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 7-polytope
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In seven-dimensional geometry, a 7-polytope is a polytope contained by 6-polytope facets. Each 5-polytope ridge being shared by exactly two 6-polytope facets, a uniform 7-polytope is one which is vertex-transitive, and constructed from uniform 6-polytope facets. Regular 7-polytopes are represented by the Schläfli symbol with u 6-polytopes facets around each 4-face, There are exactly three such convex regular 7-polytopes, - 7-simplex - 7-cube - 7-orthoplex There are no nonconvex regular 7-polytopes. The topology of any given 7-polytope is defined by its Betti numbers, the value of the Euler characteristic used to characterise polyhedra does not generalize usefully to higher dimensions, whatever their underlying topology. This inadequacy of the Euler characteristic to distinguish between different topologies in higher dimensions led to the discovery of the more sophisticated Betti numbers. Similarly, the notion of orientability of a polyhedron is insufficient to characterise the surface twistings of toroidal polytopes, There are 71 forms based on all permutations of the Coxeter-Dynkin diagrams with one or more rings. Norman Johnsons truncation names are given, bowers names and acronym are also given for cross-referencing. See also a list of A7 polytopes for symmetric Coxeter plane graphs of these polytopes, the B7 family has symmetry of order 645120. There are 127 forms based on all permutations of the Coxeter-Dynkin diagrams with one or more rings, see also a list of B7 polytopes for symmetric Coxeter plane graphs of these polytopes. The D7 family has symmetry of order 322560 and this family has 3×32−1=95 Wythoffian uniform polytopes, generated by marking one or more nodes of the D7 Coxeter-Dynkin diagram. Of these,63 are repeated from the B7 family and 32 are unique to this family, bowers names and acronym are given for cross-referencing. See also list of D7 polytopes for Coxeter plane graphs of these polytopes, the E7 Coxeter group has order 2,903,040. There are 127 forms based on all permutations of the Coxeter-Dynkin diagrams with one or more rings, see also a list of E7 polytopes for symmetric Coxeter plane graphs of these polytopes. Coxeter calls the first one a quarter 6-cubic honeycomb, however, there are 3 noncompact hyperbolic Coxeter groups of rank 7, each generating uniform honeycombs in 6-space as permutations of rings of the Coxeter diagrams. The reflective 7-dimensional uniform polytopes are constructed through a Wythoff construction process, and represented by a Coxeter-Dynkin diagram, an active mirror is represented by a ringed node. Each combination of active mirrors generates a unique uniform polytope, Uniform polytopes are named in relation to the regular polytopes in each family. Some families have two regular constructors and thus may be named in two equally valid ways, here are the primary operators available for constructing and naming the uniform 7-polytopes. The prismatic forms and bifurcating graphs can use the same truncation indexing notation, T. Gosset, On the Regular and Semi-Regular Figures in Space of n Dimensions, Messenger of Mathematics, Macmillan,1900 A. S. M
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.
7-simplex
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In 7-dimensional geometry, a 7-simplex is a self-dual regular 7-polytope. It has 8 vertices,28 edges,56 triangle faces,70 tetrahedral cells,56 5-cell 5-faces,28 5-simplex 6-faces and its dihedral angle is cos−1, or approximately 81. 79°. It can also be called an octaexon, or octa-7-tope, as an 8-facetted polytope in 7-dimensions, the name octaexon is derived from octa for eight facets in Greek and -ex for having six-dimensional facets, and -on. Jonathan Bowers gives an octaexon the acronym oca, the Cartesian coordinates of the vertices of an origin-centered regular octaexon having edge length 2 are, More simply, the vertices of the 7-simplex can be positioned in 8-space as permutations of. This construction is based on facets of the 8-orthoplex and this polytope is a facet in the uniform tessellation 331 with Coxeter-Dynkin diagram, This polytope is one of 71 uniform 7-polytopes with A7 symmetry. Polytopes of Various Dimensions Multi-dimensional Glossary
7.
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
8.
Coxeter symbol
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They can be seen as one-end-ringed Coxeter–Dynkin diagrams. The vertex figure of ki, j is i, j, rectified simplices are included in the list as limiting cases with k=0. Similarly 0i, j, k represents a graph with a central node ringed. This included the rectified 5-cell 021 in 4-space, demipenteract 121 in 5-space,221 in 6-space,321 in 7-space,421 in 8-space, E. L. Elte independently enumerated a different semiregular list in his 1912 book, The Semiregular Polytopes of the Hyperspaces. He called them semiregular polytopes of the first kind, limiting his search to one or two types of regular or semiregular k-faces, eltes enumeration included all the kij polytopes except for the 142 which has 3 types of 6-faces. The set of figures extend into honeycombs of, and families in 6,7,8 dimensional Euclidean spaces respectively, gossets list included the 521 honeycomb as the only semiregular one in his definition. The polytopes and honeycombs in this family can be seen within ADE classification, a finite polytope kij exists if 1 i +1 +1 j +1 +1 k +1 >1 or equal for Euclidean honeycombs, and less for hyperbolic honeycombs. The Coxeter group can generate up to 3 unique uniform Gosset–Elte figures with Coxeter–Dynkin diagrams with one end node ringed, by Coxeters notation, each figure is represented by kij to mean the end-node on the k-length sequence is ringed. The simplex family can be seen as a case with k=0. The family of n-simplices contain Gosset–Elte figures of the form 0ij as all rectified forms of the n-simplex and they are listed below, along with their Coxeter–Dynkin diagram, with each dimensional family drawn as a graphic orthogonal projection in the plane of the Petrie polygon of the regular simplex. Each Dn group has two Gosset–Elte figures, the n-demihypercube as 1k1, and a form of the n-orthoplex, k11. Rectified n-demihypercubes, a lower form of a birectified n-cube. Each En group from 4 to 8 has two or three Gosset–Elte figures, represented by one of the ringed, k21, 1k2, 2k1. A rectified 1k2 series can also be represented as 0k21, the 5-dimensional hyperbolic Coxeter group, L ¯4, has five order-3 branches, and can express one honeycomb,11111, and its rectification as 011111. On the regular and semi-regular figures in space of n dimensions, the Semiregular Polytopes of the Hyperspaces, Groningen, University of Groningen, ISBN 1-4181-7968-X Coxeter, H. S. M. Regular Polytopes, Dover edition, ISBN 0-486-61480-8 Norman Johnson Uniform Polytopes, Johnson, The Theory of Uniform Polytopes and Honeycombs, Ph. D
9.
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
10.
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
11.
Petrie polygon
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In geometry, a Petrie polygon for a regular polytope of n dimensions is a skew polygon such that every consecutive sides belong to one of the facets. The Petrie polygon of a polygon is the regular polygon itself. For every regular polytope there exists an orthogonal projection onto a plane such that one Petrie polygon becomes a regular polygon with the remainder of the interior to it. The plane in question is the Coxeter plane of the group of the polygon. These polygons and projected graphs are useful in visualizing symmetric structure of the regular polytopes. John Flinders Petrie was the son of Egyptologist Flinders Petrie. He was born in 1907 and as a schoolboy showed remarkable promise of mathematical ability, in periods of intense concentration he could answer questions about complicated four-dimensional objects by visualizing them. He first noted the importance of the skew polygons which appear on the surface of regular polyhedra. When my incredulity had begun to subside, he described them to me, one consisting of squares, six at each vertex, in 1938 Petrie collaborated with Coxeter, Patrick du Val, and H. T. Flather to produce The Fifty-Nine Icosahedra for publication, realizing the geometric facility of the skew polygons used by Petrie, Coxeter named them after his friend when he wrote Regular Polytopes. In 1972, a few months after his retirement, Petrie was killed by a car attempting to cross a motorway near his home in Surrey. The idea of Petrie polygons was later extended to semiregular polytopes, the Petrie polygon of the regular polyhedron has h sides, where h+2=24/. The regular duals, and, are contained within the same projected Petrie polygon, three of the Kepler–Poinsot polyhedra have hexagonal, and decagrammic, petrie polygons. The Petrie polygon projections are most useful for visualization of polytopes of dimension four and this table represents Petrie polygon projections of 3 regular families, and the exceptional Lie group En which generate semiregular and uniform polytopes for dimensions 4 to 8. Coxeter, H. S. M. Regular Polytopes, 3rd ed, Section 4.3 Flags and Orthoschemes, Section 11.3 Petrie polygons Ball, W. W. R. and H. S. M. Coxeter Mathematical Recreations and Essays, 13th ed. The Beauty of Geometry, Twelve Essays, Dover Publications LCCN 99-35678 Peter McMullen, Egon Schulte Abstract Regular Polytopes, ISBN 0-521-81496-0 Steinberg, Robert, ON THE NUMBER OF SIDES OF A PETRIE POLYGON Weisstein, Eric W. Petrie polygon. Weisstein, Eric W. Cross polytope graphs, Weisstein, Eric W. Gosset graph 3_21
12.
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
13.
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
14.
Edge 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
15.
2 51 honeycomb
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In 8-dimensional geometry, the 251 honeycomb is a space-filling uniform tessellation. It is composed of 241 polytope and 8-simplex facets arranged in an 8-demicube vertex figure and it is the final figure in the 2k1 family. It is created by a Wythoff construction upon a set of 9 hyperplane mirrors in 8-dimensional space, the facet information can be extracted from its Coxeter-Dynkin diagram. Removing the node on the branch leaves the 8-simplex. Removing the node on the end of the 5-length branch leaves the 241, the vertex figure is determined by removing the ringed node and ringing the neighboring node. The edge figure is the figure of the vertex figure. This makes the rectified 7-simplex,051. S. M, coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication,1995, ISBN 978-0-471-01003-6 H. S. M, coxeter, Regular and Semi-Regular Polytopes III
16.
Rectified 8-orthoplex
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In eight-dimensional geometry, a rectified 8-orthoplex is a convex uniform 8-polytope, being a rectification of the regular 8-orthoplex. There are unique 8 degrees of rectifications, the zeroth being the 8-orthoplex, vertices of the rectified 8-orthoplex are located at the edge-centers of the 8-orthoplex. Vertices of the birectified 8-orthoplex are located in the face centers of the 8-orthoplex. Vertices of the trirectified 8-orthoplex are located in the cell centers of the 8-orthoplex. The rectified 8-orthoplex has 112 vertices and these represent the root vectors of the simple Lie group D8. The vertices can be seen in 3 hyperplanes, with the 28 vertices rectified 7-simplexs cells on opposite sides, when combined with the 16 vertices of the 8-orthoplex, these vertices represent the 128 root vectors of the B8 and C8 simple Lie groups. The rectified 8-orthoplex is the figure for the demiocteractic honeycomb. Coxeter, Regular Polytopes, 3rd Edition, Dover New York,1973 Kaleidoscopes, coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication,1995, ISBN 978-0-471-01003-6 H. S. M, coxeter, Regular and Semi Regular Polytopes I, H. S. M. Coxeter, Regular and Semi-Regular Polytopes II, H. S. M, coxeter, Regular and Semi-Regular Polytopes III, Norman Johnson Uniform Polytopes, Manuscript N. W. Johnson, The Theory of Uniform Polytopes and Honeycombs, Ph. D, o3x3o3o3o3o3o4o - rek, o3o3x3o3o3o3o4o - bark, o3o3o3x3o3o3o4o - tark Olshevsky, George. Archived from the original on 4 February 2007, Polytopes of Various Dimensions Multi-dimensional Glossary
17.
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
18.
Rectified 6-simplex
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In six-dimensional geometry, a rectified 6-simplex is a convex uniform 6-polytope, being a rectification of the regular 6-simplex. There are three degrees of rectifications, including the zeroth, the 6-simplex itself. Vertices of the rectified 6-simplex are located at the edge-centers of the 6-simplex, vertices of the birectified 6-simplex are located in the triangular face centers of the 6-simplex. E. L. Elte identified it in 1912 as a semiregular polytope and it is also called 04,1 for its branching Coxeter-Dynkin diagram, shown as. Rectified heptapeton The vertices of the rectified 6-simplex can be most simply positioned in 7-space as permutations of and this construction is based on facets of the rectified 7-orthoplex. E. L. Elte identified it in 1912 as a semiregular polytope and it is also called 03,2 for its branching Coxeter-Dynkin diagram, shown as. Birectified heptapeton The vertices of the birectified 6-simplex can be most simply positioned in 7-space as permutations of and this construction is based on facets of the birectified 7-orthoplex. The rectified 6-simplex polytope is the figure of the 7-demicube. These polytopes are a part of 35 uniform 6-polytopes based on the Coxeter group, 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. O3x3o3o3o3o - ril, o3x3o3o3o3o - bril Olshevsky, George, archived from the original on 4 February 2007. Polytopes of Various Dimensions Multi-dimensional Glossary
19.
Birectified 6-simplex
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In six-dimensional geometry, a rectified 6-simplex is a convex uniform 6-polytope, being a rectification of the regular 6-simplex. There are three degrees of rectifications, including the zeroth, the 6-simplex itself. Vertices of the rectified 6-simplex are located at the edge-centers of the 6-simplex, vertices of the birectified 6-simplex are located in the triangular face centers of the 6-simplex. E. L. Elte identified it in 1912 as a semiregular polytope and it is also called 04,1 for its branching Coxeter-Dynkin diagram, shown as. Rectified heptapeton The vertices of the rectified 6-simplex can be most simply positioned in 7-space as permutations of and this construction is based on facets of the rectified 7-orthoplex. E. L. Elte identified it in 1912 as a semiregular polytope and it is also called 03,2 for its branching Coxeter-Dynkin diagram, shown as. Birectified heptapeton The vertices of the birectified 6-simplex can be most simply positioned in 7-space as permutations of and this construction is based on facets of the birectified 7-orthoplex. The rectified 6-simplex polytope is the figure of the 7-demicube. These polytopes are a part of 35 uniform 6-polytopes based on the Coxeter group, 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. O3x3o3o3o3o - ril, o3x3o3o3o3o - bril Olshevsky, George, archived from the original on 4 February 2007. Polytopes of Various Dimensions Multi-dimensional Glossary
20.
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
21.
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
22.
Birectified 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
23.
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
24.
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
25.
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
26.
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
27.
Birectified 8-orthoplex
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In eight-dimensional geometry, a rectified 8-orthoplex is a convex uniform 8-polytope, being a rectification of the regular 8-orthoplex. There are unique 8 degrees of rectifications, the zeroth being the 8-orthoplex, vertices of the rectified 8-orthoplex are located at the edge-centers of the 8-orthoplex. Vertices of the birectified 8-orthoplex are located in the face centers of the 8-orthoplex. Vertices of the trirectified 8-orthoplex are located in the cell centers of the 8-orthoplex. The rectified 8-orthoplex has 112 vertices and these represent the root vectors of the simple Lie group D8. The vertices can be seen in 3 hyperplanes, with the 28 vertices rectified 7-simplexs cells on opposite sides, when combined with the 16 vertices of the 8-orthoplex, these vertices represent the 128 root vectors of the B8 and C8 simple Lie groups. The rectified 8-orthoplex is the figure for the demiocteractic honeycomb. Coxeter, Regular Polytopes, 3rd Edition, Dover New York,1973 Kaleidoscopes, coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication,1995, ISBN 978-0-471-01003-6 H. S. M, coxeter, Regular and Semi Regular Polytopes I, H. S. M. Coxeter, Regular and Semi-Regular Polytopes II, H. S. M, coxeter, Regular and Semi-Regular Polytopes III, Norman Johnson Uniform Polytopes, Manuscript N. W. Johnson, The Theory of Uniform Polytopes and Honeycombs, Ph. D, o3x3o3o3o3o3o4o - rek, o3o3x3o3o3o3o4o - bark, o3o3o3x3o3o3o4o - tark Olshevsky, George. Archived from the original on 4 February 2007, Polytopes of Various Dimensions Multi-dimensional Glossary
28.
Facet-transitive
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In geometry, a polytope of dimension 3 or higher is isohedral or face-transitive when all its faces are the same. More specifically, all faces must be not merely congruent but must be transitive, in other words, for any faces A and B, there must be a symmetry of the entire solid by rotations and reflections that maps A onto B. For this reason, convex polyhedra are the shapes that will make fair dice. They can be described by their face configuration, a polyhedron which is isohedral has a dual polyhedron that is vertex-transitive. The Catalan solids, the bipyramids and the trapezohedra are all isohedral and they are the duals of the isogonal Archimedean solids, prisms and antiprisms, respectively. The Platonic solids, which are either self-dual or dual with another Platonic solid, are vertex, edge, a polyhedron which is isohedral and isogonal is said to be noble. A polyhedron is if it contains k faces within its symmetry fundamental domain. Similarly a k-isohedral tiling has k separate symmetry orbits, a monohedral polyhedron or monohedral tiling has congruent faces, as either direct or reflectively, which occur in one or more symmetry positions. An r-hedral polyhedra or tiling has r types of faces, a facet-transitive or isotopic figure is a n-dimensional polytopes or honeycomb, with its facets congruent and transitive. The dual of an isotope is an isogonal polytope, by definition, this isotopic property is common to the duals of the uniform polytopes. An isotopic 2-dimensional figure is isotoxal, an isotopic 3-dimensional figure is isohedral. An isotopic 4-dimensional figure is isochoric, edge-transitive Anisohedral tiling Peter R. Cromwell, Polyhedra, Cambridge University Press 1997, ISBN 0-521-55432-2, p.367 Transitivity Olshevsky, George. Archived from the original on 4 February 2007
29.
Intersection (set theory)
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In mathematics, the intersection A ∩ B of two sets A and B is the set that contains all elements of A that also belong to B, but no other elements. For explanation of the used in this article, refer to the table of mathematical symbols. The intersection of A and B is written A ∩ B, formally, A ∩ B = that is x ∈ A ∩ B if and only if x ∈ A and x ∈ B. For example, The intersection of the sets and is, the number 9 is not in the intersection of the set of prime numbers and the set of odd numbers. More generally, one can take the intersection of sets at once. The intersection of A, B, C, and D, Intersection is an associative operation, thus, A ∩ = ∩ C. Additionally, intersection is commutative, thus A ∩ B = B ∩ A, inside a universe U one may define the complement Ac of A to be the set of all elements of U not in A. We say that A intersects B if A intersects B at some element, a intersects B if their intersection is inhabited. We say that A and B are disjoint if A does not intersect B, in plain language, they have no elements in common. A and B are disjoint if their intersection is empty, denoted A ∩ B = ∅, for example, the sets and are disjoint, the set of even numbers intersects the set of multiples of 3 at 0,6,12,18 and other numbers. The most general notion is the intersection of a nonempty collection of sets. If M is a nonempty set whose elements are themselves sets, then x is an element of the intersection of M if, the notation for this last concept can vary considerably. Set theorists will sometimes write ⋂M, while others will instead write ⋂A∈M A, the latter notation can be generalized to ⋂i∈I Ai, which refers to the intersection of the collection. Here I is a nonempty set, and Ai is a set for every i in I. In the case that the index set I is the set of numbers, notation analogous to that of an infinite series may be seen. When formatting is difficult, this can also be written A1 ∩ A2 ∩ A3 ∩, even though strictly speaking, A1 ∩ (A2 ∩ (A3 ∩. Finally, let us note that whenever the symbol ∩ is placed before other symbols instead of them, it should be of a larger size. Note that in the section we excluded the case where M was the empty set
30.
Dual polytope
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Such dual figures remain combinatorial or abstract polyhedra, but not all are also geometric polyhedra. Starting with any given polyhedron, the dual of its dual is the original polyhedron, duality preserves the symmetries of a polyhedron. Therefore, for classes of polyhedra defined by their symmetries. Thus, the regular polyhedra – the Platonic solids and Kepler-Poinsot polyhedra – form dual pairs, the dual of an isogonal polyhedron, having equivalent vertices, is one which is isohedral, having equivalent faces. The dual of a polyhedron is also isotoxal. Duality is closely related to reciprocity or polarity, a transformation that. There are many kinds of duality, the kinds most relevant to elementary polyhedra are polar reciprocity and topological or abstract duality. The duality of polyhedra is often defined in terms of polar reciprocation about a concentric sphere. In coordinates, for reciprocation about the sphere x 2 + y 2 + z 2 = r 2, the vertex is associated with the plane x 0 x + y 0 y + z 0 z = r 2. The vertices of the dual are the reciprocal to the face planes of the original. Also, any two adjacent vertices define an edge, and these will reciprocate to two adjacent faces which intersect to define an edge of the dual and this dual pair of edges are always orthogonal to each other. If r 0 is the radius of the sphere, and r 1 and r 2 respectively the distances from its centre to the pole and its polar, then, r 1. R2 = r 02 For the more symmetrical polyhedra having an obvious centroid, it is common to make the polyhedron and sphere concentric, the choice of center for the sphere is sufficient to define the dual up to similarity. If multiple symmetry axes are present, they will intersect at a single point. Failing that, a sphere, inscribed sphere, or midsphere is commonly used. If a polyhedron in Euclidean space has an element passing through the center of the sphere, since Euclidean space never reaches infinity, the projective equivalent, called extended Euclidean space, may be formed by adding the required plane at infinity. Some theorists prefer to stick to Euclidean space and say there is no dual. Meanwhile, Wenninger found a way to represent these infinite duals, the concept of duality here is closely related to the duality in projective geometry, where lines and edges are interchanged
31.
1 33 honeycomb
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In 7-dimensional geometry,133 is a uniform honeycomb, also given by Schläfli symbol, and is composed of 132 facets. 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. Removing a node on the end of one of the 3-length branch leaves the 132, the vertex figure is determined by removing the ringed node and ringing the neighboring node. This makes the trirectified 7-simplex,033, the edge figure is determined by removing the ringed nodes of the vertex figure and ringing the neighboring node. This makes the tetrahedral duoprism, ×, each vertex of this polytope corresponds to the center of a 6-sphere in a moderately dense sphere packing, in which each sphere is tangent to 70 others, the best known for 7 dimensions is 126. The E ~7 group is related to the F ~4 by a geometric folding, E ~7 contains A ~7 as a subgroup of index 144. Both E ~7 and A ~7 can be seen as affine extension from A7 from different nodes, The E7* lattice has double the symmetry, the Voronoi cell of the E7* lattice is the 132 polytope, and voronoi tessellation the 133 honeycomb. The 133 is fourth in a series of uniform polytopes and honeycombs. The final is a noncompact hyperbolic honeycomb,134, the rectified 133 or 0331, Coxeter diagram has facets and, and vertex figure. Coxeter, Regular Polytopes, 3rd Edition, Dover New York,1973 Coxeter The Beauty of Geometry, Twelve Essays, Dover Publications,1999, ISBN 978-0-486-40919-1 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
32.
Trirectified 8-orthoplex
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In eight-dimensional geometry, a rectified 8-orthoplex is a convex uniform 8-polytope, being a rectification of the regular 8-orthoplex. There are unique 8 degrees of rectifications, the zeroth being the 8-orthoplex, vertices of the rectified 8-orthoplex are located at the edge-centers of the 8-orthoplex. Vertices of the birectified 8-orthoplex are located in the face centers of the 8-orthoplex. Vertices of the trirectified 8-orthoplex are located in the cell centers of the 8-orthoplex. The rectified 8-orthoplex has 112 vertices and these represent the root vectors of the simple Lie group D8. The vertices can be seen in 3 hyperplanes, with the 28 vertices rectified 7-simplexs cells on opposite sides, when combined with the 16 vertices of the 8-orthoplex, these vertices represent the 128 root vectors of the B8 and C8 simple Lie groups. The rectified 8-orthoplex is the figure for the demiocteractic honeycomb. Coxeter, Regular Polytopes, 3rd Edition, Dover New York,1973 Kaleidoscopes, coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication,1995, ISBN 978-0-471-01003-6 H. S. M, coxeter, Regular and Semi Regular Polytopes I, H. S. M. Coxeter, Regular and Semi-Regular Polytopes II, H. S. M, coxeter, Regular and Semi-Regular Polytopes III, Norman Johnson Uniform Polytopes, Manuscript N. W. Johnson, The Theory of Uniform Polytopes and Honeycombs, Ph. D, o3x3o3o3o3o3o4o - rek, o3o3x3o3o3o3o4o - bark, o3o3o3x3o3o3o4o - tark Olshevsky, George. Archived from the original on 4 February 2007, Polytopes of Various Dimensions Multi-dimensional Glossary
33.
Hexagon
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In geometry, a hexagon is a six sided polygon or 6-gon. The total of the angles of any hexagon is 720°. A regular hexagon has Schläfli symbol and can also be constructed as an equilateral triangle, t. A regular hexagon is defined as a hexagon that is both equilateral and equiangular and it is bicentric, meaning that it is both cyclic and tangential. The common length of the sides equals the radius of the circumscribed circle, all internal angles are 120 degrees. A regular hexagon has 6 rotational symmetries and 6 reflection symmetries, the longest diagonals of a regular hexagon, connecting diametrically opposite vertices, are twice the length of one side. Like squares and equilateral triangles, regular hexagons fit together without any gaps to tile the plane, the cells of a beehive honeycomb are hexagonal for this reason and because the shape makes efficient use of space and building materials. The Voronoi diagram of a triangular lattice is the honeycomb tessellation of hexagons. It is not usually considered a triambus, although it is equilateral, the maximal diameter, D is twice the maximal radius or circumradius, R, which equals the side length, t. The minimal diameter or the diameter of the circle, d, is twice the minimal radius or inradius. If a regular hexagon has successive vertices A, B, C, D, E, F, the regular hexagon has Dih6 symmetry, order 12. There are 3 dihedral subgroups, Dih3, Dih2, and Dih1, and 4 cyclic subgroups, Z6, Z3, Z2 and these symmetries express 9 distinct symmetries of a regular hexagon. John Conway labels these by a letter and group order, r12 is full symmetry, and a1 is no symmetry. These two forms are duals of each other and have half the order of the regular hexagon. The i4 forms are regular hexagons flattened or stretched along one symmetry direction and it can be seen as an elongated rhombus, while d2 and p2 can be seen as horizontally and vertically elongated kites. G2 hexagons, with sides parallel are also called hexagonal parallelogons. Each subgroup symmetry allows one or more degrees of freedom for irregular forms, only the g6 subgroup has no degrees of freedom but can seen as directed edges. Hexagons of symmetry g2, i4, and r12, as parallelogons can tessellate the Euclidean plane by translation, other hexagon shapes can tile the plane with different orientations
34.
Bitruncated 5-cell
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In geometry, a truncated 5-cell is a uniform 4-polytope formed as the truncation of the regular 5-cell. There are two degrees of trunctions, including a bitruncation, the truncated 5-cell, truncated pentachoron or truncated 4-simplex is bounded by 10 cells,5 tetrahedra, and 5 truncated tetrahedra. Each vertex is surrounded by 3 truncated tetrahedra and one tetrahedron, the truncated 5-cell may be constructed from the 5-cell by truncating its vertices at 1/3 the edge length. This truncates the 5 tetrahedral cells into truncated tetrahedra, and introduces 5 new tetrahedral cells positioned on the original vertices, the truncated tetrahedra are joined to each other at their hexagonal faces, and to the tetrahedra at their triangular faces. The tetrahedron-first parallel projection of the truncated 5-cell into 3-dimensional space has the following structure, one of the truncated tetrahedral cells project onto the entire envelope. One of the tetrahedral cells project onto a tetrahedron lying at the center of the envelope, four flattened tetrahedra are joined to the triangular faces of the envelope, and connected to the central tetrahedron via 4 radial edges. These are the images of the remaining 4 tetrahedral cells, between the central tetrahedron and the 4 hexagonal faces of the envelope are 4 irregular truncated tetrahedral volumes, which are the images of the 4 remaining trucated tetrahedral cells. This layout of cells in projection is analogous to the layout of faces in the face-first projection of the tetrahedron into 2-dimensional space. The truncated 5-cell is the 4-dimensional analogue of the truncated tetrahedron and these coordinates come from positive orthant facets of the truncated pentacross and bitruncated penteract respectively. The bitruncated 5-cell is a 4-dimensional polytope, or 4-polytope, composed of 10 cells in the shape of truncated tetrahedra, E. L. Elte identified it in 1912 as a semiregular polytope. Each hexagonal face of the tetrahedra is joined in complementary orientation to the neighboring truncated tetrahedron. Each edge is shared by two hexagons and one triangle, each vertex is surrounded by 4 truncated tetrahedral cells in a tetragonal disphenoid vertex figure. The bitruncated 5-cell is the intersection of two pentachora in dual configuration, as such, it is also the intersection of a penteract with the hyperplane that bisects the penteracts long diagonal orthogonally. In this sense it is a 4-dimensional analog of the regular octahedron, the 5-dimensional analog is the birectified 5-simplex, and the n -dimensional analog is the polytope whose Coxeter–Dynkin diagram is linear with rings on the middle one or two nodes. The bitruncated 5-cell is one of the two non-regular uniform 4-polytopes which are cell-transitive, the other is the bitruncated 24-cell, which is composed of 48 truncated cubes. These represent positive orthant facets of the bitruncated pentacross, another 5-space construction, centered on the origin are all 20 permutations of. The regular skew polyhedron, exists in 4-space with 4 hexagonal around each vertex and these hexagonal faces can be seen on the bitruncated 5-cell, using all 60 edges and 20 vertices. The 20 triangular faces of the bitruncated 5-cell can be seen as removed, the dual regular skew polyhedron, is similarly related to the square faces of the runcinated 5-cell
35.
Tritruncated 6-simplex
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In six-dimensional geometry, a truncated 6-simplex is a convex uniform 6-polytope, being a truncation of the regular 6-simplex. There are unique 3 degrees of truncation, vertices of the truncation 6-simplex are located as pairs on the edge of the 6-simplex. Vertices of the bitruncated 6-simplex are located on the faces of the 6-simplex. Vertices of the tritruncated 6-simplex are located inside the cells of the 6-simplex. Truncated heptapeton The vertices of the truncated 6-simplex can be most simply positioned in 7-space as permutations of and this construction is based on facets of the truncated 7-orthoplex. Bitruncated heptapeton The vertices of the bitruncated 6-simplex can be most simply positioned in 7-space as permutations of and this construction is based on facets of the bitruncated 7-orthoplex. The tritruncated 6-simplex is a uniform polytope, with 14 identical bitruncated 5-simplex facets. The tritruncated 6-simplex is the intersection of two 6-simplexes in dual configuration, and, tetradecapeton The vertices of the tritruncated 6-simplex can be most simply positioned in 7-space as permutations of. This construction is based on facets of the bitruncated 7-orthoplex, alternately it can be centered on the origin as permutations of. Note, Symmetry doubled for Ak graphs with even k due to symmetrically-ringed Coxeter-Dynkin diagram, the truncated 6-simplex is one of 35 uniform 6-polytopes based on the Coxeter group, all shown here in A6 Coxeter plane orthographic projections. Coxeter, Regular Polytopes, 3rd Edition, Dover New York,1973 Kaleidoscopes, Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication,1995, ISBN 978-0-471-01003-6 H. S. M, Coxeter, Regular and Semi Regular Polytopes I, H. S. M. Coxeter, Regular and Semi-Regular Polytopes II, H. S. M, Coxeter, Regular and Semi-Regular Polytopes III, Norman Johnson Uniform Polytopes, Manuscript N. W. Johnson, The Theory of Uniform Polytopes and Honeycombs, Ph. D, o3x3o3o3o3o - til, o3x3x3o3o3o - batal, o3o3x3x3o3o - fe Olshevsky, George. Archived from the original on 4 February 2007, Polytopes of Various Dimensions Multi-dimensional Glossary
36.
Trirectified 7-simplex
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In seven-dimensional geometry, a rectified 7-simplex is a convex uniform 7-polytope, being a rectification of the regular 7-simplex. There are four degrees of rectifications, including the zeroth. Vertices of the rectified 7-simplex are located at the edge-centers of the 7-simplex, vertices of the birectified 7-simplex are located in the triangular face centers of the 7-simplex. Vertices of the trirectified 7-simplex are located in the cell centers of the 7-simplex. The rectified 7-simplex is the figure of the 251 honeycomb. It is called 05,1 for its branching Coxeter-Dynkin diagram, E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as S17. Rectified octaexon The vertices of the rectified 7-simplex can be most simply positioned in 8-space as permutations of and this construction is based on facets of the rectified 8-orthoplex. E. L. Elte identified it in 1912 as a semiregular polytope and it is also called 04,2 for its branching Coxeter-Dynkin diagram, shown as. Birectified octaexon The vertices of the birectified 7-simplex can be most simply positioned in 8-space as permutations of and this construction is based on facets of the birectified 8-orthoplex. The trirectified 7-simplex is the intersection of two regular 7-simplexes in dual configuration, E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as S37. This polytope is the figure of the 133 honeycomb. It is called 03,3 for its branching Coxeter-Dynkin diagram, hexadecaexon The vertices of the trirectified 7-simplex can be most simply positioned in 8-space as permutations of. This construction is based on facets of the trirectified 8-orthoplex, the trirectified 7-simplex is the intersection of two regular 7-simplices in dual configuration. This characterization yields simple coordinates for the vertices of a trirectified 7-simplex in 8-space and these polytopes are three of 71 uniform 7-polytopes with A7 symmetry. Coxeter, Regular Polytopes, 3rd Edition, Dover New York,1973 Kaleidoscopes, coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication,1995, ISBN 978-0-471-01003-6 H. S. M, coxeter, Regular and Semi Regular Polytopes I, H. S. M. Coxeter, Regular and Semi-Regular Polytopes II, H. S. M, coxeter, Regular and Semi-Regular Polytopes III, Norman Johnson Uniform Polytopes, Manuscript N. W. Johnson, The Theory of Uniform Polytopes and Honeycombs, Ph. D, o3o3x3o3o3o3o - broc, o3x3o3o3o3o3o - roc, o3o3x3o3o3o3o - he Olshevsky, George
37.
Quadritruncated 8-simplex
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In eight-dimensional geometry, a truncated 8-simplex is a convex uniform 8-polytope, being a truncation of the regular 8-simplex. There are a four unique degrees of truncation, vertices of the truncation 8-simplex are located as pairs on the edge of the 8-simplex. Vertices of the bitruncated 8-simplex are located on the faces of the 8-simplex. Vertices of the tritruncated 8-simplex are located inside the cells of the 8-simplex. Truncated enneazetton The Cartesian coordinates of the vertices of the truncated 8-simplex can be most simply positioned in 8-space as permutations of and this construction is based on facets of the truncated 9-orthoplex. Bitruncated enneazetton The Cartesian coordinates of the vertices of the bitruncated 8-simplex can be most simply positioned in 8-space as permutations of and this construction is based on facets of the bitruncated 9-orthoplex. Tritruncated enneazetton The Cartesian coordinates of the vertices of the tritruncated 8-simplex can be most simply positioned in 8-space as permutations of and this construction is based on facets of the tritruncated 9-orthoplex. The quadritruncated 8-simplex an isotopic polytope, constructed from 18 tritruncated 7-simplex facets, octadecazetton The Cartesian coordinates of the vertices of the quadritruncated 8-simplex can be most simply positioned in 8-space as permutations of. This construction is based on facets of the quadritruncated 9-orthoplex and this polytope is one of 135 uniform 8-polytopes with A8 symmetry. Coxeter, Regular Polytopes, 3rd Edition, Dover New York,1973 Kaleidoscopes, coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication,1995, ISBN 978-0-471-01003-6 H. S. M, coxeter, Regular and Semi Regular Polytopes I, H. S. M. Coxeter, Regular and Semi-Regular Polytopes II, H. S. M, coxeter, Regular and Semi-Regular Polytopes III, Norman Johnson Uniform Polytopes, Manuscript N. W. Johnson, The Theory of Uniform Polytopes and Honeycombs, Ph. D, x3x3o3o3o3o3o3o - tene, o3x3x3o3o3o3o3o - batene, o3o3x3x3o3o3o3o - tatene, o3o3o3x3x3o3o3o - be Olshevsky, George. Archived from the original on 4 February 2007, Polytopes of Various Dimensions Multi-dimensional Glossary
38.
Truncated tetrahedron
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In geometry, the truncated tetrahedron is an Archimedean solid. It has 4 regular hexagonal faces,4 equilateral triangle faces,12 vertices and 18 edges and it can be constructed by truncating all 4 vertices of a regular tetrahedron at one third of the original edge length. A deeper truncation, removing a tetrahedron of half the edge length from each vertex, is called rectification. The rectification of a tetrahedron produces an octahedron, a truncated tetrahedron is the Goldberg polyhedron GIII, containing triangular and hexagonal faces. A truncated tetrahedron can be called a cube, with Coxeter diagram. There are two positions of this construction, and combining them creates the uniform compound of two truncated tetrahedra. The area A and the volume V of a tetrahedron of edge length a are. The densest packing of the Archimedean truncated tetrahedron is believed to be Φ = 207/208, in fact, if the truncation of the corners is slightly smaller than that of an Archimedean truncated tetrahedron, this new shape can be used to completely fill space. This projection is conformal, preserving angles but not areas or lengths, straight lines on the sphere are projected as circular arcs on the plane. A lower symmetry version of the tetrahedron is called a Friauf polyhedron in crystals such as complex metallic alloys. This form fits 5 Friauf polyhedra around an axis, giving a 72-degree dihedral angle on a subset of 6-6 edges, Friauf and his 1927 paper The crystal structure of the intermetallic compound MgCu2. Giant truncated tetrahedra were used for the Man the Explorer and Man the Producer theme pavilions in Expo 67 and they were made of massive girders of steel bolted together in a geometric lattice. The truncated tetrahedra were interconnected with lattice steel platforms, all of these buildings were demolished after the end of Expo 67, as they had not been built to withstand the severity of the Montreal weather over the years. Their only remnants are in the Montreal city archives, the Public Archives Of Canada, the Tetraminx puzzle has a truncated tetrahedral shape. This puzzle shows a dissection of a tetrahedron into 4 octahedra and 6 tetrahedra. It contains 4 central planes of rotations, in the mathematical field of graph theory, a truncated tetrahedral graph is a Archimedean graph, the graph of vertices and edges of the truncated tetrahedron, one of the Archimedean solids. It has 12 vertices and 18 edges and it is a connected cubic graph, and connected cubic transitive graph. It is also a part of a sequence of cantic polyhedra, in this wythoff construction the edges between the hexagons represent degenerate digons
39.
Bitruncated 5-simplex
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In five-dimensional geometry, a truncated 5-simplex is a convex uniform 5-polytope, being a truncation of the regular 5-simplex. There are unique 2 degrees of truncation, vertices of the truncation 5-simplex are located as pairs on the edge of the 5-simplex. Vertices of the bitruncation 5-simplex are located on the faces of the 5-simplex. The truncated 5-simplex has 30 vertices,75 edges,80 triangular faces,45 cells, Truncated hexateron The vertices of the truncated 5-simplex can be most simply constructed on a hyperplane in 6-space as permutations of or of. These coordinates come from facets of the truncated 6-orthoplex and bitruncated 6-cube respectively, bitruncated hexateron The vertices of the bitruncated 5-simplex can be most simply constructed on a hyperplane in 6-space as permutations of or of. These represent positive orthant facets of the bitruncated 6-orthoplex, and the tritruncated 6-cube respectively, the truncated 5-simplex is one of 19 uniform 5-polytopes based on the Coxeter group, all shown here in A5 Coxeter plane orthographic projections. Coxeter, Regular Polytopes, 3rd Edition, Dover New York,1973 Kaleidoscopes, Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication,1995, ISBN 978-0-471-01003-6 H. S. M, Coxeter, Regular and Semi Regular Polytopes I, H. S. M. Coxeter, Regular and Semi-Regular Polytopes II, H. S. M, Coxeter, Regular and Semi-Regular Polytopes III, Norman Johnson Uniform Polytopes, Manuscript N. W. Johnson, The Theory of Uniform Polytopes and Honeycombs, Ph. D, x3x3o3o3o - tix, o3x3x3o3o - bittix Glossary for hyperspace, George Olshevsky. Polytopes of Various Dimensions, Jonathan Bowers Truncated uniform polytera, Jonathan Bowers Multi-dimensional Glossary
40.
Tritruncated 7-simplex
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In seven-dimensional geometry, a truncated 7-simplex is a convex uniform 7-polytope, being a truncation of the regular 7-simplex. There are unique 3 degrees of truncation, vertices of the truncation 7-simplex are located as pairs on the edge of the 7-simplex. Vertices of the bitruncated 7-simplex are located on the faces of the 7-simplex. Vertices of the tritruncated 7-simplex are located inside the cells of the 7-simplex. In seven-dimensional geometry, a truncated 7-simplex is a convex uniform 7-polytope, truncated octaexon The vertices of the truncated 7-simplex can be most simply positioned in 8-space as permutations of. This construction is based on facets of the truncated 8-orthoplex, bitruncated octaexon The vertices of the bitruncated 7-simplex can be most simply positioned in 8-space as permutations of. This construction is based on facets of the bitruncated 8-orthoplex, tritruncated octaexon The vertices of the tritruncated 7-simplex can be most simply positioned in 8-space as permutations of. This construction is based on facets of the tritruncated 8-orthoplex and these three polytopes are from a set of 71 uniform 7-polytopes with A7 symmetry. Coxeter, Regular Polytopes, 3rd Edition, Dover New York,1973 Kaleidoscopes, coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication,1995, ISBN 978-0-471-01003-6 H. S. M, coxeter, Regular and Semi Regular Polytopes I, H. S. M. Coxeter, Regular and Semi-Regular Polytopes II, H. S. M, coxeter, Regular and Semi-Regular Polytopes III, Norman Johnson Uniform Polytopes, Manuscript N. W. Johnson, The Theory of Uniform Polytopes and Honeycombs, Ph. D, x3x3o3o3o3o3o - toc, o3x3x3o3o3o3o - roc, o3o3x3x3o3o3o - tattoc Olshevsky, George. Archived from the original on 4 February 2007, Polytopes of Various Dimensions Multi-dimensional Glossary
41.
Simplex
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In geometry, a simplex is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions. Specifically, a k-simplex is a polytope which is the convex hull of its k +1 vertices. More formally, suppose the k +1 points u 0, …, u k ∈ R k are affinely independent, then, the simplex determined by them is the set of points C =. For example, a 2-simplex is a triangle, a 3-simplex is a tetrahedron, a single point may be considered a 0-simplex, and a line segment may be considered a 1-simplex. A simplex may be defined as the smallest convex set containing the given vertices, a regular simplex is a simplex that is also a regular polytope. A regular n-simplex may be constructed from a regular -simplex by connecting a new vertex to all original vertices by the edge length. In topology and combinatorics, it is common to “glue together” simplices to form a simplicial complex, the associated combinatorial structure is called an abstract simplicial complex, in which context the word “simplex” simply means any finite set of vertices. A 1-simplex is a line segment, the convex hull of any nonempty subset of the n+1 points that define an n-simplex is called a face of the simplex. In particular, the hull of a subset of size m+1 is an m-simplex. The 0-faces are called the vertices, the 1-faces are called the edges, the -faces are called the facets, in general, the number of m-faces is equal to the binomial coefficient. Consequently, the number of m-faces of an n-simplex may be found in column of row of Pascals triangle, a simplex A is a coface of a simplex B if B is a face of A. Face and facet can have different meanings when describing types of simplices in a simplicial complex, see simplical complex for more detail. The regular simplex family is the first of three regular polytope families, labeled by Coxeter as αn, the two being the cross-polytope family, labeled as βn, and the hypercubes, labeled as γn. A fourth family, the infinite tessellation of hypercubes, he labeled as δn, an -simplex can be constructed as a join of an n-simplex and a point. An -simplex can be constructed as a join of an m-simplex, the two simplices are oriented to be completely normal from each other, with translation in a direction orthogonal to both of them. A 1-simplex is a joint of two points, ∨ =2, a general 2-simplex is the join of 3 points, ∨∨. An isosceles triangle is the join of a 1-simplex and a point, a general 3-simplex is the join of 4 points, ∨∨∨. A 3-simplex with mirror symmetry can be expressed as the join of an edge and 2 points, a 3-simplex with triangular symmetry can be expressed as the join of an equilateral triangle and 1 point,3. ∨ or ∨
42.
Rectified 7-simplex
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In seven-dimensional geometry, a rectified 7-simplex is a convex uniform 7-polytope, being a rectification of the regular 7-simplex. There are four degrees of rectifications, including the zeroth. Vertices of the rectified 7-simplex are located at the edge-centers of the 7-simplex, vertices of the birectified 7-simplex are located in the triangular face centers of the 7-simplex. Vertices of the trirectified 7-simplex are located in the cell centers of the 7-simplex. The rectified 7-simplex is the figure of the 251 honeycomb. It is called 05,1 for its branching Coxeter-Dynkin diagram, E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as S17. Rectified octaexon The vertices of the rectified 7-simplex can be most simply positioned in 8-space as permutations of and this construction is based on facets of the rectified 8-orthoplex. E. L. Elte identified it in 1912 as a semiregular polytope and it is also called 04,2 for its branching Coxeter-Dynkin diagram, shown as. Birectified octaexon The vertices of the birectified 7-simplex can be most simply positioned in 8-space as permutations of and this construction is based on facets of the birectified 8-orthoplex. The trirectified 7-simplex is the intersection of two regular 7-simplexes in dual configuration, E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as S37. This polytope is the figure of the 133 honeycomb. It is called 03,3 for its branching Coxeter-Dynkin diagram, hexadecaexon The vertices of the trirectified 7-simplex can be most simply positioned in 8-space as permutations of. This construction is based on facets of the trirectified 8-orthoplex, the trirectified 7-simplex is the intersection of two regular 7-simplices in dual configuration. This characterization yields simple coordinates for the vertices of a trirectified 7-simplex in 8-space and these polytopes are three of 71 uniform 7-polytopes with A7 symmetry. Coxeter, Regular Polytopes, 3rd Edition, Dover New York,1973 Kaleidoscopes, coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication,1995, ISBN 978-0-471-01003-6 H. S. M, coxeter, Regular and Semi Regular Polytopes I, H. S. M. Coxeter, Regular and Semi-Regular Polytopes II, H. S. M, coxeter, Regular and Semi-Regular Polytopes III, Norman Johnson Uniform Polytopes, Manuscript N. W. Johnson, The Theory of Uniform Polytopes and Honeycombs, Ph. D, o3o3x3o3o3o3o - broc, o3x3o3o3o3o3o - roc, o3o3x3o3o3o3o - he Olshevsky, George
43.
Birectified 7-simplex
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In seven-dimensional geometry, a rectified 7-simplex is a convex uniform 7-polytope, being a rectification of the regular 7-simplex. There are four degrees of rectifications, including the zeroth. Vertices of the rectified 7-simplex are located at the edge-centers of the 7-simplex, vertices of the birectified 7-simplex are located in the triangular face centers of the 7-simplex. Vertices of the trirectified 7-simplex are located in the cell centers of the 7-simplex. The rectified 7-simplex is the figure of the 251 honeycomb. It is called 05,1 for its branching Coxeter-Dynkin diagram, E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as S17. Rectified octaexon The vertices of the rectified 7-simplex can be most simply positioned in 8-space as permutations of and this construction is based on facets of the rectified 8-orthoplex. E. L. Elte identified it in 1912 as a semiregular polytope and it is also called 04,2 for its branching Coxeter-Dynkin diagram, shown as. Birectified octaexon The vertices of the birectified 7-simplex can be most simply positioned in 8-space as permutations of and this construction is based on facets of the birectified 8-orthoplex. The trirectified 7-simplex is the intersection of two regular 7-simplexes in dual configuration, E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as S37. This polytope is the figure of the 133 honeycomb. It is called 03,3 for its branching Coxeter-Dynkin diagram, hexadecaexon The vertices of the trirectified 7-simplex can be most simply positioned in 8-space as permutations of. This construction is based on facets of the trirectified 8-orthoplex, the trirectified 7-simplex is the intersection of two regular 7-simplices in dual configuration. This characterization yields simple coordinates for the vertices of a trirectified 7-simplex in 8-space and these polytopes are three of 71 uniform 7-polytopes with A7 symmetry. Coxeter, Regular Polytopes, 3rd Edition, Dover New York,1973 Kaleidoscopes, coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication,1995, ISBN 978-0-471-01003-6 H. S. M, coxeter, Regular and Semi Regular Polytopes I, H. S. M. Coxeter, Regular and Semi-Regular Polytopes II, H. S. M, coxeter, Regular and Semi-Regular Polytopes III, Norman Johnson Uniform Polytopes, Manuscript N. W. Johnson, The Theory of Uniform Polytopes and Honeycombs, Ph. D, o3o3x3o3o3o3o - broc, o3x3o3o3o3o3o - roc, o3o3x3o3o3o3o - he Olshevsky, George
44.
Truncated 7-simplex
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In seven-dimensional geometry, a truncated 7-simplex is a convex uniform 7-polytope, being a truncation of the regular 7-simplex. There are unique 3 degrees of truncation, vertices of the truncation 7-simplex are located as pairs on the edge of the 7-simplex. Vertices of the bitruncated 7-simplex are located on the faces of the 7-simplex. Vertices of the tritruncated 7-simplex are located inside the cells of the 7-simplex. In seven-dimensional geometry, a truncated 7-simplex is a convex uniform 7-polytope, truncated octaexon The vertices of the truncated 7-simplex can be most simply positioned in 8-space as permutations of. This construction is based on facets of the truncated 8-orthoplex, bitruncated octaexon The vertices of the bitruncated 7-simplex can be most simply positioned in 8-space as permutations of. This construction is based on facets of the bitruncated 8-orthoplex, tritruncated octaexon The vertices of the tritruncated 7-simplex can be most simply positioned in 8-space as permutations of. This construction is based on facets of the tritruncated 8-orthoplex and these three polytopes are from a set of 71 uniform 7-polytopes with A7 symmetry. Coxeter, Regular Polytopes, 3rd Edition, Dover New York,1973 Kaleidoscopes, coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication,1995, ISBN 978-0-471-01003-6 H. S. M, coxeter, Regular and Semi Regular Polytopes I, H. S. M. Coxeter, Regular and Semi-Regular Polytopes II, H. S. M, coxeter, Regular and Semi-Regular Polytopes III, Norman Johnson Uniform Polytopes, Manuscript N. W. Johnson, The Theory of Uniform Polytopes and Honeycombs, Ph. D, x3x3o3o3o3o3o - toc, o3x3x3o3o3o3o - roc, o3o3x3x3o3o3o - tattoc Olshevsky, George. Archived from the original on 4 February 2007, Polytopes of Various Dimensions Multi-dimensional Glossary
45.
Cantellated 7-simplex
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In seven-dimensional geometry, a cantellated 7-simplex is a convex uniform 7-polytope, being a cantellation of the regular 7-simplex. There are unique 6 degrees of cantellation for the 7-simplex, including truncations, small rhombated octaexon The vertices of the cantellated 7-simplex can be most simply positioned in 8-space as permutations of. This construction is based on facets of the cantellated 8-orthoplex, small birhombated octaexon The vertices of the bicantellated 7-simplex can be most simply positioned in 8-space as permutations of. This construction is based on facets of the bicantellated 8-orthoplex, small trirhombihexadecaexon The vertices of the tricantellated 7-simplex can be most simply positioned in 8-space as permutations of. This construction is based on facets of the tricantellated 8-orthoplex, great rhombated octaexon The vertices of the cantitruncated 7-simplex can be most simply positioned in 8-space as permutations of. This construction is based on facets of the cantitruncated 8-orthoplex, great birhombated octaexon The vertices of the bicantitruncated 7-simplex can be most simply positioned in 8-space as permutations of. This construction is based on facets of the bicantitruncated 8-orthoplex, great trirhombihexadecaexon The vertices of the tricantitruncated 7-simplex can be most simply positioned in 8-space as permutations of. This construction is based on facets of the tricantitruncated 8-orthoplex and this polytope is one of 71 uniform 7-polytopes with A7 symmetry. Coxeter, Regular Polytopes, 3rd Edition, Dover New York,1973 Kaleidoscopes, coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication,1995, ISBN 978-0-471-01003-6 H. S. M, coxeter, Regular and Semi Regular Polytopes I, H. S. M. Coxeter, Regular and Semi-Regular Polytopes II, H. S. M, coxeter, Regular and Semi-Regular Polytopes III, Norman Johnson Uniform Polytopes, Manuscript N. W. Johnson, The Theory of Uniform Polytopes and Honeycombs, Ph. D, x3o3x3o3o3o3o - saro, o3x3o3x3o3o3o - sabro, o3o3x3o3x3o3o - stiroh, x3x3x3o3o3o3o - garo, o3x3x3x3o3o3o - gabro, o3o3x3x3x3o3o - gatroh Olshevsky, George. Archived from the original on 4 February 2007, Polytopes of Various Dimensions Multi-dimensional Glossary
46.
Bitruncated 7-simplex
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In seven-dimensional geometry, a truncated 7-simplex is a convex uniform 7-polytope, being a truncation of the regular 7-simplex. There are unique 3 degrees of truncation, vertices of the truncation 7-simplex are located as pairs on the edge of the 7-simplex. Vertices of the bitruncated 7-simplex are located on the faces of the 7-simplex. Vertices of the tritruncated 7-simplex are located inside the cells of the 7-simplex. In seven-dimensional geometry, a truncated 7-simplex is a convex uniform 7-polytope, truncated octaexon The vertices of the truncated 7-simplex can be most simply positioned in 8-space as permutations of. This construction is based on facets of the truncated 8-orthoplex, bitruncated octaexon The vertices of the bitruncated 7-simplex can be most simply positioned in 8-space as permutations of. This construction is based on facets of the bitruncated 8-orthoplex, tritruncated octaexon The vertices of the tritruncated 7-simplex can be most simply positioned in 8-space as permutations of. This construction is based on facets of the tritruncated 8-orthoplex and these three polytopes are from a set of 71 uniform 7-polytopes with A7 symmetry. Coxeter, Regular Polytopes, 3rd Edition, Dover New York,1973 Kaleidoscopes, coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication,1995, ISBN 978-0-471-01003-6 H. S. M, coxeter, Regular and Semi Regular Polytopes I, H. S. M. Coxeter, Regular and Semi-Regular Polytopes II, H. S. M, coxeter, Regular and Semi-Regular Polytopes III, Norman Johnson Uniform Polytopes, Manuscript N. W. Johnson, The Theory of Uniform Polytopes and Honeycombs, Ph. D, x3x3o3o3o3o3o - toc, o3x3x3o3o3o3o - roc, o3o3x3x3o3o3o - tattoc Olshevsky, George. Archived from the original on 4 February 2007, Polytopes of Various Dimensions Multi-dimensional Glossary