1.
Orthogonal projection
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In linear algebra and functional analysis, a projection is a linear transformation P from a vector space to itself such that 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
2.
Coxeter plane
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In mathematics, the Coxeter number h is the order of a Coxeter element of an irreducible Coxeter group. Note that this assumes a finite Coxeter group. For infinite Coxeter groups, there are multiple classes of Coxeter elements. There are many different ways to define the Coxeter number h of a root system. A Coxeter element is a product of all simple reflections, the product depends on the order in which they are taken, but different orderings produce conjugate elements, which have the same order. The Coxeter number is the number of roots divided by the rank, the number of reflections in the Coxeter group is half the number of roots. The Coxeter number is the order of any Coxeter element, if the highest root is ∑miαi for simple roots αi, then the Coxeter number is 1 + ∑mi The dimension of the corresponding Lie algebra is n, where n is the rank and h is the Coxeter number. The Coxeter number is the highest degree of an invariant of the Coxeter group acting on polynomials. Notice that if m is a degree of a fundamental invariant then so is h +2 − m, the eigenvalues of a Coxeter element are the numbers e2πi/h as m runs through the degrees of the fundamental invariants. Since this starts with m =2, these include the primitive hth root of unity, ζh = e2πi/h, an example, has h=30, so 64*30/g =12 -3 -6 -5 + 4/3 + 4/5 = 2/15, so g = 1920*15/2= 960*15 =14400. Coxeter elements of A n −1 ≅ S n, considered as the group on n elements, are n-cycles, for simple reflections the adjacent transpositions, …. The dihedral group Dihm is generated by two reflections that form an angle of 2 π /2 m, and thus their product is a rotation by 2 π / m. For a given Coxeter element w, there is a unique plane P on which w acts by rotation by 2π/h and this is called the Coxeter plane and is the plane on which P has eigenvalues e2πi/h and e−2πi/h = e2πi/h. This plane was first systematically studied in, and subsequently used in to provide uniform proofs about properties of Coxeter elements, for polytopes, a vertex may map to zero, as depicted below. Projections onto the Coxeter plane are depicted below for the Platonic solids, in three dimensions, the symmetry of a regular polyhedron, with one directed petrie polygon marked, defined as a composite of 3 reflections, has rotoinversion symmetry Sh, order h. Adding a mirror, the symmetry can be doubled to symmetry, Dhd. In orthogonal 2D projection, this becomes dihedral symmetry, Dihh, in four dimension, the symmetry of a regular polychoron, with one directed petrie polygon marked is a double rotation, defined as a composite of 4 reflections, with symmetry +1/h, order h. In five dimension, the symmetry of a regular polyteron, with one directed petrie polygon marked, is represented by the composite of 5 reflections
3.
Geometry
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Geometry is a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space. A mathematician who works in the field of geometry is called a geometer, Geometry arose independently in a number of early cultures as a practical way for dealing with lengths, areas, and volumes. Geometry began to see elements of mathematical science emerging in the West as early as the 6th century BC. By the 3rd century BC, geometry was put into a form by Euclid, whose treatment, Euclids Elements. Geometry arose independently in India, with texts providing rules for geometric constructions appearing as early as the 3rd century BC, islamic scientists preserved Greek ideas and expanded on them during the Middle Ages. By the early 17th century, geometry had been put on a solid footing by mathematicians such as René Descartes. Since then, and into modern times, geometry has expanded into non-Euclidean geometry and manifolds, while geometry has evolved significantly throughout the years, there are some general concepts that are more or less fundamental to geometry. These include the concepts of points, lines, planes, surfaces, angles, contemporary geometry has many subfields, Euclidean geometry is geometry in its classical sense. The mandatory educational curriculum of the majority of nations includes the study of points, lines, planes, angles, triangles, congruence, similarity, solid figures, circles, Euclidean geometry also has applications in computer science, crystallography, and various branches of modern mathematics. Differential geometry uses techniques of calculus and linear algebra to problems in geometry. It has applications in physics, including in general relativity, topology is the field concerned with the properties of geometric objects that are unchanged by continuous mappings. In practice, this often means dealing with large-scale properties of spaces, convex geometry investigates convex shapes in the Euclidean space and its more abstract analogues, often using techniques of real analysis. It has close connections to convex analysis, optimization and functional analysis, algebraic geometry studies geometry through the use of multivariate polynomials and other algebraic techniques. It has applications in areas, including cryptography and string theory. Discrete geometry is concerned mainly with questions of relative position of simple objects, such as points. It shares many methods and principles with combinatorics, Geometry has applications to many fields, including art, architecture, physics, as well as to other branches of mathematics. The earliest recorded beginnings of geometry can be traced to ancient Mesopotamia, the earliest known texts on geometry are the Egyptian Rhind Papyrus and Moscow Papyrus, the Babylonian clay tablets such as Plimpton 322. For example, the Moscow Papyrus gives a formula for calculating the volume of a truncated pyramid, later clay tablets demonstrate that Babylonian astronomers implemented trapezoid procedures for computing Jupiters position and motion within time-velocity space
4.
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
5.
Rectification (geometry)
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In Euclidean geometry, rectification or complete-truncation is the process of truncating a polytope by marking the midpoints of all its edges, and cutting off its vertices at those points. The resulting polytope will be bounded by vertex figure facets and the facets of the original polytope. A rectification operator is denoted by the symbol r, for example, r is the rectified cube. Conway polyhedron notation uses ambo for this operator, in graph theory this operation creates a medial graph. Rectification is the point of a truncation process. The highest degree of rectification creates the dual polytope, a rectification truncates edges to points. A birectification truncates faces to points, a trirectification truncates cells to points, and so on. New vertices are placed at the center of the edges of the original polygon, each platonic solid and its dual have the same rectified polyhedron. The rectified polyhedron turns out to be expressible as the intersection of the original platonic solid with an appropriated scaled concentric version of its dual, the rectified octahedron, whose dual is the cube, is the cuboctahedron. The rectified icosahedron, whose dual is the dodecahedron, is the icosidodecahedron, a rectified square tiling is a square tiling. A rectified triangular tiling or hexagonal tiling is a trihexagonal tiling, examples If a polyhedron is not regular, the edge midpoints surrounding a vertex may not be coplanar. The resulting medial graph remains polyhedral, so by Steinitzs theorem it can be represented as a polyhedron, the Conway polyhedron notation equivalent to rectification is ambo, represented by a. Applying twice aa, is Conways expand operation, e, which is the same as Johnsons cantellation operation, t0,2 generated from regular polyhedral, each Convex regular 4-polytope has a rectified form as a uniform 4-polytope. Its rectification will have two types, a rectified polyhedron left from the original cells and polyhedron as new cells formed by each truncated vertex. A rectified is not the same as a rectified, however, a further truncation, called bitruncation, is symmetric between a 4-polytope and its dual. Examples A first rectification truncates edges down to points, If a polytope is regular, this form is represented by an extended Schläfli symbol notation t1 or r. A second rectification, or birectification, truncates faces down to points, If regular it has notation t2 or 2r. For polyhedra, a birectification creates a dual polyhedron, higher degree rectifications can be constructed for higher dimensional polytopes
6.
5-simplex
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In five-dimensional geometry, a 5-simplex is a self-dual regular 5-polytope. It has six vertices,15 edges,20 triangle faces,15 tetrahedral cells and it has a dihedral angle of cos−1, or approximately 78. 46°. It can also be called a hexateron, or hexa-5-tope, as a 6-facetted polytope in 5-dimensions, the name hexateron is derived from hexa- for having six facets and teron for having four-dimensional facets. By Jonathan Bowers, a hexateron is given the acronym hix, the hexateron can be constructed from a 5-cell by adding a 6th vertex such that it is equidistant from all the other vertices of the 5-cell. These construction can be seen as facets of the 6-orthoplex or rectified 6-cube respectively and it is first in a dimensional series of uniform polytopes and honeycombs, expressed by Coxeter as 13k series. A degenerate 4-dimensional case exists as 3-sphere tiling, a tetrahedral dihedron and it is first in a dimensional series of uniform polytopes and honeycombs, expressed by Coxeter as 3k1 series. A degenerate 4-dimensional case exists as 3-sphere tiling, a tetrahedral hosohedron, the 5-simplex, as 220 polytope is first in dimensional series 22k. The regular 5-simplex is one of 19 uniform polytera based on the Coxeter group, the 5-simplex can also be considered a 5-cell pyramid, constructed as a 5-cell base in a 4-space hyperplane, and an apex point above the hyperplane. The five sides of the pyramid are made of 5-cell cells, T. Gosset, On the Regular and Semi-Regular Figures in Space of n Dimensions, Messenger of Mathematics, Macmillan,1900 H. S. M. Coxeter, Coxeter, Regular Polytopes, Dover edition, ISBN 0-486-61480-8, p.296, Table I, Regular Polytopes, three regular polytopes in n-dimensions H. S. M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York,1973, p.296, Table I, Regular Polytopes, Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication,1995, ISBN 978-0-471-01003-6 H. S. M, Coxeter, Regular and Semi Regular Polytopes I, H. S. M. Coxeter, Regular and Semi-Regular Polytopes II, H. S. M, johnson, The Theory of Uniform Polytopes and Honeycombs, Ph. D. 5D uniform polytopes x3o3o3o3o - hix, archived from the original on 4 February 2007. Polytopes of Various Dimensions, Jonathan Bowers Multi-dimensional Glossary
7.
Coxeter diagram
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In geometry, a Coxeter–Dynkin diagram is a graph with numerically labeled edges representing the spatial relations between a collection of mirrors. It describes a kaleidoscopic construction, each node represents a mirror. An unlabeled branch implicitly represents order-3, each diagram represents a Coxeter group, and Coxeter groups are classified by their associated diagrams. Dynkin diagrams correspond to and are used to root systems. Branches of a Coxeter–Dynkin diagram are labeled with a number p. When p =2 the angle is 90° and the mirrors have no interaction, if a branch is unlabeled, it is assumed to have p =3, representing an angle of 60°. Two parallel mirrors have a branch marked with ∞, in principle, n mirrors can be represented by a complete graph in which all n /2 branches are drawn. In practice, nearly all interesting configurations of mirrors include a number of right angles, diagrams can be labeled by their graph structure. The first forms studied by Ludwig Schläfli are the orthoschemes which have linear graphs that generate regular polytopes, plagioschemes are simplices represented by branching graphs, and cycloschemes are simplices represented by cyclic graphs. Every Coxeter diagram has a corresponding Schläfli matrix with matrix elements ai, j = aj, as a matrix of cosines, it is also called a Gramian matrix after Jørgen Pedersen Gram. All Coxeter group Schläfli matrices are symmetric because their root vectors are normalized. It is related closely to the Cartan matrix, used in the similar but directed graph Dynkin diagrams in the cases of p =2,3,4, and 6. The determinant of the Schläfli matrix, called the Schläflian, and its sign determines whether the group is finite, affine and this rule is called Schläflis Criterion. The eigenvalues of the Schläfli matrix determines whether a Coxeter group is of type, affine type. The indefinite type is further subdivided, e. g. into hyperbolic. However, there are multiple non-equivalent definitions for hyperbolic Coxeter groups and we use the following definition, A Coxeter group with connected diagram is hyperbolic if it is neither of finite nor affine type, but every proper connected subdiagram is of finite or affine type. A hyperbolic Coxeter group is compact if all subgroups are finite, Finite and affine groups are also called elliptical and parabolic respectively. Hyperbolic groups are also called Lannér, after F. Lannér who enumerated the compact groups in 1950
8.
5-cell
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In geometry, the 5-cell is a four-dimensional object bounded by 5 tetrahedral cells. It is also known as a C5, pentachoron, pentatope, pentahedroid and it is a 4-simplex, the simplest possible convex regular 4-polytope, and is analogous to the tetrahedron in three dimensions and the triangle in two dimensions. The pentachoron is a four dimensional pyramid with a tetrahedral base, the regular 5-cell is bounded by regular tetrahedra, and is one of the six regular convex 4-polytopes, represented by Schläfli symbol. Pentachoron 4-simplex Pentatope Pentahedroid Pen Hyperpyramid, tetrahedral pyramid The 5-cell is self-dual and its maximal intersection with 3-dimensional space is the triangular prism. Its dihedral angle is cos−1, or approximately 75. 52°, the 5-cell can be constructed from a tetrahedron by adding a 5th vertex such that it is equidistant from all the other vertices of the tetrahedron. The simplest set of coordinates is, with edge length 2√2, a 5-cell can be constructed as a Boerdijk–Coxeter helix of five chained tetrahedra, folded into a 4-dimensional ring. The 10 triangle faces can be seen in a 2D net within a triangular tiling, with 6 triangles around every vertex, the purple edges represent the Petrie polygon of the 5-cell. The A4 Coxeter plane projects the 5-cell into a regular pentagon, the four sides of the pyramid are made of tetrahedron cells. Many uniform 5-polytopes have tetrahedral pyramid vertex figures, Other uniform 5-polytopes have irregular 5-cell vertex figures, the symmetry of a vertex figure of a uniform polytope is represented by removing the ringed nodes of the Coxeter diagram. The compound of two 5-cells in dual configurations can be seen in this A5 Coxeter plane projection, with a red and this compound has symmetry, order 240. The intersection of these two 5-cells is a uniform birectified 5-cell, the pentachoron is the simplest of 9 uniform polychora constructed from the Coxeter group. It is in the sequence of regular polychora, the tesseract, 120-cell, of Euclidean 4-space, all of these have a tetrahedral vertex figure. It is similar to three regular polychora, the tesseract, 600-cell of Euclidean 4-space, and the order-6 tetrahedral honeycomb of hyperbolic space, all of these have a tetrahedral cell. T. Gosset, On the Regular and Semi-Regular Figures in Space of n Dimensions, Messenger of Mathematics, Macmillan,1900 H. S. M. Coxeter, Coxeter, Regular Polytopes, Dover edition, ISBN 0-486-61480-8, p.296, Table I, Regular Polytopes, three regular polytopes in n-dimensions H. S. M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York,1973, p.296, Table I, Regular Polytopes, Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication,1995, ISBN 978-0-471-01003-6 H. S. M, Coxeter, Regular and Semi Regular Polytopes I, H. S. M. Coxeter, Regular and Semi-Regular Polytopes II, H. S. M, johnson, The Theory of Uniform Polytopes and Honeycombs, Ph. D
9.
Rectified 5-cell
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In four-dimensional geometry, the rectified 5-cell is a uniform 4-polytope composed of 5 regular tetrahedral and 5 regular octahedral cells. Each edge has one tetrahedron and two octahedra, each vertex has two tetrahedra and three octahedra. In total it has 30 triangle faces,30 edges, and 10 vertices, each vertex is surrounded by 3 octahedra and 2 tetrahedra, the vertex figure is a triangular prism. The vertex figure of the rectified 5-cell is a triangular prism. Together with the simplex and 24-cell, this shape and its dual was one of the first 2-simple 2-simplicial 4-polytopes known and this means that all of its two-dimensional faces, and all of the two-dimensional faces of its dual, are triangles. In 1997, Tom Braden found another pair of examples. The birectified 5-cell can be seen as the intersection of two regular 5-cells in dual positions and it is one of three semiregular 4-polytope made of two or more cells which are Platonic solids, discovered by Thorold Gosset in his 1900 paper. He called it a tetroctahedric for being made of tetrahedron and octahedron cells, E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as tC5. These construction can be seen as positive orthant facets of the rectified pentacross or birectified penteract respectively and this polytope is the vertex figure of the 5-demicube, and the edge figure of the uniform 221 polytope. It is also one of 9 Uniform 4-polytopes constructed from the Coxeter group, the rectified 5-cell is second in a dimensional series of semiregular polytopes. Each progressive uniform polytope is constructed as the figure of the previous polytope. Thorold Gosset identified this series in 1900 as containing all regular polytope facets, the Coxeter symbol for the rectified 5-cell is 021. Semiregular k 21 polytope T. Gosset, On the Regular and Semi-Regular Figures in Space of n Dimensions, Messenger of Mathematics, Macmillan,1900 J. H. Conway and M. J. T. Guy, Four-Dimensional Archimedean Polytopes, Proceedings of the Colloquium on Convexity at Copenhagen, 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. John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 Rectified 5-cell - data, convex uniform polychora based on the pentachoron - Model 2, George Olshevsky
10.
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
11.
Octahedron
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In geometry, an octahedron is a polyhedron with eight faces, twelve edges, and six vertices. A regular octahedron is a Platonic solid composed of eight equilateral triangles, a regular octahedron is the dual polyhedron of a cube. It is a square bipyramid in any of three orthogonal orientations and it is also a triangular antiprism in any of four orientations. An octahedron is the case of the more general concept of a cross polytope. A regular octahedron is a 3-ball in the Manhattan metric, the second and third correspond to the B2 and A2 Coxeter planes. The octahedron can also be represented as a tiling. 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 octahedron with edge length √2 can be placed with its center at the origin and its vertices on the coordinate axes, the Cartesian coordinates of the vertices are then. In an x–y–z Cartesian coordinate system, the octahedron with center coordinates, additionally the inertia tensor of the stretched octahedron is I =. These reduce to the equations for the regular octahedron when x m = y m = z m = a 22, the interior of the compound of two dual tetrahedra is an octahedron, and this compound, called the stella octangula, is its first and only stellation. Correspondingly, an octahedron is the result of cutting off from a regular tetrahedron. One can also divide the edges of an octahedron in the ratio of the mean to define the vertices of an icosahedron. There are five octahedra that define any given icosahedron in this fashion, octahedra and tetrahedra can be alternated to form a vertex, edge, and face-uniform tessellation of space, called the octet truss by Buckminster Fuller. This is the only such tiling save the regular tessellation of cubes, another is a tessellation of octahedra and cuboctahedra. The octahedron is unique among the Platonic solids in having a number of faces meeting at each vertex. Consequently, it is the member of that group to possess mirror planes that do not pass through any of the faces. Using the standard nomenclature for Johnson solids, an octahedron would be called a square bipyramid, truncation of two opposite vertices results in a square bifrustum. The octahedron is 4-connected, meaning that it takes the removal of four vertices to disconnect the remaining vertices and it is one of only four 4-connected simplicial well-covered polyhedra, meaning that all of the maximal independent sets of its vertices have the same size
12.
Triangle
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A triangle is a polygon with three edges and three vertices. It is one of the shapes in geometry. A triangle with vertices A, B, and C is denoted △ A B C, in Euclidean geometry any three points, when non-collinear, determine a unique triangle and a unique plane. This article is about triangles in Euclidean geometry except where otherwise noted, triangles can be classified according to the lengths of their sides, An equilateral triangle has all sides the same length. An equilateral triangle is also a polygon with all angles measuring 60°. An isosceles triangle has two sides of equal length, some mathematicians define an isosceles triangle to have exactly two equal sides, whereas others define an isosceles triangle as one with at least two equal sides. The latter definition would make all equilateral triangles isosceles triangles, the 45–45–90 right triangle, which appears in the tetrakis square tiling, is isosceles. A scalene triangle has all its sides of different lengths, equivalently, it has all angles of different measure. Hatch marks, also called tick marks, are used in diagrams of triangles, a side can be marked with a pattern of ticks, short line segments in the form of tally marks, two sides have equal lengths if they are both marked with the same pattern. In a triangle, the pattern is no more than 3 ticks. Similarly, patterns of 1,2, or 3 concentric arcs inside the angles are used to indicate equal angles, triangles can also be classified according to their internal angles, measured here in degrees. A right triangle has one of its interior angles measuring 90°, the side opposite to the right angle is the hypotenuse, the longest side of the triangle. The other two sides are called the legs or catheti of the triangle, special right triangles are right triangles with additional properties that make calculations involving them easier. One of the two most famous is the 3–4–5 right triangle, where 32 +42 =52, in this situation,3,4, and 5 are a Pythagorean triple. The other one is a triangle that has 2 angles that each measure 45 degrees. Triangles that do not have an angle measuring 90° are called oblique triangles, a triangle with all interior angles measuring less than 90° is an acute triangle or acute-angled triangle. If c is the length of the longest side, then a2 + b2 > c2, a triangle with one interior angle measuring more than 90° is an obtuse triangle or obtuse-angled triangle. If c is the length of the longest side, then a2 + b2 < c2, a triangle with an interior angle of 180° is degenerate
13.
Vertex figure
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In geometry, a vertex figure, broadly speaking, is the figure exposed when a corner of a polyhedron or polytope is sliced off. Take some vertex of a polyhedron, mark a point somewhere along each connected edge. Draw lines across the faces, joining adjacent points. When done, these form a complete circuit, i. e. a polygon. This polygon is the vertex figure, more precise formal definitions can vary quite widely, according to circumstance. For example Coxeter varies his definition as convenient for the current area of discussion, most of the following definitions of a vertex figure apply equally well to infinite tilings, or space-filling tessellation with polytope cells. Make a slice through the corner of the polyhedron, cutting all the edges connected to the vertex. The cut surface is the vertex figure and this is perhaps the most common approach, and the most easily understood. Different authors make the slice in different places, Wenninger cuts each edge a unit distance from the vertex, as does Coxeter. For uniform polyhedra the Dorman Luke construction cuts each connected edge at its midpoint, other authors make the cut through the vertex at the other end of each edge. For irregular polyhedra, these approaches may produce a figure that does not lie in a plane. A more general approach, valid for convex polyhedra, is to make the cut along any plane which separates the given vertex from all the other vertices. Cromwell makes a cut or scoop, centered on the vertex. The cut surface or vertex figure is thus a spherical polygon marked on this sphere, many combinatorial and computational approaches treat a vertex figure as the ordered set of points of all the neighboring vertices to the given vertex. In the theory of polytopes, the vertex figure at a given vertex V comprises all the elements which are incident on the vertex, edges, faces. More formally it is the -section Fn/V, where Fn is the greatest face and this set of elements is elsewhere known as a vertex star. A vertex figure for an n-polytope is an -polytope, for example, a vertex figure for a polyhedron is a polygon figure, and the vertex figure for a 4-polytope is a polyhedron. Each edge of the vertex figure exists on or inside of a face of the original polytope connecting two vertices from an original face
14.
Tetrahedral prism
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In geometry, a tetrahedral prism is a convex uniform 4-polytope. This 4-polytope has 6 polyhedral cells,2 tetrahedra connected by 4 triangular prisms and it has 14 faces,8 triangular and 6 square. It has 16 edges and 8 vertices and it is one of 18 uniform polyhedral prisms created by using uniform prisms to connect pairs of parallel Platonic solids and Archimedean solids. Tetrahedral dyadic prism Tepe Tetrahedral hyperprism Digonal antiprismatic prism Digonal antiprismatic hyperprism The tetrahedral prism is bounded by two tetrahedra and four triangular prisms, the triangular prisms are joined to each other via their square faces, and are joined to the two tetrahedra via their triangular faces. The tetrahedron-first orthographic projection of the prism into 3D space has a tetrahedral projection envelope. Both tetrahedral cells project onto this tetrahedron, while the triangular prisms project to its faces, the triangular-prism-first orthographic projection of the tetrahedral prism into 3D space has a projection envelope in the shape of a triangular prism. The two tetrahedral cells are projected onto the ends of the prism, each with a vertex that projects to the center of the respective triangular face. An edge connects two vertices through the center of the projection. The prism may be divided into three triangular prisms that meet at this edge, these 3 volumes correspond with the images of three of the four triangular prismic cells. The last triangular prismic cell projects onto the projection envelope. The edge-first orthographic projection of the prism into 3D space is identical to its triangular-prism-first parallel projection. The square-face-first orthographic projection of the prism into 3D space has a cubical envelope. Each triangular prismic cell projects onto half of the cubical volume, the tetrahedral cells project onto the top and bottom faces of the cube. It is the first in an series of uniform antiprismatic prisms. The tetrahedral prism, -131, is first in a series of uniform polytopes. The tetrahedral prism is the figure for the second, the rectified 5-simplex. The fifth figure is a Euclidean honeycomb,331, and the final is a noncompact hyperbolic honeycomb,431, each uniform polytope in the sequence is the vertex figure of the next. John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 Norman Johnson Uniform Polytopes, convex uniform prismatic polychora - Model 48, George Olshevsky
15.
Coxeter group
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In mathematics, a Coxeter group, named after H. S. M. Coxeter, is an abstract group that admits a formal description in terms of reflections. Indeed, the finite Coxeter groups are precisely the finite Euclidean reflection groups, however, not all Coxeter groups are finite, and not all can be described in terms of symmetries and Euclidean reflections. Coxeter groups were introduced as abstractions of reflection groups, and finite Coxeter groups were classified in 1935, Coxeter groups find applications in many areas of mathematics. Examples of finite Coxeter groups include the groups of regular polytopes. The condition m i j = ∞ means no relation of the form m should be imposed, the pair where W is a Coxeter group with generators S = is called a Coxeter system. Note that in general S is not uniquely determined by W, for example, the Coxeter groups of type B3 and A1 × A3 are isomorphic but the Coxeter systems are not equivalent. A number of conclusions can be drawn immediately from the above definition, the relation m i i =1 means that 1 =2 =1 for all i, as such the generators are involutions. If m i j =2, then the r i and r j commute. This follows by observing that x x = y y =1, in order to avoid redundancy among the relations, it is necessary to assume that m i j = m j i. This follows by observing that y y =1, together with m =1 implies that m = m y y = y m y = y y =1. Alternatively, k and k are elements, as y k y −1 = k y y −1 = k. The Coxeter matrix is the n × n, symmetric matrix with entries m i j, indeed, every symmetric matrix with positive integer and ∞ entries and with 1s on the diagonal such that all nondiagonal entries are greater than 1 serves to define a Coxeter group. The Coxeter matrix can be encoded by a Coxeter diagram. The vertices of the graph are labelled by generator subscripts, vertices i and j are adjacent if and only if m i j ≥3. An edge is labelled with the value of m i j whenever the value is 4 or greater, in particular, two generators commute if and only if they are not connected by an edge. Furthermore, if a Coxeter graph has two or more connected components, the group is the direct product of the groups associated to the individual components. Thus the disjoint union of Coxeter graphs yields a product of Coxeter groups. The Coxeter matrix, M i j, is related to the n × n Schläfli matrix C with entries C i j = −2 cos , but the elements are modified, being proportional to the dot product of the pairwise generators
16.
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
17.
Convex polytope
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A convex polytope is a special case of a polytope, having the additional property that it is also a convex set of points in the n-dimensional space Rn. Some authors use the terms polytope and convex polyhedron interchangeably. In addition, some require a polytope to be a bounded set. The terms bounded/unbounded convex polytope will be used whenever the boundedness is critical to the discussed issue. Yet other texts treat a convex n-polytope as a surface or -manifold, Convex polytopes play an important role both in various branches of mathematics and in applied areas, most notably in linear programming. A comprehensive and influential book in the subject, called Convex Polytopes, was published in 1967 by Branko Grünbaum, in 2003 the 2nd edition of the book was published, with significant additional material contributed by new writers. In Grünbaums book, and in other texts in discrete geometry. Grünbaum points out that this is solely to avoid the repetition of the word convex. A polytope is called if it is an n-dimensional object in Rn. Many examples of bounded convex polytopes can be found in the article polyhedron, a convex polytope may be defined in a number of ways, depending on what is more suitable for the problem at hand. Grünbaums definition is in terms of a set of points in space. Other important definitions are, as the intersection of half-spaces and as the hull of a set of points. This is equivalent to defining a bounded convex polytope as the hull of a finite set of points. Such a definition is called a vertex representation, for a compact convex polytope, the minimal V-description is unique and it is given by the set of the vertices of the polytope. A convex polytope may be defined as an intersection of a number of half-spaces. Such definition is called a half-space representation, there exist infinitely many H-descriptions of a convex polytope. However, for a convex polytope, the minimal H-description is in fact unique and is given by the set of the facet-defining halfspaces. A closed half-space can be written as an inequality, a 1 x 1 + a 2 x 2 + ⋯ + a n x n ≤ b where n is the dimension of the space containing the polytope under consideration
18.
Isogonal figure
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In geometry, a polytope is isogonal or vertex-transitive if, loosely speaking, all its vertices are equivalent. That implies that each vertex is surrounded by the kinds of face in the same or reverse order. Technically, we say that for any two vertices there exists a symmetry of the polytope mapping the first isometrically onto the second. Other ways of saying this are that the group of automorphisms of the polytope is transitive on its vertices, all vertices of a finite n-dimensional isogonal figure exist on an -sphere. The term isogonal has long used for polyhedra. Vertex-transitive is a synonym borrowed from modern ideas such as symmetry groups, all regular polygons, apeirogons and regular star polygons are isogonal. The dual of a polygon is an isotoxal polygon. Some even-sided polygons and apeirogons which alternate two edge lengths, for example a rectangle, are isogonal, all planar isogonal 2n-gons have dihedral symmetry with reflection lines across the mid-edge points. An isogonal polyhedron and 2D tiling has a kind of vertex. An isogonal polyhedron with all faces is also a uniform polyhedron. Geometrically distorted variations of uniform polyhedra and tilings can also be given the vertex configuration, isogonal polyhedra and 2D tilings may be further classified, Regular if it is also isohedral and isotoxal, this implies that every face is the same kind of regular polygon. Quasi-regular if it is also isotoxal but not isohedral, semi-regular if every face is a regular polygon but it is not isohedral or isotoxal. Uniform if every face is a polygon, i. e. it is regular, quasiregular or semi-regular. Noble if it is also isohedral and these definitions can be extended to higher-dimensional polytopes and tessellations. Most generally, all uniform polytopes are isogonal, for example, the dual of an isogonal polytope is called an isotope which is transitive on its facets. A polytope or tiling may be called if its vertices form k transitivity classes. A more restrictive term, k-uniform is defined as a figure constructed only from regular polygons. They can be represented visually with colors by different uniform colorings, edge-transitive Face-transitive Peter R. Cromwell, Polyhedra, Cambridge University Press 1997, ISBN 0-521-55432-2, p.369 Transitivity Grünbaum, Branko, Shephard, G. C
19.
Isotoxal figure
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In geometry, a polytope, or a tiling, is isotoxal or edge-transitive if its symmetries act transitively on its edges. The term isotoxal is derived from the Greek τοξον meaning arc, an isotoxal polygon is an equilateral polygon, but not all equilateral polygons are isotoxal. The duals of isotoxal polygons are isogonal polygons, in general, an isotoxal 2n-gon will have Dn dihedral symmetry. A rhombus is a polygon with D2 symmetry. All regular polygons are isotoxal, having double the symmetry order. A regular 2n-gon is a polygon and can be marked with alternately colored vertices. An isotoxal polyhedron or tiling must be either isogonal or isohedral or both, regular polyhedra are isohedral, isogonal and isotoxal. Quasiregular polyhedra are isogonal and isotoxal, but not isohedral, their duals are isohedral and isotoxal, not every polyhedron or 2-dimensional tessellation constructed from regular polygons is isotoxal. An isotoxal polyhedron has the dihedral angle for all edges. There are nine convex isotoxal polyhedra formed from the Platonic solids,8 formed by the Kepler–Poinsot polyhedra, cS1 maint, Multiple names, authors list Coxeter, Harold Scott MacDonald, Longuet-Higgins, M. S. Miller, J. C. P. Uniform polyhedra, Philosophical Transactions of the Royal Society of London, mathematical and Physical Sciences,246, 401–450, doi,10. 1098/rsta.1954.0003, ISSN 0080-4614, JSTOR91532, MR0062446
20.
Five-dimensional space
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A five-dimensional space is a space with five dimensions. If interpreted physically, that is one more than the three spatial dimensions and the fourth dimension of time used in relativitistic physics. It is an abstraction which occurs frequently in mathematics, where it is a legitimate construct, in physics and mathematics, a sequence of N numbers can be understood to represent a location in an N-dimensional space. Whether or not the universe is five-dimensional is a topic of debate, although their approaches were later found to be at least partially inaccurate, the concept provided a basis for further research over the past century. To explain why this dimension would not be observable, Klein suggested that the fifth dimension would be rolled up into a tiny. While not detectable, it would imply a connection between seemingly unrelated forces. Superstring theory then evolved into a generalized approach known as M-theory. M-theory suggested a potentially observable extra dimension in addition to the ten essential dimensions which would allow for the existence of superstrings, the other 10 dimensions are compacted, or rolled up, to a size below the subatomic level. Kaluza–Klein theory today is seen as essentially a gauge theory, with the gauge being the circle group, the fifth dimension is difficult to directly observe, though the Large Hadron Collider provides an opportunity to record indirect evidence of its existence. Mathematical approaches were developed in the early 20th century that viewed the fifth dimension as a theoretical construct and these theories make reference to Hilbert space, a concept that postulates an infinite number of mathematical dimensions to allow for a limitless number of quantum states. They suggested that electromagnetism resulted from a field that is “polarized” in the fifth dimension. The main novelty of Einstein and Bergmann was to consider the fifth dimension as a physical entity, rather than an excuse to combine the metric tensor. But they then reneged, modifying the theory to break its five-dimensional symmetry, minkowski space and Maxwells equations in vacuum can be embedded in a five-dimensional Riemann curvature tensor. For example, holograms are three-dimensional pictures placed on a two-dimensional surface, similarly, in general relativity, the fourth dimension is manifested in observable three dimensions as the curvature path of a moving infinitesimal particle. T Hooft has speculated that the dimension is really the spacetime fabric. According to Klein’s definition, a geometry is the study of the invariant properties of a spacetime, therefore, the geometry of the 5th dimension studies the invariant properties of such space-time, as we move within it, expressed in formal equations. In five or more dimensions, only three regular polytopes exist, in five dimensions, they are, The 5-simplex of the simplex family, with 6 vertices,15 edges,20 faces,15 cells, and 6 hypercells. The 5-cube of the family, with 32 vertices,80 edges,80 faces,40 cells
21.
Vertex (geometry)
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In geometry, a vertex is a point where two or more curves, lines, or edges meet. As a consequence of this definition, the point where two lines meet to form an angle and the corners of polygons and polyhedra are vertices. A vertex is a point of a polygon, polyhedron, or other higher-dimensional polytope. However, in theory, vertices may have fewer than two incident edges, which is usually not allowed for geometric vertices. However, a smooth approximation to a polygon will also have additional vertices. A polygon vertex xi of a simple polygon P is a principal polygon vertex if the diagonal intersects the boundary of P only at x and x, there are two types of principal vertices, ears and mouths. A principal vertex xi of a simple polygon P is called an ear if the diagonal that bridges xi lies entirely in P, according to the two ears theorem, every simple polygon has at least two ears. A principal vertex xi of a simple polygon P is called a mouth if the diagonal lies outside the boundary of P. Any convex polyhedrons surface has Euler characteristic V − E + F =2, where V is the number of vertices, E is the number of edges and this equation is known as Eulers polyhedron formula. Thus the number of vertices is 2 more than the excess of the number of edges over the number of faces, for example, a cube has 12 edges and 6 faces, and hence 8 vertices
22.
Edge (geometry)
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For edge in graph theory, see Edge In geometry, an edge is a particular type of line segment joining two vertices in a polygon, polyhedron, or higher-dimensional polytope. In a polygon, an edge is a segment on the boundary. In a polyhedron or more generally a polytope, an edge is a segment where two faces meet. A segment joining two vertices while passing through the interior or exterior is not an edge but instead is called a diagonal. In graph theory, an edge is an abstract object connecting two vertices, unlike polygon and polyhedron edges which have a concrete geometric representation as a line segment. However, any polyhedron can be represented by its skeleton or edge-skeleton, conversely, the graphs that are skeletons of three-dimensional polyhedra can be characterized by Steinitzs theorem as being exactly the 3-vertex-connected planar graphs. Any convex polyhedrons surface has Euler characteristic V − E + F =2, where V is the number of vertices, E is the number of edges and this equation is known as Eulers polyhedron formula. Thus the number of edges is 2 less than the sum of the numbers of vertices and faces, for example, a cube has 8 vertices and 6 faces, and hence 12 edges. In a polygon, two edges meet at each vertex, more generally, by Balinskis theorem, at least d edges meet at every vertex of a convex polytope. Similarly, in a polyhedron, exactly two faces meet at every edge, while in higher dimensional polytopes three or more two-dimensional faces meet at every edge. Thus, the edges of a polygon are its facets, the edges of a 3-dimensional convex polyhedron are its ridges, archived from the original on 4 February 2007
23.
Face (geometry)
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In solid geometry, a face is a flat surface that forms part of the boundary of a solid object, a three-dimensional solid bounded exclusively by flat faces is a polyhedron. In more technical treatments of the geometry of polyhedra and higher-dimensional polytopes, in elementary geometry, a face is a polygon on the boundary of a polyhedron. Other names for a polygonal face include side of a polyhedron, for example, any of the six squares that bound a cube is a face of the cube. Sometimes face is used to refer to the 2-dimensional features of a 4-polytope. With this meaning, the 4-dimensional tesseract has 24 square faces, some other polygons, which are not faces, are also important for polyhedra and tessellations. These include Petrie polygons, vertex figures and facets, 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 F is the number of faces. This equation is known as Eulers polyhedron formula, thus the number of faces is 2 more than the excess of the number of edges over the number of vertices. For example, a cube has 12 edges and 8 vertices, in higher-dimensional geometry the faces of a polytope are features of all dimensions. A face of dimension k is called a k-face, for example, the polygonal faces of an ordinary polyhedron are 2-faces. In set theory, the set of faces of a polytope includes the polytope itself, for any n-polytope, −1 ≤ k ≤ n. For example, with meaning, the faces of a cube include the empty set, its vertices, edges and squares. Formally, a face of a polytope P is the intersection of P with any closed halfspace whose boundary is disjoint from the interior of P, from this definition it follows that the set of faces of a polytope includes the polytope itself and the empty set. In other areas of mathematics, such as the theories of abstract polytopes and star polytopes, abstract theory still requires that the set of faces include the polytope itself and the empty set. A cell is an element of a 4-dimensional polytope or 3-dimensional tessellation. Cells are facets for 4-polytopes and 3-honeycombs, examples, In higher-dimensional geometry, the facets of a n-polytope are the -faces of dimension one less than the polytope itself. A polytope is bounded by its facets, for example, The facets of a line segment are its 0-faces or vertices. The facets of a polygon are its 1-faces or edges, the facets of a polyhedron or plane tiling are its 2-faces. The facets of a 4D polytope or 3-honeycomb are its 3-faces, the facets of a 5D polytope or 4-honeycomb are its 4-faces
24.
Cell (geometry)
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In solid geometry, a face is a flat surface that forms part of the boundary of a solid object, a three-dimensional solid bounded exclusively by flat faces is a polyhedron. In more technical treatments of the geometry of polyhedra and higher-dimensional polytopes, in elementary geometry, a face is a polygon on the boundary of a polyhedron. Other names for a polygonal face include side of a polyhedron, for example, any of the six squares that bound a cube is a face of the cube. Sometimes face is used to refer to the 2-dimensional features of a 4-polytope. With this meaning, the 4-dimensional tesseract has 24 square faces, some other polygons, which are not faces, are also important for polyhedra and tessellations. These include Petrie polygons, vertex figures and facets, 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 F is the number of faces. This equation is known as Eulers polyhedron formula, thus the number of faces is 2 more than the excess of the number of edges over the number of vertices. For example, a cube has 12 edges and 8 vertices, in higher-dimensional geometry the faces of a polytope are features of all dimensions. A face of dimension k is called a k-face, for example, the polygonal faces of an ordinary polyhedron are 2-faces. In set theory, the set of faces of a polytope includes the polytope itself, for any n-polytope, −1 ≤ k ≤ n. For example, with meaning, the faces of a cube include the empty set, its vertices, edges and squares. Formally, a face of a polytope P is the intersection of P with any closed halfspace whose boundary is disjoint from the interior of P, from this definition it follows that the set of faces of a polytope includes the polytope itself and the empty set. In other areas of mathematics, such as the theories of abstract polytopes and star polytopes, abstract theory still requires that the set of faces include the polytope itself and the empty set. A cell is an element of a 4-dimensional polytope or 3-dimensional tessellation. Cells are facets for 4-polytopes and 3-honeycombs, examples, In higher-dimensional geometry, the facets of a n-polytope are the -faces of dimension one less than the polytope itself. A polytope is bounded by its facets, for example, The facets of a line segment are its 0-faces or vertices. The facets of a polygon are its 1-faces or edges, the facets of a polyhedron or plane tiling are its 2-faces. The facets of a 4D polytope or 3-honeycomb are its 3-faces, the facets of a 5D polytope or 4-honeycomb are its 4-faces
25.
4-face
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In solid geometry, a face is a flat surface that forms part of the boundary of a solid object, a three-dimensional solid bounded exclusively by flat faces is a polyhedron. In more technical treatments of the geometry of polyhedra and higher-dimensional polytopes, in elementary geometry, a face is a polygon on the boundary of a polyhedron. Other names for a polygonal face include side of a polyhedron, for example, any of the six squares that bound a cube is a face of the cube. Sometimes face is used to refer to the 2-dimensional features of a 4-polytope. With this meaning, the 4-dimensional tesseract has 24 square faces, some other polygons, which are not faces, are also important for polyhedra and tessellations. These include Petrie polygons, vertex figures and facets, 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 F is the number of faces. This equation is known as Eulers polyhedron formula, thus the number of faces is 2 more than the excess of the number of edges over the number of vertices. For example, a cube has 12 edges and 8 vertices, in higher-dimensional geometry the faces of a polytope are features of all dimensions. A face of dimension k is called a k-face, for example, the polygonal faces of an ordinary polyhedron are 2-faces. In set theory, the set of faces of a polytope includes the polytope itself, for any n-polytope, −1 ≤ k ≤ n. For example, with meaning, the faces of a cube include the empty set, its vertices, edges and squares. Formally, a face of a polytope P is the intersection of P with any closed halfspace whose boundary is disjoint from the interior of P, from this definition it follows that the set of faces of a polytope includes the polytope itself and the empty set. In other areas of mathematics, such as the theories of abstract polytopes and star polytopes, abstract theory still requires that the set of faces include the polytope itself and the empty set. A cell is an element of a 4-dimensional polytope or 3-dimensional tessellation. Cells are facets for 4-polytopes and 3-honeycombs, examples, In higher-dimensional geometry, the facets of a n-polytope are the -faces of dimension one less than the polytope itself. A polytope is bounded by its facets, for example, The facets of a line segment are its 0-faces or vertices. The facets of a polygon are its 1-faces or edges, the facets of a polyhedron or plane tiling are its 2-faces. The facets of a 4D polytope or 3-honeycomb are its 3-faces, the facets of a 5D polytope or 4-honeycomb are its 4-faces
26.
Rectified 6-orthoplex
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In six-dimensional geometry, a rectified 6-orthoplex is a convex uniform 6-polytope, being a rectification of the regular 6-orthoplex. There are unique 6 degrees of rectifications, the zeroth being the 6-orthoplex, vertices of the rectified 6-orthoplex are located at the edge-centers of the 6-orthoplex. Vertices of the birectified 6-orthoplex are located in the face centers of the 6-orthoplex. The rectified 6-orthoplex is the figure for the demihexeractic honeycomb. Cartesian coordinates for the vertices of a rectified hexacross, centered at the origin, edge length 2 are all permutations of, The 60 vertices represent the root vectors of the simple Lie group D6. The vertices can be seen in 3 hyperplanes, with the 15 vertices rectified 5-simplexs cells on opposite sides, when combined with the 12 vertices of the 6-orthoplex, these vertices represent the 72 root vectors of the B6 and C6 simple Lie groups. The birectified 6-orthoplex can tessellation space in the trirectified 6-cubic 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. 6D uniform polytopes o3x3o3o3o4o - rag, archived from the original on 4 February 2007. Polytopes of Various Dimensions Multi-dimensional Glossary
27.
Birectified 6-cube
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In six-dimensional geometry, a rectified 6-cube is a convex uniform 6-polytope, being a rectification of the regular 6-cube. There are unique 6 degrees of rectifications, the zeroth being the 6-cube, vertices of the rectified 6-cube are located at the edge-centers of the 6-cube. Vertices of the birectified 6-ocube are located in the face centers of the 6-cube. Rectified hexeract The rectified 6-cube may be constructed from the 6-cube by truncating its vertices at the midpoints of its edges, 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. O3x3o3o3o4o - rax, o3o3x3o3o4o - brox, Weisstein, Eric W. Hypercube, archived from the original on 4 February 2007. Polytopes of Various Dimensions Multi-dimensional Glossary
28.
F-vector
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In mathematics, a simplicial complex is a set composed of points, line segments, triangles, and their n-dimensional counterparts. Simplicial complexes should not be confused with the abstract notion of a simplicial set appearing in modern simplicial homotopy theory. The purely combinatorial counterpart to a complex is an abstract simplicial complex. A simplicial complex K is a set of simplices that satisfies the conditions,1. Any face of a simplex from K is also in K.2, the intersection of any two simplices σ1, σ2 ∈ K is either ∅ or a face of both σ1 and σ2. Note that the empty set is a face of every simplex, see also the definition of an abstract simplicial complex, which loosely speaking is a simplicial complex without an associated geometry. A simplicial k-complex K is a complex where the largest dimension of any simplex in K equals k. For instance, a simplicial 2-complex must contain at least one triangle, a pure or homogeneous simplicial k-complex K is a simplicial complex where every simplex of dimension less than k is a face of some simplex σ ∈ K of dimension exactly k. Informally, a pure 1-complex looks like its made of a bunch of lines, an example of a non-homogeneous complex is a triangle with a line segment attached to one of its vertices. A facet is any simplex in a complex that is not a face of any larger simplex, a pure simplicial complex can be thought of as a complex where all facets have the same dimension. Sometimes the term face is used to refer to a simplex of a complex, for a simplicial complex embedded in a k-dimensional space, the k-faces are sometimes referred to as its cells. The term cell is used in a broader sense to denote a set homeomorphic to a simplex. The underlying space, sometimes called the carrier of a complex is the union of its simplices. Let K be a complex and let S be a collection of simplices in K. The closure of S is the smallest simplicial subcomplex of K that contains each simplex in S. Cl S is obtained by adding to S each face of every simplex in S. The star of S is the union of the stars of each simplex in S, for a single simplex s, the star of s is the set of simplices having a face in s. The link of S equals Cl St S − St Cl S and it is the closed star of S minus the stars of all faces of S. In algebraic topology, simplicial complexes are useful for concrete calculations
29.
Wythoff construction
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In geometry, a Wythoff construction, named after mathematician Willem Abraham Wythoff, is a method for constructing a uniform polyhedron or plane tiling. It is often referred to as Wythoffs kaleidoscopic construction and it is based on the idea of tiling a sphere, with spherical triangles – see Schwarz triangles. This construction arranges three mirrors at the sides of a triangle, like in a kaleidoscope, however, different from a kaleidoscope, the mirrors are not parallel, but intersect at a single point. They therefore enclose a spherical triangle on the surface of any sphere centered on that point, if the angles of the spherical triangle are chosen appropriately, the triangles will tile the sphere, one or more times. If one places a vertex at a point inside the spherical triangle enclosed by the mirrors. For a spherical triangle ABC we have four possibilities which will produce a uniform polyhedron and this produces a polyhedron with Wythoff symbol a|b c, where a equals π divided by the angle of the triangle at A, and similarly for b and c. A vertex is placed at a point on line AB so that it bisects the angle at C and this produces a polyhedron with Wythoff symbol a b|c. A vertex is placed so that it is on the incenter of ABC and this produces a polyhedron with Wythoff symbol a b c|. The vertex is at a point such that, when it is rotated around any of the corners by twice the angle at that point. Only even-numbered reflections of the vertex are used. The polyhedron has the Wythoff symbol |a b c, the process in general also applies for higher-dimensional regular polytopes, including the 4-dimensional uniform 4-polytopes. Uniform polytopes that cannot be created through a Wythoff mirror construction are called non-Wythoffian and they generally can be derived from Wythoffian forms either by alternation or by insertion of alternating layers of partial figures. Both of these types of figures will contain rotational symmetry, sometimes snub forms are considered Wythoffian, even though they can only be constructed by the alternation of omnitruncated forms. Wythoff symbol - a symbol for the Wythoff construction of uniform polyhedra, coxeter-Dynkin diagram - a generalized symbol for the Wythoff construction of uniform polytopes and honeycombs. Coxeter Regular Polytopes, Third edition, Dover edition, ISBN 0-486-61480-8 Coxeter The Beauty of Geometry, Twelve Essays, Dover Publications,1999, ISBN 0-486-40919-8 HarEl, Z. W. A. Wythoff, A relation between the polytopes of the C600-family, Koninklijke Akademie van Wetenschappen te Amsterdam, Proceedings of the Section of Sciences,20 966–970, archived from the original on 4 February 2007. Displays Uniform Polyhedra using Wythoffs construction method Description of Wythoff Constructions Jenn, software that generates views of polyhedra and polychora from symmetry groups
30.
Triangular pyramid
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In geometry, a pyramid is a polyhedron formed by connecting a polygonal base and a point, called the apex. Each base edge and apex form a triangle, called a lateral face and it is a conic solid with polygonal base. A pyramid with a base has n +1 vertices, n +1 faces. A right pyramid has its apex directly above the centroid of its base, nonright pyramids are called oblique pyramids. A regular pyramid has a polygon base and is usually implied to be a right pyramid. When unspecified, a pyramid is usually assumed to be a square pyramid. A triangle-based pyramid is often called a tetrahedron. Among oblique pyramids, like acute and obtuse triangles, a pyramid can be called if its apex is above the interior of the base and obtuse if its apex is above the exterior of the base. A right-angled pyramid has its apex above an edge or vertex of the base, in a tetrahedron these qualifiers change based on which face is considered the base. Pyramids are a subclass of the prismatoids, pyramids can be doubled into bipyramids by adding a second offset point on the other side of the base plane. A right pyramid with a base has isosceles triangle sides, with symmetry is Cnv or. It can be given an extended Schläfli symbol ∨, representing a point, a join operation creates a new edge between all pairs of vertices of the two joined figures. The trigonal or triangular pyramid with all equilateral triangles faces becomes the regular tetrahedron, a lower symmetry case of the triangular pyramid is C3v, which has an equilateral triangle base, and 3 identical isosceles triangle sides. The square and pentagonal pyramids can also be composed of convex polygons. Right pyramids with regular star polygon bases are called star pyramids, for example, the pentagrammic pyramid has a pentagram base and 5 intersecting triangle sides. A right pyramid can be named as ∨P, where is the point, ∨ is a join operator. It has C1v symmetry from two different base-apex orientations, and C2v in its full symmetry, a rectangular right pyramid, written as ∨, and a rhombic pyramid, as ∨, both have symmetry C2v. The volume of a pyramid is V =13 b h and this works for any polygon, regular or non-regular, and any location of the apex, provided that h is measured as the perpendicular distance from the plane containing the base
31.
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
32.
Stereographic projection
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In geometry, the stereographic projection is a particular mapping that projects a sphere onto a plane. The projection is defined on the sphere, except at one point. Where it is defined, the mapping is smooth and bijective and it is conformal, meaning that it preserves angles. It is neither isometric nor area-preserving, that is, it preserves neither distances nor the areas of figures, intuitively, then, the stereographic projection is a way of picturing the sphere as the plane, with some inevitable compromises. In practice, the projection is carried out by computer or by using a special kind of graph paper called a stereographic net, shortened to stereonet. The stereographic projection was known to Hipparchus, Ptolemy and probably earlier to the Egyptians and it was originally known as the planisphere projection. Planisphaerium by Ptolemy is the oldest surviving document that describes it, one of its most important uses was the representation of celestial charts. The term planisphere is still used to refer to such charts, in the 16th and 17th century, the equatorial aspect of the stereographic projection was commonly used for maps of the Eastern and Western Hemispheres. It is believed that already the map created in 1507 by Gualterius Lud was in stereographic projection, as were later the maps of Jean Roze, Rumold Mercator, in star charts, even this equatorial aspect had been utilised already by the ancient astronomers like Ptolemy. François dAguilon gave the stereographic projection its current name in his 1613 work Opticorum libri sex philosophis juxta ac mathematicis utiles, in 1695, Edmond Halley, motivated by his interest in star charts, published the first mathematical proof that this map is conformal. He used the recently established tools of calculus, invented by his friend Isaac Newton and this section focuses on the projection of the unit sphere from the north pole onto the plane through the equator. Other formulations are treated in later sections, the unit sphere in three-dimensional space R3 is the set of points such that x2 + y2 + z2 =1. Let N = be the pole, and let M be the rest of the sphere. The plane z =0 runs through the center of the sphere, for any point P on M, there is a unique line through N and P, and this line intersects the plane z =0 in exactly one point P′. Define the stereographic projection of P to be this point P′ in the plane, in Cartesian coordinates on the sphere and on the plane, the projection and its inverse are given by the formulas =, =. In spherical coordinates on the sphere and polar coordinates on the plane, here, φ is understood to have value π when R =0. Also, there are ways to rewrite these formulas using trigonometric identities. In cylindrical coordinates on the sphere and polar coordinates on the plane, the projection is not defined at the projection point N =
33.
Dihedral symmetry
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In mathematics, a dihedral group is the group of symmetries of a regular polygon, which includes rotations and reflections. Dihedral groups are among the simplest examples of groups, and they play an important role in group theory, geometry. The notation for the group of order n differs in geometry. In geometry, Dn or Dihn refers to the symmetries of the n-gon, in abstract algebra, Dn refers to the dihedral group of order n. The geometric convention is used in this article, a regular polygon with n sides has 2 n different symmetries, n rotational symmetries and n reflection symmetries. Usually, we take n ≥3 here. The associated rotations and reflections make up the dihedral group D n, if n is odd, each axis of symmetry connects the midpoint of one side to the opposite vertex. If n is even, there are n/2 axes of symmetry connecting the midpoints of opposite sides, in either case, there are n axes of symmetry and 2 n elements in the symmetry group. Reflecting in one axis of symmetry followed by reflecting in another axis of symmetry produces a rotation through twice the angle between the axes, as with any geometric object, the composition of two symmetries of a regular polygon is again a symmetry of this object. With composition of symmetries to produce another as the binary operation, the following Cayley table shows the effect of composition in the group D3. R0 denotes the identity, r1 and r2 denote counterclockwise rotations by 120° and 240° respectively, for example, s2s1 = r1, because the reflection s1 followed by the reflection s2 results in a rotation of 120°. The order of elements denoting the composition is right to left, the composition operation is not commutative. In all cases, addition and subtraction of subscripts are to be performed using modular arithmetic with modulus n, if we center the regular polygon at the origin, then elements of the dihedral group act as linear transformations of the plane. This lets us represent elements of Dn as matrices, with composition being matrix multiplication and this is an example of a group representation. For example, the elements of the group D4 can be represented by the eight matrices. In general, the matrices for elements of Dn have the following form, rk is a rotation matrix, expressing a counterclockwise rotation through an angle of 2πk/n. Sk is a reflection across a line makes an angle of πk/n with the x-axis. Further equivalent definitions of Dn are, D1 is isomorphic to Z2, D2 is isomorphic to K4, the Klein four-group. D1 and D2 are exceptional in that, D1 and D2 are the only abelian dihedral groups, Dn is a subgroup of the symmetric group Sn for n ≥3
34.
Coxeter
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Harold Scott MacDonald Donald Coxeter, FRS, FRSC, CC was a British-born Canadian geometer. Coxeter is regarded as one of the greatest geometers of the 20th century and he was born in London but spent most of his adult life in Canada. He was always called Donald, from his third name MacDonald, in his youth, Coxeter composed music and was an accomplished pianist at the age of 10. He felt that mathematics and music were intimately related, outlining his ideas in a 1962 article on Mathematics and he worked for 60 years at the University of Toronto and published twelve books. He was most noted for his work on regular polytopes and higher-dimensional geometries and he was a champion of the classical approach to geometry, in a period when the tendency was to approach geometry more and more via algebra. Coxeter went up to Trinity College, Cambridge in 1926 to read mathematics, there he earned his BA in 1928, and his doctorate in 1931. In 1932 he went to Princeton University for a year as a Rockefeller Fellow, where he worked with Hermann Weyl, Oswald Veblen, returning to Trinity for a year, he attended Ludwig Wittgensteins seminars on the philosophy of mathematics. In 1934 he spent a year at Princeton as a Procter Fellow. In 1936 Coxeter moved to the University of Toronto, flather, and John Flinders Petrie published The Fifty-Nine Icosahedra with University of Toronto Press. In 1940 Coxeter edited the eleventh edition of Mathematical Recreations and Essays and he was elevated to professor in 1948. Coxeter was elected a Fellow of the Royal Society of Canada in 1948 and he also inspired some of the innovations of Buckminster Fuller. Coxeter, M. S. Longuet-Higgins and J. C. P. Miller were the first to publish the full list of uniform polyhedra, since 1978, the Canadian Mathematical Society have awarded the Coxeter–James Prize in his honor. He was made a Fellow of the Royal Society in 1950, in 1990, he became a Foreign Member of the American Academy of Arts and Sciences and in 1997 was made a Companion of the Order of Canada. In 1973 he got the Jeffery–Williams Prize,1940, Regular and Semi-Regular Polytopes I, Mathematische Zeitschrift 46, 380-407, MR2,10 doi,10. 1007/BF011814491942, Non-Euclidean Geometry, University of Toronto Press, MAA. 1954, Uniform Polyhedra, Philosophical Transactions of the Royal Society A246, arthur Sherk, Peter McMullen, Anthony C. Thompson and Asia Ivić Weiss, editors, Kaleidoscopes — Selected Writings of H. S. M. John Wiley and Sons ISBN 0-471-01003-01999, The Beauty of Geometry, Twelve Essays, Dover Publications, LCCN 99-35678, ISBN 0-486-40919-8 Davis, Chandler, Ellers, Erich W, the Coxeter Legacy, Reflections and Projections. King of Infinite Space, Donald Coxeter, the Man Who Saved Geometry, www. donaldcoxeter. com www. math. yorku. ca/dcoxeter webpages dedicated to him Jarons World, Shapes in Other Dimensions, Discover mag. Apr 2007 The Mathematics in the Art of M. C, escher video of a lecture by H. S. M
35.
3 31 honeycomb
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In 7-dimensional geometry, the 331 honeycomb is a uniform honeycomb, also given by Schläfli symbol and is composed of 321 and 7-simplex facets, with 56 and 576 of them respectively around each vertex. It is created by a Wythoff construction upon a set of 8 hyperplane mirrors in 7-dimensional space, the facet information can be extracted from its Coxeter-Dynkin diagram. The edge figure is determined by removing the ringed node and ringing the neighboring node, the face figure is determined by removing the ringed node and ringing the neighboring node. The cell figure is determined by removing the ringed node of the face figure, each vertex of this tessellation is the center of a 6-sphere in the densest known packing in 7 dimensions, its kissing number is 126, represented by the vertices of its vertex figure 231. The 331 honeycombs vertex arrangement is called the E7 lattice, E ~7 contains A ~7 as a subgroup of index 144. The Voronoi cell of the E7* lattice is the 132 polytope and it is in a dimensional series of uniform polytopes and honeycombs, expressed by Coxeter as 3k1 series. A degenerate 4-dimensional case exists as 3-sphere tiling, a tetrahedral hosohedron, Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication,1995, ISBN 978-0-471-01003-6 GoogleBook H. S. M, Coxeter, Regular and Semi-Regular Polytopes III, R. T. Worley, The Voronoi Region of E7*. Conway, John H. Sloane, Neil J. A, p124-125,8.2 The 7-dimensinoal lattices, E7 and E7*
36.
Four-dimensional space
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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
37.
Six-dimensional space
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Six-dimensional space is any space that has six dimensions, six degrees of freedom, and that needs six pieces of data, or coordinates, to specify a location in this space. There are a number of these, but those of most interest are simpler ones that model some aspect of the environment. Of particular interest is six-dimensional Euclidean space, in which 6-polytopes, six-dimensional elliptical space and hyperbolic spaces are also studied, with constant positive and negative curvature. Formally, six-dimensional Euclidean space, ℝ6, is generated by considering all real 6-tuples as 6-vectors in this space, as such it has the properties of all Euclidean spaces, so it is linear, has a metric and a full set of vector operations. In particular the dot product between two 6-vectors is readily defined, and can be used to calculate the metric,6 ×6 matrices can be used to describe transformations such as rotations that keep the origin fixed. More generally, any space that can be described locally with six coordinates, one example is the surface of the 6-sphere, S6. This is the set of all points in seven-dimensional Euclidean space ℝ7 that are equidistant from the origin and this constraint reduces the number of coordinates needed to describe a point on the 6-sphere by one, so it has six dimensions. Such non-Euclidean spaces are far more common than Euclidean spaces, a polytope in six dimensions is called a 6-polytope. The most studied are the regular polytopes, of which there are three in six dimensions, the 6-simplex, 6-cube, and 6-orthoplex. A wider family are the uniform 6-polytopes, constructed from fundamental domains of reflection. Each uniform polytope is defined by a ringed Coxeter-Dynkin diagram, the 6-demicube is a unique polytope from the D6 family, and 221 and 122 polytopes from the E6 family. The 5-sphere, or hypersphere in six dimensions, is the five dimensional surface equidistant from a point and it has symbol S5, and the equation for the 5-sphere, radius r, centre the origin is S5 =. The volume of space bounded by this 5-sphere is V6 = π3 r 66 which is 5.16771 × r6. The 6-sphere, or hypersphere in seven dimensions, is the six-dimensional surface equidistant from a point and it has symbol S6, and the equation for the 6-sphere, radius r, centre the origin is S6 =. The volume of the bounded by this 6-sphere is V7 =16 π3 r 7105 which is 4.72477 × r7. In three dimensional space a transformation has six degrees of freedom, three translations along the three coordinate axes and three from the rotation group SO. Often these transformations are handled separately as they have different geometrical structures. In screw theory angular and linear velocity are combined into one six-dimensional object, a similar object called a wrench combines forces and torques in six dimensions
38.
Seven-dimensional space
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In mathematics, a sequence of n real numbers can be understood as a location in n-dimensional space. When n =7, the set of all locations is called 7-dimensional space. Often such a space is studied as a space, without any notion of distance. Seven-dimensional Euclidean space is seven-dimensional space equipped with a Euclidean metric, more generally, the term may refer to a seven-dimensional vector space over any field, such as a seven-dimensional complex vector space, which has 14 real dimensions. It may also refer to a manifold such as a 7-sphere. Seven-dimensional spaces have a number of properties, many of them related to the octonions. An especially distinctive property is that a product can be defined only in three or seven dimensions. This is related to Hurwitzs theorem, which prohibits the existence of structures like the quaternions and octonions in dimensions other than 2,4. The first exotic spheres ever discovered were seven-dimensional, a polytope in seven dimensions is called a 7-polytope. The most studied are the regular polytopes, of which there are three in seven dimensions, the 7-simplex, 7-cube, and 7-orthoplex. A wider family are the uniform 7-polytopes, constructed from fundamental domains of reflection. Each uniform polytope is defined by a ringed Coxeter-Dynkin diagram, the 7-demicube is a unique polytope from the D7 family, and 321,231, and 132 polytopes from the E7 family. The 6-sphere or hypersphere in seven-dimensional Euclidean space is the six-dimensional surface equidistant from a point and it has symbol S6, with formal definition for the 6-sphere with radius r of S6 =. The volume of the bounded by this 6-sphere is V7 =16 π3105 r 7 which is 4.72477 × r7. A cross product, that is a valued, bilinear, anticommutative. Along with the usual cross product in three dimensions it is the only such product, except for trivial products. In 1956, John Milnor constructed an exotic sphere in 7 dimensions, in 1963 he showed that the exact number of such structures is 28. Euclidean geometry List of geometry topics List of regular polytopes H. S. M, dover,1973 J. W. Milnor, On manifolds homeomorphic to the 7-sphere
39.
Eight-dimensional space
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In mathematics, a sequence of n real numbers can be understood as a location in n-dimensional space. When n =8, the set of all locations is called 8-dimensional space. Often such spaces are studied as vector spaces, without any notion of distance, eight-dimensional Euclidean space is eight-dimensional space equipped with a Euclidean metric, which is defined by the dot product. More generally the term may refer to a vector space over any field, such as an eight-dimensional complex vector space. It may also refer to a manifold such as an 8-sphere. A polytope in eight dimensions is called an 8-polytope, the most studied are the regular polytopes, of which there are only three in eight dimensions, the 8-simplex, 8-cube, and 8-orthoplex. A broader family are the uniform 8-polytopes, constructed from fundamental domains of reflection. Each uniform polytope is defined by a ringed Coxeter-Dynkin diagram, the 8-demicube is a unique polytope from the D8 family, and 421,241, and 142 polytopes from the E8 family. The 7-sphere or hypersphere in eight dimensions is the seven-dimensional surface equidistant from a point and it has symbol S7, with formal definition for the 7-sphere with radius r of S7 =. The volume of the bounded by this 7-sphere is V8 = π424 R8 which is 4.05871 × r8. The kissing number problem has been solved in eight dimensions, thanks to the existence of the 421 polytope, the kissing number in eight dimensions is 240. The octonions are a division algebra over the real numbers. Mathematically they can be specified by 8-tuplets of real numbers, so form an 8-dimensional vector space over the reals, a normed algebra is one with a product that satisfies ∥ x y ∥ ≤ ∥ x ∥ ∥ y ∥ for all x and y in the algebra. A normed division algebra additionally must be finite-dimensional, and have the property that every non-zero vector has a multiplicative inverse. Hurwitzs theorem prohibits such a structure from existing in other than 1,2,4. The complexified quaternions C ⊗ H, or biquaternions, are an eight-dimensional algebra dating to William Rowan Hamiltons work in the 1850s and this algebra is equivalent to the Clifford algebra C ℓ2 and the Pauli algebra. It has also proposed as a practical or pedagogical tool for doing calculations in special relativity. Coxeter, Regular Polytopes, 3rd Edition, Dover New York,1973 Kaleidoscopes, Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C
40.
Nine-dimensional space
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In mathematics, a sequence of n real numbers can be understood as a point in n-dimensional space. When n =9, the set of all locations is called 9-dimensional space. Often such spaces are studied as vector spaces, without any notion of distance, nine-dimensional Euclidean space is nine-dimensional space equipped with a Euclidean metric, which is defined by the dot product. More generally, the term may refer to a vector space over any field, such as a nine-dimensional complex vector space. It may also refer to a manifold such as a 9-sphere. A polytope in nine dimensions is called an 9-polytope, the most studied are the regular polytopes, of which there are only three in nine dimensions, the 9-simplex, 9-cube, and 9-orthoplex. A broader family are the uniform 9-polytopes, constructed from fundamental domains of reflection. Each uniform polytope is defined by a ringed Coxeter-Dynkin diagram, the 9-demicube is a unique polytope from the D9 family. H. S. M. Coxeter, H. S. M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York,1973 Kaleidoscopes, Selected Writings of H. S. M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication,1995, ISBN 978-0-471-01003-6 Wiley, Kaleidoscopes, Selected Writings of H. S. M
41.
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
42.
Coxeter notation
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The notation is named after H. S. M. Coxeter, and has been more comprehensively defined by Norman Johnson. For Coxeter groups defined by pure reflections, there is a correspondence between the bracket notation and Coxeter-Dynkin diagram. The numbers in the notation represent the mirror reflection orders in the branches of the Coxeter diagram. It uses the same simplification, suppressing 2s between orthogonal mirrors, the Coxeter notation is simplified with exponents to represent the number of branches in a row for linear diagram. So the An group is represented by, to imply n nodes connected by n-1 order-3 branches, example A2 = = or represents diagrams or. Coxeter initially represented bifurcating diagrams with vertical positioning of numbers, but later abbreviated with an exponent notation, like, Coxeter allowed for zeros as special cases to fit the An family, like A3 = = = =, like = =. Coxeter groups formed by cyclic diagrams are represented by parenthesese inside of brackets, if the branch orders are equal, they can be grouped as an exponent as the length the cycle in brackets, like =, representing Coxeter diagram or. More complicated looping diagrams can also be expressed with care, the paracompact complete graph diagram or, is represented as with the superscript as the symmetry of its regular tetrahedron coxeter diagram. The Coxeter diagram usually leaves order-2 branches undrawn, but the bracket notation includes an explicit 2 to connect the subgraphs, so the Coxeter diagram = A2×A2 = 2A2 can be represented by × =2 =. For the affine and hyperbolic groups, the subscript is one less than the number of nodes in each case, Coxeters notation represents rotational/translational symmetry by adding a + superscript operator outside the brackets which cuts the order of the group in half. This is called a direct subgroup because what remains are only direct isometries without reflective symmetry, + operators can also be applied inside of the brackets, and creates semidirect subgroups that include both reflective and nonreflective generators. Semidirect subgroups can only apply to Coxeter group subgroups that have even order branches next to it, the subgroup index is 2n for n + operators. So the snub cube, has symmetry +, and the tetrahedron, has symmetry. Johnson extends the + operator to work with a placeholder 1 nodes, in general this operation only applies to mirrors bounded by all even-order branches. The 1 represents a mirror so can be seen as, or, like diagram or, the effect of a mirror removal is to duplicate connecting nodes, which can be seen in the Coxeter diagrams, =, or in bracket notation, = =. Each of these mirrors can be removed so h = = = and this can be shown in a Coxeter diagram by adding a + symbol above the node, = =. If both mirrors are removed, a subgroup is generated, with the branch order becoming a gyration point of half the order, q = = +. For example, = = = ×, order 4. = +, the opposite to halving is doubling which adds a mirror, bisecting a fundamental domain, and doubling the group order
43.
Group order
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In group theory, a branch of mathematics, the term order is used in two unrelated senses, The order of a group is its cardinality, i. e. the number of elements in its set. Also, the order, sometimes period, of an element a of a group is the smallest positive integer m such that am = e, if no such m exists, a is said to have infinite order. The ordering relation of a partially or totally ordered group and this article is about the first sense of order. The order of a group G is denoted by ord or | G |, the symmetric group S3 has the following multiplication table. This group has six elements, so ord =6, by definition, the order of the identity, e, is 1. Each of s, t, and w squares to e, completing the enumeration, both u and v have order 3, for u2 = v and u3 = vu = e, and v2 = u and v3 = uv = e. The order of a group and that of an element tend to speak about the structure of the group, roughly speaking, the more complicated the factorization of the order the more complicated the group. If the order of group G is 1, then the group is called a trivial group, given an element a, ord =1 if and only if a is the identity. If every element in G is the same as its inverse, then ord =2 and consequently G is abelian since a b = −1 = b −1 a −1 = b a by Elementary group theory. The converse of this statement is not true, for example, the cyclic group Z6 of integers modulo 6 is abelian, but the number 2 has order 3,2 +2 +2 =6 ≡0. The relationship between the two concepts of order is the following, if we write ⟨ a ⟩ = for the subgroup generated by a, for any integer k, we have ak = e if and only if ord divides k. In general, the order of any subgroup of G divides the order of G, more precisely, if H is a subgroup of G, then ord / ord =, where is called the index of H in G, an integer. As an immediate consequence of the above, we see that the order of every element of a group divides the order of the group. For example, in the symmetric group shown above, where ord =6, the following partial converse is true for finite groups, if d divides the order of a group G and d is a prime number, then there exists an element of order d in G. The statement does not hold for composite orders, e. g. the Klein four-group does not have an element of order four) and this can be shown by inductive proof. The consequences of the include, the order of a group G is a power of a prime p if. If a has order, then all powers of a have infinite order as well. If a has order, we have the following formula for the order of the powers of a
44.
Rectified 5-simplex
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In five-dimensional geometry, a rectified 5-simplex is a convex uniform 5-polytope, being a rectification of the regular 5-simplex. There are three degrees of rectifications, including the zeroth, the 5-simplex itself. Vertices of the rectified 5-simplex are located at the edge-centers of the 5-simplex, vertices of the birectified 5-simplex are located in the triangular face centers of the 5-simplex. In five dimensional geometry, a rectified 5-simplex, is a uniform 5-polytope with 15 vertices,60 edges,80 triangular faces,45 cells and it is also called 03,1 for its branching Coxeter-Dynkin diagram, shown as. E. L. Elte identified it in 1912 as a semiregular polytope, the rectified 5-simplex,031, is second in a dimensional series of uniform polytopes, expressed by Coxeter as 13k series. The fifth figure is a Euclidean honeycomb,331, and the final is a noncompact hyperbolic honeycomb,431, each progressive uniform polytope is constructed from the previous as its vertex figure. Rectified hexateron The vertices of the rectified 5-simplex can be more simply positioned on a hyperplane in 6-space as permutations of or and these construction can be seen as facets of the rectified 6-orthoplex or birectified 6-cube respectively. The birectified 5-simplex is isotopic, with all 12 of its facets as rectified 5-cells and it has 20 vertices,90 edges,120 triangular faces,60 cells. E. L. Elte identified it in 1912 as a semiregular polytope and it is also called 02,2 for its branching Coxeter-Dynkin diagram, shown as. It is seen in the figure of the 6-dimensional 122. Birectified hexateron dodecateron The A5 projection has an appearance to Metatrons Cube. The birectified 5-simplex is the intersection of two regular 5-simplexes in dual configuration, the vertices of a birectification exist at the center of the faces of the original polytope. It is also the intersection of a 6-cube with the hyperplane that bisects the 6-cubes long diagonal orthogonally, in this sense it is the 5-dimensional analog of the regular hexagon, octahedron, and bitruncated 5-cell. This characterization yields simple coordinates for the vertices of a birectified 5-simplex in 6-space, the vertices of the birectified 5-simplex can also be positioned on a hyperplane in 6-space as permutations of. This construction can be seen as facets of the birectified 6-orthoplex, the birectified 5-simplex,022, is second in a dimensional series of uniform polytopes, expressed by Coxeter as k22 series. The birectified 5-simplex is the figure for the third, the 122. The fourth figure is a Euclidean honeycomb,222, and the final is a noncompact hyperbolic honeycomb,322, each progressive uniform polytope is constructed from the previous as its vertex figure. This polytope is the figure of the 6-demicube, and the edge figure of the uniform 231 polytope
45.
6-demicube
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In geometry, a 6-demicube or demihexteract is a uniform 6-polytope, constructed from a 6-cube with alternated vertices truncated. It is part of an infinite family of uniform polytopes called demihypercubes. E. L. Elte identified it in 1912 as a semiregular polytope, Coxeter named this polytope as 131 from its Coxeter diagram, with a ring on one of the 1-length branches. It can named similarly by a 3-dimensional exponential Schläfli symbol or, cartesian coordinates for the vertices of a demihexeract centered at the origin are alternate halves of the hexeract, with an odd number of plus signs. The fifth figure is a Euclidean honeycomb,331, and the final is a noncompact hyperbolic honeycomb,431, each progressive uniform polytope is constructed from the previous as its vertex figure. It is also the second in a series of uniform polytopes and honeycombs. The next figure is the Euclidean honeycomb 133 and the final is a noncompact hyperbolic honeycomb,134. Coxeter, Coxeter, Regular Polytopes, Dover edition, ISBN 0-486-61480-8, p.296, Table I, Regular Polytopes, three regular polytopes in n-dimensions H. S. M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York,1973, p.296, Table I, Regular Polytopes, Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication,1995, ISBN 978-0-471-01003-6 H. S. M, Coxeter, Regular and Semi Regular Polytopes I, H. S. M. Coxeter, Regular and Semi-Regular Polytopes II, H. S. M, Coxeter, Regular and Semi-Regular Polytopes III, John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 Klitzing, Richard. 6D uniform polytopes x3o3o *b3o3o3o – hax, archived from the original on 4 February 2007
46.
2 31 polytope
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In 7-dimensional geometry,231 is a uniform polytope, constructed from the E7 group. Its Coxeter symbol is 231, describing its bifurcating Coxeter-Dynkin diagram, the rectified 231 is constructed by points at the mid-edges of the 231. The 231 is composed of 126 vertices,2016 edges,10080 faces,20160 cells,16128 4-faces,4788 5-faces,632 6-faces and its vertex figure is a 6-demicube. Its 126 vertices represent the vectors of the simple Lie group E7. This polytope is the figure for a uniform tessellation of 7-dimensional space,331. E. L. Elte named it V126 in his 1912 listing of semiregular polytopes and it was called 231 by Coxeter for its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of the 2-node sequence. Pentacontihexa-pentacosiheptacontihexa-exon - 56-576 facetted polyexon It is created by a Wythoff construction upon a set of 7 hyperplane mirrors in 7-dimensional space, the facet information can be extracted from its Coxeter-Dynkin diagram. Removing the node on the branch leaves the 6-simplex. There are 576 of these facets and these facets are centered on the locations of the vertices of the 321 polytope. Removing the node on the end of the 3-length branch leaves the 221, there are 56 of these facets. These facets are centered on the locations of the vertices of the 132 polytope, the vertex figure is determined by removing the ringed node and ringing the neighboring node. The rectified 231 is a rectification of the 231 polytope, creating new vertices on the center of edge of the 231, rectified pentacontihexa-pentacosiheptacontihexa-exon - as a rectified 56-576 facetted polyexon It is created by a Wythoff construction upon a set of 7 hyperplane mirrors in 7-dimensional space. The facet information can be extracted from its Coxeter-Dynkin diagram, removing the node on the short branch leaves the rectified 6-simplex. Removing the node on the end of the 2-length branch leaves the, removing the node on the end of the 3-length branch leaves the rectified 221. The vertex figure is determined by removing the ringed node and ringing the neighboring node, list of E7 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 *c3o3o3o - laq, o3x3o3o *c3o3o3o - rolaq