol. Nonagon List of regular star polygons Bibliography John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 Nonagram -- from Wolfram MathWorld
Enneagram (geometry)
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Enneagrams shown as sequential stellations
99.
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
100.
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
101.
Uniform polytope
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A uniform polytope of dimension three or higher is a vertex-transitive polytope bounded by uniform facets. The uniform polytopes in two dimensions are the regular polygons and this is a generalization of the older category of semiregular polytopes, but also includes the regular polytopes. Further, star regular faces and vertex figures are allowed, which expand the possible solutions. A strict definition requires uniform polytopes to be finite, while a more expansive definition allows uniform honeycombs of Euclidean, nearly every uniform polytope can be generated by a Wythoff construction, and represented by a Coxeter diagram. Notable exceptions include the antiprism in four dimensions. Equivalently, the Wythoffian polytopes can be generated by applying basic operations to the regular polytopes in that dimension and this approach was first used by Johannes Kepler, and is the basis of the Conway polyhedron notation. Regular n-polytopes have n orders of rectification, the zeroth rectification is the original form. The th rectification is the dual, an extended Schläfli symbol can be used for representing rectified forms, with a single subscript, k-th rectification = tk = kr. Truncation operations that can be applied to regular n-polytopes in any combination, the resulting Coxeter diagram has two ringed nodes, and the operation is named for the distance between them. Truncation cuts vertices, cantellation cuts edges, runcination cuts faces, each higher operation also cuts lower ones too, so a cantellation also truncates vertices. T0,1 or t, Truncation - applied to polygons, a truncation removes vertices, and inserts a new facet in place of each former vertex. Faces are truncated, doubling their edges and it can be seen as rectifying its rectification. A cantellation truncates both vertices and edges and replaces them with new facets, cells are replaced by topologically expanded copies of themselves. There are higher cantellations also, bicantellation t1,3 or r2r, tricantellation t2,4 or r3r, quadricantellation t3,5 or r4r, etc. t0,1,2 or tr, Cantitruncation - applied to polyhedra and higher. It can be seen as a truncation of its rectification, a cantitruncation truncates both vertices and edges and replaces them with new facets. Cells are replaced by topologically expanded copies of themselves, runcination truncates vertices, edges, and faces, replacing them each with new facets. 4-faces are replaced by topologically expanded copies of themselves, There are higher runcinations also, biruncination t1,4, triruncination t2,5, etc. t0,4 or 2r2r, Sterication - applied to Uniform 5-polytopes and higher. It can be seen as birectifying its birectification, Sterication truncates vertices, edges, faces, and cells, replacing each with new facets
Uniform polytope
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Truncated triangle is a uniform hexagon, with
Coxeter diagram.
102.
E6 (mathematics)
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The designation E6 comes from the Cartan–Killing classification of the complex simple Lie algebras. This classifies Lie algebras into four infinite series labeled An, Bn, Cn, Dn, and five exceptional cases labeled E6, E7, E8, F4, the E6 algebra is thus one of the five exceptional cases. The fundamental group of the form, compact real form, or any algebraic version of E6 is the cyclic group Z/3Z. Its fundamental representation is 27-dimensional, and a basis is given by the 27 lines on a cubic surface, the dual representation, which is inequivalent, is also 27-dimensional. In particle physics, E6 plays a role in some grand unified theories, there is a unique complex Lie algebra of type E6, corresponding to a complex group of complex dimension 78. The complex adjoint Lie group E6 of complex dimension 78 can be considered as a simple real Lie group of real dimension 156, the split form, EI, which has maximal compact subgroup Sp/, fundamental group of order 2 and outer automorphism group of order 2. The quasi-split form EII, which has maximal compact subgroup SU × SU/, fundamental group cyclic of order 6, EIII, which has maximal compact subgroup SO × Spin/, fundamental group Z and trivial outer automorphism group. EIV, which has maximal compact subgroup F4, trivial fundamental group cyclic, the EIV form of E6 is the group of collineations of the octonionic projective plane OP2. It is also the group of determinant-preserving linear transformations of the exceptional Jordan algebra, the exceptional Jordan algebra is 27-dimensional, which explains why the compact real form of E6 has a 27-dimensional complex representation. Over finite fields, the Lang–Steinberg theorem implies that H1 =0, meaning that E6 has exactly one twisted form, known as 2E6, the Dynkin diagram for E6 is given by, which may also be drawn as or. Although they span a space, it is much more symmetrical to consider them as vectors in a six-dimensional subspace of a nine-dimensional space. Two 16-dimensional subalgebras that transform as a Weyl spinor of spin and these have a non-zero last entry. 1 generator which is their chirality generator, and is the sixth Cartan generator, the Lie algebra E6 has an F4 subalgebra, which is the fixed subalgebra of an outer automorphism, and an SU × SU × SU subalgebra. Other maximal subalgebras which have an importance in physics and can be read off the Dynkin diagram, are the algebras of SO × U, in addition to the 78-dimensional adjoint representation, there are two dual 27-dimensional vector representations. The characters of finite dimensional representations of the real and complex Lie algebras, the fundamental representations have dimensions 27,351,2925,351,27 and 78. The E6 polytope is the hull of the roots of E6. It therefore exists in 6 dimensions, its symmetry group contains the Coxeter group for E6 as an index 2 subgroup, the groups of type E6 over arbitrary fields were introduced by Dickson. The points over a field with q elements of the algebraic group E6, whether of the adjoint or simply connected form
E6 (mathematics)
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Algebraic structure → Group theory
Group theory
103.
E7 (mathematics)
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The E7 algebra is thus one of the five exceptional cases. The fundamental group of the form, compact real form, or any algebraic version of E7 is the cyclic group Z/2Z. The dimension of its representation is 56. There is a unique complex Lie algebra of type E7, corresponding to a group of complex dimension 133. The complex adjoint Lie group E7 of complex dimension 133 can be considered as a simple real Lie group of real dimension 266. This has fundamental group Z/2Z, has maximal compact subgroup the compact form of E7, the split form, EV, which has maximal compact subgroup SU/, fundamental group cyclic of order 4 and outer automorphism group of order 2. EVI, which has maximal compact subgroup SU·SO/, fundamental group non-cyclic of order 4, EVII, which has maximal compact subgroup SO·E6/, infinite cyclic findamental group and outer automorphism group of order 2. For a complete list of forms of simple Lie algebras. The compact real form of E7 is the group of the 64-dimensional exceptional compact Riemannian symmetric space EVI. This can be seen using a construction known as the magic square, due to Hans Freudenthal. The Tits–Koecher construction produces forms of the E7 Lie algebra from Albert algebras, over finite fields, the Lang–Steinberg theorem implies that H1 =0, meaning that E7 has no twisted forms, see below. The Dynkin diagram for E7 is given by, even though the roots span a 7-dimensional space, it is more symmetric and convenient to represent them as vectors lying in a 7-dimensional subspace of an 8-dimensional vector space. The roots are all the 8×7 permutations of and all the permutations of Note that the 7-dimensional subspace is the subspace where the sum of all the eight coordinates is zero. The simple roots are We have ordered them so that their corresponding nodes in the Dynkin diagram are ordered left to right with the side node last. Given the E7 Cartan matrix and a Dynkin diagram node ordering of, the Weyl group of E7 is of order 2903040, it is the direct product of the cyclic group of order 2 and the unique simple group of order 1451520. E7 has an SU subalgebra, as is evident by noting that in the 8-dimensional description of the root system, in addition to the 133-dimensional adjoint representation, there is a 56-dimensional vector representation, to be found in the E8 adjoint representation. The characters of finite dimensional representations of the real and complex Lie algebras, there exist non-isomorphic irreducible representation of dimensions 1903725824,16349520330, etc. The fundamental representations are those with dimensions 133,8645,365750,27664,1539,56 and 912, E7 is the automorphism group of the following pair of polynomials in 56 non-commutative variables
E7 (mathematics)
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Algebraic structure → Group theory
Group theory
104.
F4 (mathematics)
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In mathematics, F4 is the name of a Lie group and also its Lie algebra f4. It is one of the five exceptional simple Lie groups, F4 has rank 4 and dimension 52. The compact form is connected and its outer automorphism group is the trivial group. The compact real form of F4 is the group of a 16-dimensional Riemannian manifold known as the octonionic projective plane OP2. This can be seen using a construction known as the magic square, due to Hans Freudenthal. There are 3 real forms, a one, a split one. They are the groups of the three real Albert algebras. The F4 Lie algebra may be constructed by adding 16 generators transforming as a spinor to the 36-dimensional Lie algebra so, in older books and papers, F4 is sometimes denoted by E4. The Dynkin diagram for F4 is and its Weyl/Coxeter group G = W is the symmetry group of the 24-cell, it is a solvable group of order 1152. It has minimal faithful degree μ =24 which is realized by the action on the 24-cell, the F4 lattice is a four-dimensional body-centered cubic lattice. They form a ring called the Hurwitz quaternion ring, the 24 Hurwitz quaternions of norm 1 form the vertices of a 24-cell centered at the origin. One choice of roots for F4, is given by the rows of the following matrix. Invariant, F4 is the group of automorphisms of the set of 3 polynomials in 27 variables. Another way of writing these invariants is as Tr, Tr and Tr of the hermitian octonion matrix, the characters of finite dimensional representations of the real and complex Lie algebras and Lie groups are all given by the Weyl character formula. There are two non-isomorphic irreducible representations of dimensions 1053,160056,4313088, etc, the fundamental representations are those with dimensions 52,1274,273,26. The Exceptional Simple Lie Algebras F and E. Proc
F4 (mathematics)
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Algebraic structure → Group theory
Group theory
105.
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
106.
Tetrahedron
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In geometry, a tetrahedron, also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all the ordinary convex polyhedra, the tetrahedron is the three-dimensional case of the more general concept of a Euclidean simplex. The tetrahedron is one kind of pyramid, which is a polyhedron with a polygon base. In the case of a tetrahedron the base is a triangle, like all convex polyhedra, a tetrahedron can be folded from a single sheet of paper. For any tetrahedron there exists a sphere on which all four vertices lie, a regular tetrahedron is one in which all four faces are equilateral triangles. It is one of the five regular Platonic solids, which have known since antiquity. In a regular tetrahedron, not only are all its faces the same size and shape, regular tetrahedra alone do not tessellate, but if alternated with regular octahedra they form the alternated cubic honeycomb, which is a tessellation. The regular tetrahedron is self-dual, which means that its dual is another regular tetrahedron, the compound figure comprising two such dual tetrahedra form a stellated octahedron or stella octangula. This form has Coxeter diagram and Schläfli symbol h, the tetrahedron in this case has edge length 2√2. Inverting these coordinates generates the dual tetrahedron, and the together form the stellated octahedron. In other words, if C is the centroid of the base and this follows from the fact that the medians of a triangle intersect at its centroid, and this point divides each of them in two segments, one of which is twice as long as the other. The vertices of a cube can be grouped into two groups of four, each forming a regular tetrahedron, the symmetries of a regular tetrahedron correspond to half of those of a cube, those that map the tetrahedra to themselves, and not to each other. The tetrahedron is the only Platonic solid that is not mapped to itself by point inversion, the regular tetrahedron has 24 isometries, forming the symmetry group Td, isomorphic to the symmetric group, S4. The first corresponds to the A2 Coxeter plane, the two skew perpendicular opposite edges of a regular tetrahedron define a set of parallel planes. When one of these intersects the tetrahedron the resulting cross section is a rectangle. When the intersecting plane is one of the edges the rectangle is long. When halfway between the two edges the intersection is a square, the aspect ratio of the rectangle reverses as you pass this halfway point. For the midpoint square intersection the resulting boundary line traverses every face of the tetrahedron similarly, if the tetrahedron is bisected on this plane, both halves become wedges
Tetrahedron
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(Click here for rotating model)
Tetrahedron
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4-sided die
107.
Regular dodecahedron
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It is one of the five Platonic solids. It has 12 faces,20 vertices,30 edges, and 160 diagonals. Note that, given a regular dodecahedron of edge length one, ru is the radius of a sphere about a cube of edge length ϕ. In perspective projection, viewed above a face, the regular dodecahedron can be seen as a linear-edged schlegel diagram. These projections are used in showing the four-dimensional 120-cell, a regular 4-dimensional polytope, constructed from 120 dodecahedra. The regular dodecahedron can also be represented as a spherical tiling, the following Cartesian coordinates define the 20 vertices of a regular dodecahedron centered at the origin and suitably scaled and oriented, where ϕ =1 + √5/2 is the golden ratio ≈1.618. The edge length is 2/ϕ = √5 −1, the containing sphere has a radius of √3. 5650512°. A137218 If the original regular dodecahedron has edge length 1, its dual icosahedron has edge length ϕ, If the five Platonic solids are built with same volume, the regular dodecahedron has the shortest edges. The map-coloring number of a regular dodecahedrons faces is 4, the distance between the vertices on the same face not connected by an edge is ϕ times the edge length. If two edges share a vertex, then the midpoints of those edges form an equilateral triangle with the body center. The regular dodecahedron is the third in a set of truncated trapezohedra which can be constructed by truncating the two axial vertices of a pentagonal trapezohedron. The stellations of the regular dodecahedron make up three of the four Kepler–Poinsot polyhedra, a rectified regular dodecahedron forms an icosidodecahedron. The regular dodecahedron has icosahedral symmetry Ih, Coxeter group, order 120, when a regular dodecahedron is inscribed in a sphere, it occupies more of the spheres volume than an icosahedron inscribed in the same sphere. A regular dodecahedron has 12 faces and 20 vertices, whereas a regular icosahedron has 20 faces and 12 vertices, a cube can be embedded within a regular dodecahedron, affixed to eight of its equidistant vertices, in five different positions. In fact, five cubes may overlap and interlock inside the regular dodecahedron to result in the compound of five cubes, the ratio of the edge of a regular dodecahedron to the edge of a cube embedded inside such a regular dodecahedron is 1, ϕ, or,1. The ratio of a regular dodecahedrons volume to the volume of a cube embedded inside such a regular dodecahedron is 1, 2/2 + ϕ, or 1 + ϕ/2,1, or,4. For example, a cube with a volume of 64. Thus, the difference in volume between the regular dodecahedron and the enclosed cube is always one half the volume of the cube times ϕ
Regular dodecahedron
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(Click here for rotating model)
Regular dodecahedron
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Roman dodecahedron
Regular dodecahedron
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Ho-Mg-Zn
quasicrystal
Regular dodecahedron
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A climbing wall consisting of three dodecahedral pieces
108.
Regular icosahedron
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In geometry, a regular icosahedron is a convex polyhedron with 20 faces,30 edges and 12 vertices. It is one of the five Platonic solids, and also the one with the most sides and it has five equilateral triangular faces meeting at each vertex. It is represented by its Schläfli symbol, or sometimes by its vertex figure as 3.3.3.3.3 or 35 and it is the dual of the dodecahedron, which is represented by, having three pentagonal faces around each vertex. A regular icosahedron is a pentagonal bipyramid and a biaugmented pentagonal antiprism in any of six orientations. The name comes from Greek εἴκοσι, meaning twenty, and ἕδρα, the plural can be either icosahedrons or icosahedra. The surface area A and the volume V of a regular icosahedron of edge length a are, note that these vertices form five sets of three concentric, mutually orthogonal golden rectangles, whose edges form Borromean rings. If the original icosahedron has edge length 1, its dual dodecahedron has edge length √5 − 1/2 = 1/ϕ = ϕ −1, the 12 edges of a regular octahedron can be subdivided in the golden ratio so that the resulting vertices define a regular icosahedron. The locations of the vertices of a regular icosahedron can be described using spherical coordinates, if two vertices are taken to be at the north and south poles, then the other ten vertices are at latitude ±arctan ≈ ±26. 57°. These ten vertices are at evenly spaced longitudes, alternating between north and south latitudes 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, an icosahedron has 43,380 distinct nets. To color the icosahedron, such that no two adjacent faces have the color, requires at least 3 colors. A problem dating back to the ancient Greeks is to determine which of two shapes has larger volume, an icosahedron inscribed in a sphere, or a dodecahedron inscribed in the same sphere, the problem was solved by Hero, Pappus, and Fibonacci, among others. Apollonius of Perga discovered the result that the ratio of volumes of these two shapes is the same as the ratio of their surface areas. Both volumes have formulas involving the golden ratio, but taken to different powers, as it turns out, the icosahedron occupies less of the spheres volume than the dodecahedron. The following construction of the icosahedron avoids tedious computations in the number field ℚ necessary in more elementary approaches, the existence of the icosahedron amounts to the existence of six equiangular lines in ℝ3. Indeed, intersecting such a system of lines with a Euclidean sphere centered at their common intersection yields the twelve vertices of a regular icosahedron as can easily be checked. Conversely, supposing the existence of an icosahedron, lines defined by its six pairs of opposite vertices form an equiangular system. In order to such an equiangular system, we start with this 6 ×6 square matrix
Regular icosahedron
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(Click here for rotating model)
Regular icosahedron
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Model of an icosahedron made with metallic spheres and magnetic connectors
Regular icosahedron
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Gold nanoparticle viewed in electron microscope.
Regular icosahedron
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Structure of γ-boron.
109.
Uniform 4-polytope
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In geometry, a uniform 4-polytope is a 4-polytope which is vertex-transitive and whose cells are uniform polyhedra, and faces are regular polygons. 47 non-prismatic convex uniform 4-polytopes, one set of convex prismatic forms. There are also a number of non-convex star forms. Regular star 4-polytopes 1852, Ludwig Schläfli also found 4 of the 10 regular star 4-polytopes, discounting 6 with cells or vertex figures and this construction enumerated 45 semiregular 4-polytopes. 1912, E. L. Elte independently expanded on Gossets list with the publication The Semiregular Polytopes of the Hyperspaces, polytopes with one or two types of semiregular facets, Convex uniform polytopes,1940, The search was expanded systematically by H. S. M. Coxeter in his publication Regular and Semi-Regular Polytopes,1966 Norman Johnson completes his Ph. D. dissertation The Theory of Uniform Polytopes and Honeycombs under advisor Coxeter, completes the basic theory of uniform polytopes for dimensions 4 and higher. 1986 Coxeter published a paper Regular and Semi-Regular Polytopes II which included analysis of the unique snub 24-cell structure, 1998-2000, The 4-polytopes were systematically named by Norman Johnson, and given by George Olshevskys online indexed enumeration. Johnson named the 4-polytopes as polychora, like polyhedra for 3-polytopes, from the Greek roots poly,2004, A proof that the Conway-Guy set is complete was published by Marco Möller in his dissertation, Vierdimensionale Archimedische Polytope. Möller reproduced Johnsons naming system in his listing,2008, The Symmetries of Things was published by John H. He used his own ijk-ambo naming scheme for the indexed ring permutations beyond truncation and bitruncation, nonregular uniform star 4-polytopes, 2000-2005, In a collaborative search, up to 2005 a total of 1845 uniform 4-polytopes had been identified by Jonathan Bowers and George Olshevsky. Regular 4-polytopes are a subset of the uniform 4-polytopes, which satisfy additional requirements, Regular 4-polytopes can be expressed with Schläfli symbol have cells of type, faces of type, edge figures, and vertex figures. The existence of a regular 4-polytope is constrained by the existence of the regular polyhedra which becomes cells, there are 64 convex uniform 4-polytopes, including the 6 regular convex 4-polytopes, and excluding the infinite sets of the duoprisms and the antiprismatic hyperprisms. 5 are polyhedral prisms based on the Platonic solids 13 are polyhedral prisms based on the Archimedean solids 9 are in the self-dual regular A4 group family,9 are in the self-dual regular F4 group family. 15 are in the regular B4 group family 15 are in the regular H4 group family,1 special snub form in the group family. 1 special non-Wythoffian 4-polytopes, the grand antiprism, TOTAL,68 −4 =64 These 64 uniform 4-polytopes are indexed below by George Olshevsky. Repeated symmetry forms are indexed in brackets, in addition to the 64 above, there are 2 infinite prismatic sets that generate all of the remaining convex forms, Set of uniform antiprismatic prisms - sr× - Polyhedral prisms of two antiprisms. Set of uniform duoprisms - × - A product of two polygons, the 5-cell has diploid pentachoric symmetry, of order 120, isomorphic to the permutations of five elements, because all pairs of vertices are related in the same way. Facets are given, grouped in their Coxeter diagram locations by removing specified nodes, there is one small index subgroup +, order 60, or its doubling +, order 120, defining a omnisnub 5-cell which is listed for completeness, but is not uniform
Uniform 4-polytope
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Schlegel diagram for the
truncated 120-cell with
tetrahedral cells visible
110.
16-cell
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In four-dimensional geometry, a 16-cell is a regular convex 4-polytope. It is one of the six regular convex 4-polytopes first described by the Swiss mathematician Ludwig Schläfli in the mid-19th century and it is also called C16, hexadecachoron, or hexdecahedroid. It is a part of an family of polytopes, called cross-polytopes or orthoplexes. The dual polytope is the tesseract, conways name for a cross-polytope is orthoplex, for orthant complex. The 16-cell has 16 cells as the tesseract has 16 vertices and it is bounded by 16 cells, all of which are regular tetrahedra. It has 32 triangular faces,24 edges, and 8 vertices, the 24 edges bound 6 squares lying in the 6 coordinate planes. The eight vertices of the 16-cell are, all vertices are connected by edges except opposite pairs. The Schläfli symbol of the 16-cell is and its vertex figure is a regular octahedron. There are 8 tetrahedra,12 triangles, and 6 edges meeting at every vertex and its edge figure is a square. There are 4 tetrahedra and 4 triangles meeting at every edge, the 16-cell can be decomposed into two similar disjoint circular chains of eight tetrahedrons each, four edges long. Each chain, when stretched out straight, forms a Boerdijk–Coxeter helix and this decomposition can be seen in a 4-4 duoantiprism construction of the 16-cell, or, Schläfli symbol ⨂ or ss, symmetry, order 64. The 16-cell can be dissected into two octahedral pyramids, which share a new octahedron base through the 16-cell center, one can tessellate 4-dimensional Euclidean space by regular 16-cells. This is called the 16-cell honeycomb and has Schläfli symbol, hence, the 16-cell has a dihedral angle of 120°. The dual tessellation, 24-cell honeycomb, is made of by regular 24-cells, together with the tesseractic honeycomb, these are the only three regular tessellations of R4. Each 16-cell has 16 neighbors with which it shares a tetrahedron,24 neighbors with which it only an edge. Twenty-four 16-cells meet at any vertex in this tessellation. A 16-cell can constructed from two Boerdijk–Coxeter helixes of eight chained tetrahedra, each folded into a 4-dimensional ring, the 16 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 16-cell, the cell-first parallel projection of the 16-cell into 3-space has a cubical envelope
16-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.
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)
113.
Uniform 5-polytope
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In geometry, a uniform 5-polytope is a five-dimensional uniform polytope. By definition, a uniform 5-polytope is vertex-transitive and constructed from uniform 4-polytope facets, the complete set of convex uniform 5-polytopes has not been determined, but most can be made as Wythoff constructions from a small set of symmetry groups. These construction operations are represented by the permutations of rings of the Coxeter diagrams, Regular polytopes,1852, Ludwig Schläfli proved in his manuscript Theorie der vielfachen Kontinuität that there are exactly 3 regular polytopes in 5 or more dimensions. Convex uniform polytopes, 1940-1988, The search was expanded systematically by H. S. M, Coxeter in his publication Regular and Semi-Regular Polytopes I, II, and III. 1966, Norman W. Johnson completed his Ph. D, There are exactly three such regular polytopes, all convex, - 5-simplex - 5-cube - 5-orthoplex There are no nonconvex regular polytopes in 5 or more dimensions. There are 104 known convex uniform 5-polytopes, plus a number of families of duoprism prisms. All except the grand antiprism prism are based on Wythoff constructions, the 5-simplex is the regular form in the A5 family. The 5-cube and 5-orthoplex are the forms in the B5 family. The bifurcating graph of the D6 family contains the pentacross, as well as a 5-demicube which is an alternated 5-cube, one non-Wythoffian - The grand antiprism prism is the only known non-Wythoffian convex uniform 5-polytope, constructed from two grand antiprisms connected by polyhedral prisms. That brings the tally to, 19+31+8+46+1=105 In addition there are, Infinitely many uniform 5-polytope constructions based on duoprism prismatic families, Infinitely many uniform 5-polytope constructions based on duoprismatic families, ×, ×, ×. There are 19 forms based on all permutations of the Coxeter diagrams with one or more rings and they are named by Norman Johnson from the Wythoff construction operations upon regular 5-simplex. The A5 family has symmetry of order 720,7 of the 19 figures, with symmetrically ringed Coxeter diagrams have doubled symmetry, order 1440. The coordinates of uniform 5-polytopes with 5-simplex symmetry can be generated as permutations of simple integers in 6-space, the B5 family has symmetry of order 3840. This family has 25−1=31 Wythoffian uniform polytopes generated by marking one or more nodes of the Coxeter diagram, for simplicity it is divided into two subgroups, each with 12 forms, and 7 middle forms which equally belong in both. The 5-cube family of 5-polytopes are given by the hulls of the base points listed in the following table, with all permutations of coordinates. Each base point generates a distinct uniform 5-polytope, all coordinates correspond with uniform 5-polytopes of edge length 2. The D5 family has symmetry of order 1920 and this family has 23 Wythoffian uniform polyhedra, from 3x8-1 permutations of the D5 Coxeter diagram with one or more rings. 15 are repeated from the B5 family and 8 are unique to this family, There are 5 finite categorical uniform prismatic families of polytopes based on the nonprismatic uniform 4-polytopes, This prismatic family has 9 forms, The A1 x A4 family has symmetry of order 240
Uniform 5-polytope
114.
Uniform 6-polytope
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In six-dimensional geometry, a uniform polypeton is a six-dimensional uniform polytope. A uniform polypeton is vertex-transitive, and all facets are uniform 5-polytopes, the complete set of convex uniform polypeta has not been determined, but most can be made as Wythoff constructions from a small set of symmetry groups. These construction operations are represented by the permutations of rings of the Coxeter-Dynkin diagrams, each combination of at least one ring on every connected group of nodes in the diagram produces a uniform 6-polytope. The simplest uniform polypeta are regular polytopes, the 6-simplex, the 6-cube, Regular polytopes,1852, Ludwig Schläfli proved in his manuscript Theorie der vielfachen Kontinuität that there are exactly 3 regular polytopes in 5 or more dimensions. Convex uniform polytopes,1940, The search was expanded systematically by H. S. M, Coxeter in his publication Regular and Semi-Regular Polytopes. Nonregular uniform star polytopes, Ongoing, Thousands of nonconvex uniform polypeta are known, participating researchers include Jonathan Bowers, Richard Klitzing and Norman Johnson. Uniform 6-polytopes with reflective symmetry can be generated by these four Coxeter groups, There are four fundamental reflective symmety groups which generate 153 unique uniform 6-polytopes. Uniform prism There are 6 categorical uniform prisms based on the uniform 5-polytopes, Uniform duoprism There are 11 categorical uniform duoprismatic families of polytopes based on Cartesian products of lower-dimensional uniform polytopes. Each combination of at least one ring on every connected group produces a uniform prismatic 6-polytope, in addition, there are 105 uniform 6-polytope constructions based on prisms of the uniform 5-polytopes. In addition, there are many uniform 6-polytope based on. There are 32+4−1=35 forms, derived by marking one or more nodes of the Coxeter-Dynkin diagram and they are named by Norman Johnson from the Wythoff construction operations upon regular 6-simplex. Bowers-style acronym names are given in parentheses for cross-referencing, the A6 family has symmetry of order 5040. The coordinates of uniform 6-polytopes with 6-simplex symmetry can be generated as permutations of simple integers in 7-space, see also list of A6 polytopes for graphs of these polytopes. There are 63 forms based on all permutations of the Coxeter-Dynkin diagrams with one or more rings, the B6 family has symmetry of order 46080. They are named by Norman Johnson from the Wythoff construction operations upon the regular 6-cube, Bowers names and acronym names are given for cross-referencing. See also list of B6 polytopes for graphs of these polytopes, the D6 family has symmetry of order 23040. This family has 3×16−1=47 Wythoffian uniform polytopes, generated by marking one or more nodes of the D6 Coxeter-Dynkin diagram, of these,31 are repeated from the B6 family and 16 are unique to this family. The 16 unique forms are enumerated below, bowers-style acronym names are given for cross-referencing
Uniform 6-polytope
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6-simplex
115.
6-orthoplex
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In geometry, a 6-orthoplex, or 6-cross polytope, is a regular 6-polytope with 12 vertices,60 edges,160 triangle faces,240 tetrahedron cells,192 5-cell 4-faces, and 64 5-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 6-hypercube, or hexeract, hexacross, derived from combining the family name cross polytope with hex for six in Greek. A lowest symmetry construction is based on a dual of a 6-orthotope, cartesian coordinates for the vertices of a 6-orthoplex, centered at the origin are, Every vertex pair is connected by an edge, except opposites. This polytope is one of 63 uniform 6-polytopes generated from the B6 Coxeter plane, 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.1966 Klitzing, Richard. 6D uniform polytopes x3o3o3o3o4o - gee, archived from the original on 4 February 2007. Polytopes of Various Dimensions Multi-dimensional Glossary
6-orthoplex
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Orthogonal projection inside
Petrie polygon
116.
2 21 polytope
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In 6-dimensional geometry, the 221 polytope is a uniform 6-polytope, constructed within the symmetry of the E6 group. It was discovered by Thorold Gosset, published in his 1900 paper and he called it an 6-ic semi-regular figure. It is also called the Schläfli polytope and its Coxeter symbol is 221, describing its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of one of the 2-node sequences. He also studied its connection with the 27 lines on the cubic surface, the rectified 221 is constructed by points at the mid-edges of the 221. The birectified 221 is constructed by points at the face centers of the 221. The 221 has 27 vertices, and 99 facets,27 5-orthoplexes and 72 5-simplices and its vertex figure is a 5-demicube. For visualization this 6-dimensional polytope is often displayed in a special skewed orthographic projection direction that fits its 27 vertices within a 12-gonal regular polygon and its 216 edges are drawn between 2 rings of 12 vertices, and 3 vertices projected into the center. Higher elements can also be extracted and drawn on this projection, the Schläfli graph contains the 1-skeleton of this polytope. E. L. Elte named it V27 in his 1912 listing of semiregular polytopes, icosihepta-heptacontidi-peton - 27-72 facetted polypeton The 27 vertices can be expressed in 8-space as an edge-figure of the 421 polytope, Its construction is based on the E6 group. The facet information can be extracted from its Coxeter-Dynkin diagram, removing the node on the short branch leaves the 5-simplex. Removing the node on the end of the 2-length branch leaves the 5-orthoplex in its alternated form, every simplex facet touches an 5-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, vertices are colored by their multiplicity in this projection, in progressive order, red, orange, yellow. The number of vertices by color are given in parentheses, the 221 is related to the 24-cell by a geometric folding of the E6/F4 Coxeter-Dynkin diagrams. 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 221. This polytope can tessellate Euclidean 6-space, forming the 222 honeycomb with this Coxeter-Dynkin diagram, the regular complex polygon 333, in C2 has a real representation as the 221 polytope, in 4-dimensional space. It is called a Hessian polyhedron after Edmund Hess and it has 27 vertices,72 3-edges, and 2733 faces. Its complex reflection group is 333, order 648, the 221 is fourth 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
2 21 polytope
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2 21
117.
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
118.
7-simplex
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In 7-dimensional geometry, a 7-simplex is a self-dual regular 7-polytope. It has 8 vertices,28 edges,56 triangle faces,70 tetrahedral cells,56 5-cell 5-faces,28 5-simplex 6-faces and its dihedral angle is cos−1, or approximately 81. 79°. It can also be called an octaexon, or octa-7-tope, as an 8-facetted polytope in 7-dimensions, the name octaexon is derived from octa for eight facets in Greek and -ex for having six-dimensional facets, and -on. Jonathan Bowers gives an octaexon the acronym oca, the Cartesian coordinates of the vertices of an origin-centered regular octaexon having edge length 2 are, More simply, the vertices of the 7-simplex can be positioned in 8-space as permutations of. This construction is based on facets of the 8-orthoplex and this polytope is a facet in the uniform tessellation 331 with Coxeter-Dynkin diagram, This polytope is one of 71 uniform 7-polytopes with A7 symmetry. Polytopes of Various Dimensions Multi-dimensional Glossary
7-simplex
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Model created using straws (edges) and plasticine balls (vertices) in
triakis tetrahedral envelope
7-simplex
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Orthogonal projection inside
Petrie polygon
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-orthoplex
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It has two constructive forms, the first being regular with Schläfli symbol, and the second with alternately labeled facets, with Schläfli symbol or Coxeter symbol 511. It is a part of an family of polytopes, called cross-polytopes or orthoplexes. The dual polytope is an 8-hypercube, or octeract. A lowest symmetry construction is based on a dual of an 8-orthotope, cartesian coordinates for the vertices of an 8-cube, centered at the origin are, Every vertex pair is connected by an edge, except opposites. It is used in its alternated form 511 with the 8-simplex to form the 521 honeycomb, 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. 8D uniform polytopes x3o3o3o3o3o3o4o - ek, archived from the original on 4 February 2007. Polytopes of Various Dimensions Multi-dimensional Glossary
8-orthoplex
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Orthogonal projection inside
Petrie polygon
123.
8-demicube
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In geometry, a demiocteract or 8-demicube is a uniform 8-polytope, constructed from the 8-hypercube, octeract, with alternated vertices truncated. It is part of an infinite family of uniform polytopes called demihypercubes. E. L. Elte identified it in 1912 as a semiregular polytope, Coxeter named this polytope as 151 from its Coxeter diagram, with a ring on one of the 1-length branches, and Schläfli symbol or. Cartesian coordinates for the vertices of an 8-demicube centered at the origin are alternate halves of the 8-cube and this polytope is the vertex figure for the uniform tessellation,251 with Coxeter-Dynkin diagram, H. S. M. Coxeter, Coxeter, Regular Polytopes, Dover edition, ISBN 0-486-61480-8, p.296, Table I, Regular Polytopes, three regular polytopes in n-dimensions H. S. M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York,1973, p.296, Table I, Regular Polytopes, Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication,1995, ISBN 978-0-471-01003-6 H. S. M, Coxeter, Regular and Semi Regular Polytopes I, H. S. M. Coxeter, Regular and Semi-Regular Polytopes II, H. S. M, Coxeter, Regular and Semi-Regular Polytopes III, John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 Olshevsky, George. Archived from the original on 4 February 2007
8-demicube
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Petrie polygon projection
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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4_{21}
125.
4 21 polytope
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In 8-dimensional geometry, the 421 is a semiregular uniform 8-polytope, constructed within the symmetry of the E8 group. It was discovered by Thorold Gosset, published in his 1900 paper and he called it an 8-ic semi-regular figure. Its Coxeter symbol is 421, describing its bifurcating Coxeter-Dynkin diagram, the rectified 421 is constructed by points at the mid-edges of the 421. The birectified 421 is constructed by points at the face centers of the 421. The trirectified 421 is constructed by points at the centers of the 421. The 421 is composed of 17,280 7-simplex and 2,160 7-orthoplex facets and its vertex figure is the 321 polytope. For visualization this 8-dimensional polytope is often displayed in a special skewed orthographic projection direction that fits its 240 vertices within a regular triacontagon and its 6720 edges are drawn between the 240 vertices. Specific higher elements can also be extracted and drawn on this projection, as its 240 vertices represent the root vectors of the simple Lie group E8, the polytope is sometimes referred to as the E8 polytope. The vertices of this polytope can be obtained by taking the 240 integral octonions of norm 1, because the octonions are a nonassociative normed division algebra, these 240 points have a multiplication operation making them not into a group but rather a loop, in fact a Moufang loop. This polytope was discovered by Thorold Gosset, who described it in his 1900 paper as an 8-ic semi-regular figure and it is the last finite semiregular figure in his enumeration, semiregular to him meaning that it contained only regular facets. E. L. Elte named it V240 in his 1912 listing of semiregular polytopes, Coxeter called it 421 because its Coxeter-Dynkin diagram has three branches of length 4,2, and 1, with a single node on the terminal node of the 4 branch. Dischiliahectohexaconta-myriaheptachiliadiacosioctaconta-zetton - 2160-17280 facetted polyzetton It is created by a Wythoff construction upon a set of 8 hyperplane mirrors in 8-dimensional space and these 56 points are the vertices of a 321 polytope in 7 dimensions. These 126 points are the vertices of a 231 polytope in 7 dimensions. Each vertex also has 56 third nearest neighbors, which are the negatives of its nearest neighbors, there are 17,280 simplex facets and 2160 orthoplex facets. Since every 7-simplex has 7 6-simplex facets, each incident to no other 6-simplex, since every 7-orthoplex has 128 6-simplex facets, half of which are not incident to 7-simplexes, the 421 polytope has 138,240 6-simplex faces that are not facets of 7-simplexes. The 421 polytope thus has two kinds of 6-simplex faces, not interchanged by symmetries of this polytope, the total number of 6-simplex faces is 259200. The vertex figure of a polytope is obtained by removing the ringed node. These graphs represent orthographic projections in the E8, E7, E6, the vertex colors are by overlapping multiplicity in the projection, colored by increasing order of multiplicities as red, orange, yellow, green
4 21 polytope
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The 4 21 graph created as
string art.
4 21 polytope
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4 21
4 21 polytope
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The 4 21 polytope can be projected into 3-space as a physical vertex-edge model. Pictured here as 2 concentric
600-cells (at the golden ratio) using
Zome tools. (Not all of the 3360 edges of length √2(√5-1) are represented.)
126.
Uniform 9-polytope
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In nine-dimensional geometry, a nine-dimensional polytope or 9-polytope is a polytope contained by 8-polytope facets. Each 7-polytope ridge being shared by exactly two 8-polytope facets, a uniform 9-polytope is one which is vertex-transitive, and constructed from uniform 8-polytope facets. Regular 9-polytopes can be represented by the Schläfli symbol, with w 8-polytope facets around each peak, There are exactly three such convex regular 9-polytopes, - 9-simplex - 9-cube - 9-orthoplex There are no nonconvex regular 9-polytopes. The topology of any given 9-polytope is defined by its Betti numbers, 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, - 9-orthoplex,611 - The A9 family has symmetry of order 3628800. There are 256+16-1=271 forms based on all permutations of the Coxeter-Dynkin diagrams with one or more rings, bowers-style acronym names are given in parentheses for cross-referencing. There are 511 forms based on all permutations of the Coxeter-Dynkin diagrams with one or more rings, eleven cases are shown below, Nine rectified forms and 2 truncations. Bowers-style acronym names are given in parentheses for cross-referencing, bowers-style acronym names are given in parentheses for cross-referencing. The D9 family has symmetry of order 92,897,280 and this family has 3×128−1=383 Wythoffian uniform polytopes, generated by marking one or more nodes of the D9 Coxeter-Dynkin diagram. Of these,255 are repeated from the B9 family and 128 are unique to this family, bowers-style acronym names are given in parentheses for cross-referencing. However, there are 4 noncompact hyperbolic Coxeter groups of rank 9, 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 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 9-polytope
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9-simplex
127.
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
128.
10-cube
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In geometry, a 10-cube is a ten-dimensional hypercube. It can be named by its Schläfli symbol, being composed of 3 9-cubes around each 8-face and it is a part of an infinite family of polytopes, called hypercubes. The dual of a dekeract can be called a 10-orthoplex or decacross, cartesian coordinates for the vertices of a dekeract centered at the origin and edge length 2 are while the interior of the same consists of all points with −1 < xi <1. Applying an alternation operation, deleting alternating vertices of the dekeract, creates another uniform polytope, called a 10-demicube, which has 20 demienneractic and 512 enneazettonic 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, 10D uniform polytopes o3o3o3o3o3o3o3o3o4x - deker. Archived from the original on 4 February 2007, multi-dimensional Glossary, hypercube Garrett Jones Sloanes A135289, Hypercubes, 10-cube. The On-Line Encyclopedia of Integer Sequences
10-cube
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Orthogonal projection inside
Petrie polygon Orange vertices are doubled, and central yellow one has four
129.
10-demicube
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In geometry, a 10-demicube or demidekeract is a uniform 10-polytope, constructed from the 10-cube with alternated vertices truncated. It is part of an infinite family of uniform polytopes called demihypercubes. E. L. Elte identified it in 1912 as a semiregular polytope, Coxeter named this polytope as 171 from its Coxeter diagram, with a ring on one of the 1-length branches, and Schläfli symbol or. Cartesian coordinates for the vertices of a demidekeract centered at the origin are alternate halves of the dekeract, with an odd number of plus signs. 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 x3o3o *b3o3o3o3o3o3o3o - hede, archived from the original on 4 February 2007
10-demicube
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Petrie polygon projection
130.
Polytope
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In elementary geometry, a polytope is a geometric object with flat sides, and may exist in any general number of dimensions n as an n-dimensional polytope or n-polytope. For example, a polygon is a 2-polytope and a three-dimensional polyhedron is a 3-polytope. Polytopes in more than three dimensions were first discovered by Ludwig Schläfli, the German term polytop was coined by the mathematician Reinhold Hoppe, and was introduced to English mathematicians as polytope by Alicia Boole Stott. The term polytope is nowadays a broad term that covers a class of objects. Many of these definitions are not equivalent, resulting in different sets of objects being called polytopes and they represent different approaches to generalizing the convex polytopes to include other objects with similar properties. In this approach, a polytope may be regarded as a tessellation or decomposition of some given manifold, an example of this approach defines a polytope as a set of points that admits a simplicial decomposition. However this definition does not allow star polytopes with interior structures, the discovery of star polyhedra and other unusual constructions led to the idea of a polyhedron as a bounding surface, ignoring its interior. A polyhedron is understood as a surface whose faces are polygons, a 4-polytope as a hypersurface whose facets are polyhedra and this approach is used for example in the theory of abstract polytopes. In certain fields of mathematics, the terms polytope and polyhedron are used in a different sense and this terminology is typically confined to polytopes and polyhedra that are convex. A polytope comprises elements of different dimensionality such as vertices, edges, faces, cells, terminology for these is not fully consistent across different authors. For example, some authors use face to refer to an -dimensional element while others use face to denote a 2-face specifically, authors may use j-face or j-facet to indicate an element of j dimensions. Some use edge to refer to a ridge, while H. S. M. Coxeter uses cell to denote an -dimensional element, the terms adopted in this article are given in the table below, An n-dimensional polytope is bounded by a number of -dimensional facets. These facets are themselves polytopes, whose facets are -dimensional ridges of the original polytope, Every ridge arises as the intersection of two facets. Ridges are once again polytopes whose facets give rise to -dimensional boundaries of the original polytope and these bounding sub-polytopes may be referred to as faces, or specifically j-dimensional faces or j-faces. A 0-dimensional face is called a vertex, and consists of a single point, a 1-dimensional face is called an edge, and consists of a line segment. A 2-dimensional face consists of a polygon, and a 3-dimensional face, sometimes called a cell, the convex polytopes are the simplest kind of polytopes, and form the basis for several different generalizations of the concept of polytopes. A convex polytope is defined as the intersection of a set of half-spaces. This definition allows a polytope to be neither bounded nor finite, Polytopes are defined in this way, e. g. in linear programming
Polytope
–
A
polygon is a 2-dimensional polytope.
131.
Uniform 1 k2 polytope
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In geometry, 1k2 polytope is a uniform polytope in n-dimensions constructed from the En Coxeter group. The family was named by their Coxeter symbol 1k2 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-demicube in 5-dimensions, each polytope is constructed from 1k-1,2 and -demicube facets. Each has a figure of a polytope is a birectified n-simplex. The sequence ends with k=6, as a tessellation of 9-dimensional hyperbolic 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 1 k2 polytope
132.
Uniform 2 k1 polytope
–
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
133.
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