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8-polytope
In Eight-dimensional space, eight-dimensional geometry, an eight-dimensional polytope or 8-polytope is a polytope contained by 7-polytope facets. Each 6-polytope Ridge (geometry), ridge being shared by exactly two 7-polytope Facet (mathematics), facets. A uniform 8-polytope is one which is vertex-transitive, and constructed from uniform 7-polytope facets. Regular 8-polytopes Regular 8-polytopes can be represented by the Schläfli symbol , with v 7-polytope Facet (mathematics), facets around each Peak (geometry), peak. There are exactly three such List of regular polytopes#Convex 4, convex regular 8-polytopes: # - 8-simplex # - 8-cube # - 8-orthoplex There are no nonconvex regular 8-polytopes. Characteristics The topology of any given 8-polytope is defined by its Betti numbers and torsion coefficient (topology), torsion coefficients.Richeson, D.; ''Euler's Gem: The Polyhedron Formula and the Birth of Topoplogy'', Princeton, 2008. The value of the Euler characteristic u ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cantellated 8-orthoplex
In eight-dimensional geometry, an eight-dimensional polytope or 8-polytope is a polytope contained by 7-polytope facets. Each 6-polytope ridge being shared by exactly two 7-polytope facets. A uniform 8-polytope is one which is vertex-transitive, and constructed from uniform 7-polytope facets. Regular 8-polytopes Regular 8-polytopes can be represented by the Schläfli symbol , with v 7-polytope facets around each peak. There are exactly three such convex regular 8-polytopes: # - 8-simplex # - 8-cube # - 8-orthoplex There are no nonconvex regular 8-polytopes. Characteristics The topology of any given 8-polytope is defined by its Betti numbers and torsion coefficients.Richeson, D.; ''Euler's Gem: The Polyhedron Formula and the Birth of Topoplogy'', Princeton, 2008. The value of the Euler characteristic used to characterise polyhedra does not generalize usefully to higher dimensions, and is zero for all 8-polytopes, whatever their underlying topology. This inadequa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cantellated 8-simplex
In eight-dimensional geometry, a cantellated 8-simplex is a convex uniform 8-polytope, being a cantellation of the regular 8-simplex. There are six unique cantellations for the 8-simplex, including permutations of truncation. Cantellated 8-simplex Alternate names * Small rhombated enneazetton (acronym: srene) (Jonathan Bowers) Coordinates The Cartesian coordinates of the vertices of the ''cantellated 8-simplex'' can be most simply positioned in 9-space as permutations of (0,0,0,0,0,0,1,1,2). This construction is based on facets of the cantellated 9-orthoplex. Images Bicantellated 8-simplex Alternate names * Small birhombated enneazetton (acronym: sabrene) (Jonathan Bowers) Coordinates The Cartesian coordinates of the vertices of the ''bicantellated 8-simplex'' can be most simply positioned in 9-space as permutations of (0,0,0,0,0,1,1,2,2). This construction is based on facets of the bicantellated 9-orthoplex. Images Tricantellated 8-simplex Alternate ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Truncated 8-simplex
In eight-dimensional geometry, a truncated 8-simplex is a convex uniform 8-polytope, being a truncation of the regular 8-simplex. There are 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 triangular faces of the 8-simplex. Vertices of the tritruncated 8-simplex are located inside the tetrahedral cells of the 8-simplex. Truncated 8-simplex Alternate names * Truncated enneazetton (Acronym: tene) (Jonathan Bowers) Coordinates The Cartesian coordinates of the vertices of the ''truncated 8-simplex'' can be most simply positioned in 9-space as permutations of (0,0,0,0,0,0,0,1,2). This construction is based on facets of the truncated 9-orthoplex. Images Bitruncated 8-simplex Alternate names * Bitruncated enneazetton (Acronym: batene) (Jonathan Bowers) Coordinates The Cartesian coordinates of the vertices of the ''bitruncated 8-simplex'' can b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stericated 8-simplex
In eight-dimensional geometry, a stericated 8-simplex is a convex uniform 8-polytope with 4th order truncations (sterication) of the regular 8-simplex. There are 16 unique sterications for the 8-simplex including permutations of truncation, cantellation, and runcination. Stericated 8-simplex Coordinates The Cartesian coordinates of the vertices of the ''stericated 8-simplex'' can be most simply positioned in 9-space as permutations of (0,0,0,0,1,1,1,1,2). This construction is based on facets of the stericated 9-orthoplex. Images Bistericated 8-simplex Coordinates The Cartesian coordinates of the vertices of the ''bistericated 8-simplex'' can be most simply positioned in 9-space as permutations of (0,0,0,1,1,1,1,2,2). This construction is based on facets of the bistericated 9-orthoplex. Images Steritruncated 8-simplex Images Bisteritruncated 8-simplex Images Stericantellated 8-simplex Images Bistericantellated 8-simplex Images Sterica ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rectified 8-simplex
In eight-dimensional geometry, a rectified 8-simplex is a convex uniform 8-polytope, being a Rectification (geometry), rectification of the regular 8-simplex. There are unique 3 degrees of rectifications in regular 8-polytopes. Vertices of the rectified 8-simplex are located at the edge-centers of the 8-simplex. Vertices of the birectified 8-simplex are located in the triangular face centers of the 8-simplex. Vertices of the trirectified 8-simplex are located in the tetrahedron, tetrahedral cell centers of the 8-simplex. Rectified 8-simplex Emanuel Lodewijk Elte, E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as S. It is also called 06,1 for its branching Coxeter-Dynkin diagram, shown as . Coordinates The Cartesian coordinates of the vertices of the ''rectified 8-simplex'' can be most simply positioned in 9-space as permutations of (0,0,0,0,0,0,0,1,1). This construction is based on Facet (geometry), facets of the rectified 9-orthoplex. Images ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pentellated 8-simplex
In eight-dimensional geometry, a pentellated 8-simplex is a convex uniform 8-polytope with 5th order truncations of the regular 8-simplex. There are two unique pentellations of the 8-simplex. Including truncations, cantellations, runcinations, and sterications, there are 32 more pentellations. These polytopes are a part of a family 135 uniform 8-polytopes with A8 symmetry. A8, 7has order 9 factorial symmetry, or 362880. The bipentalled form is symmetrically ringed, doubling the symmetry order to 725760, and is represented the double-bracketed group 37. The A8 Coxeter plane projection shows order symmetry for the pentellated 8-simplex, while the bipentellated 8-simple is doubled to 8symmetry. Pentellated 8-simplex Coordinates The Cartesian coordinates of the vertices of the ''pentellated 8-simplex'' can be most simply positioned in 9-space as permutations of (0,0,0,0,1,1,1,1,2). This construction is based on facets of the pentellated 9-orthoplex. Images Bipentella ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Runcinated 8-simplex
In eight-dimensional geometry, a runcinated 8-simplex is a convex uniform 8-polytope with 3rd order truncations (runcination) of the regular 8-simplex. There are eleven unique runcinations of the 8-simplex, including permutations of truncation and cantellation. The ''triruncinated 8-simplex'' and ''triruncicantitruncated 8-simplex'' have a doubled symmetry, showing 8order reflectional symmetry in the A8 Coxeter plane. Runcinated 8-simplex Alternate names * Runcinated enneazetton * Small prismated enneazetton (Acronym: spene) (Jonathan Bowers) Coordinates The Cartesian coordinates of the vertices of the ''runcinated 8-simplex'' can be most simply positioned in 9-space as permutations of (0,0,0,0,0,1,1,1,2). This construction is based on facets of the runcinated 9-orthoplex. Images Biruncinated 8-simplex Alternate names * Biruncinated enneazetton * Small biprismated enneazetton (Acronym: sabpene) (Jonathan Bowers) Coordinates The Cartesian coordinates of t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Heptellated 8-simplex
In eight-dimensional geometry, a heptellated 8-simplex is a convex uniform 8-polytope, including 7th-order truncations (heptellation) from the regular 8-simplex. There are 35 unique heptellations for the 8-simplex, including all permutations of truncations, cantellations, runcinations, sterications, pentellations, and hexications. The simplest heptellated 8-simplex is also called an expanded 8-simplex, with only the first and last nodes ringed, is constructed by an expansion operation applied to the regular 8-simplex. The highest form, the ''heptihexipentisteriruncicantitruncated 8-simplex'' is more simply called a ''omnitruncated 8-simplex'' with all of the nodes ringed. Heptellated 8-simplex Alternate names * Expanded 8-simplex * Small exated enneazetton (soxeb) (Jonathan Bowers) Coordinates The vertices of the ''heptellated 8-simplex'' can bepositioned in 8-space as permutations of (0,1,1,1,1,1,1,1,2). This construction is based on facets of the heptellated 9-orthople ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
8-simplex T06
In geometry, an 8-simplex is a self-dual regular 8-polytope. It has 9 vertices, 36 edges, 84 triangle faces, 126 tetrahedral cells, 126 5-cell 4-faces, 84 5-simplex 5-faces, 36 6-simplex 6-faces, and 9 7-simplex 7-faces. Its dihedral angle is cos−1(1/8), or approximately 82.82°. It can also be called an enneazetton, or ennea-8-tope, as a 9- facetted polytope in eight-dimensions. The name ''enneazetton'' is derived from ''ennea'' for nine facets in Greek and ''-zetta'' for having seven-dimensional facets, and ''-on''. As a configuration This configuration matrix represents the 8-simplex. The rows and columns correspond to vertices, edges, faces, cells, 4-faces, 5-faces, 6-faces and 7-faces. The diagonal numbers say how many of each element occur in the whole 8-simplex. The nondiagonal numbers say how many of the column's element occur in or at the row's element. This self-dual simplex's matrix is identical to its 180 degree rotation. \begin\begin 9 & 8 & 28 & 56 & 70 & 5 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Truncated 8-orthoplex
In eight-dimensional geometry, a truncated 8-orthoplex is a convex uniform 8-polytope, being a truncation of the regular 8-orthoplex. There are 7 truncation for the 8-orthoplex. Vertices of the truncation 8-orthoplex are located as pairs on the edge of the 8-orthoplex. Vertices of the bitruncated 8-orthoplex are located on the triangular faces of the 8-orthoplex. Vertices of the tritruncated 7-orthoplex are located inside the tetrahedral cells of the 8-orthoplex. The final truncations are best expressed relative to the 8-cube. Truncated 8-orthoplex Alternate names * Truncated octacross (acronym tek) (Jonthan Bowers) Construction There are two Coxeter group In mathematics, a Coxeter group, named after H. S. M. Coxeter, is an abstract group that admits a formal description in terms of reflections (or kaleidoscopic mirrors). Indeed, the finite Coxeter groups are precisely the finite Euclidean refle ...s associated with the ''truncated 8-orthoplex'', one with the C8 o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rectified 8-orthoplex
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, and the 7th and last being the 8-cube. 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 triangular face centers of the 8-orthoplex. Vertices of the trirectified 8-orthoplex are located in the tetrahedral cell centers of the 8-orthoplex. Rectified 8-orthoplex The rectified 8-orthoplex has 112 vertices. 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, and 56 vertices of an expanded 7-simplex passing through the center. When combined with the 16 vertices of the 8-orthoplex, these vertices represent the 128 root vectors of the B8 and C8 simple Lie ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
8-simplex T07
In geometry, an 8-simplex is a self-dual regular 8-polytope. It has 9 vertices, 36 edges, 84 triangle faces, 126 tetrahedral cells, 126 5-cell 4-faces, 84 5-simplex 5-faces, 36 6-simplex 6-faces, and 9 7-simplex 7-faces. Its dihedral angle is cos−1(1/8), or approximately 82.82°. It can also be called an enneazetton, or ennea-8-tope, as a 9- facetted polytope in eight-dimensions. The name ''enneazetton'' is derived from ''ennea'' for nine facets in Greek and ''-zetta'' for having seven-dimensional facets, and ''-on''. As a configuration This configuration matrix represents the 8-simplex. The rows and columns correspond to vertices, edges, faces, cells, 4-faces, 5-faces, 6-faces and 7-faces. The diagonal numbers say how many of each element occur in the whole 8-simplex. The nondiagonal numbers say how many of the column's element occur in or at the row's element. This self-dual simplex's matrix is identical to its 180 degree rotation. \begin\begin 9 & 8 & 28 & 56 & 70 & 5 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
