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7-polytope
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 whose symmetry group is transitive on vertices and whose facets are uniform 6-polytopes. Regular 7-polytopes 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. Characteristics The topology of any given 7-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, whatever their underlying topology. This inadequacy of the Euler characteristic to reliably distinguish ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hexicated 7-simplex
In seven-dimensional geometry, a hexicated 7-simplex is a convex uniform 7-polytope, including 6th-order truncations (hexication) from the regular 7-simplex. There are 20 unique hexications for the 7-simplex, including all permutations of truncations, cantellations, runcinations, sterications, and pentellations. The simple hexicated 7-simplex is also called an expanded 7-simplex, with only the first and last nodes ringed, is constructed by an Expansion (geometry), expansion operation applied to the regular 7-simplex. The highest form, the ''hexipentisteriruncicantitruncated 7-simplex'' is more simply called a ''omnitruncated 7-simplex'' with all of the nodes ringed. Hexicated 7-simplex In seven-dimensional geometry, a hexicated 7-simplex is a convex uniform 7-polytope, a hexication (6th order truncation) of the regular 7-simplex, or alternately can be seen as an Expansion (geometry), expansion operation. Root vectors Its 56 vertices represent the root vectors of the simple Li ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pentellated 7-orthoplex
In seven-dimensional geometry, a pentellated 7-orthoplex is a convex uniform 7-polytope with 5th order truncations (pentellation) of the regular 7-orthoplex. There are 32 unique of the 7-orthoplex with permutations of truncations, cantellations, runcinations, and . 16 are more simply constructed relative to the 7-cube. These polytopes are a part of a set of 127 uniform 7-polytope 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 whose symmetry group is transitive on vertices and whose ...s with B7 symmetry. Pentellated 7-orthoplex Alternate names * Small hecatonicosoctaexon (acronym: Staz) (Jonathan Bowers) Coordinates Coordinates are permutations of (0,1,1,1,1,1,2) Images Pentitruncated 7-orthoplex Alternate names * Teritruncated hecatonicosoctaexon (acronym: Tetaz) (Jonathan Bowers) Images Coordinates Coordinates are ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pentellated 7-simplex
In seven-dimensional geometry, a pentellated 7-simplex is a convex uniform 7-polytope with 5th order truncations (pentellation) of the regular 7-simplex. There are 16 unique pentellations of the 7-simplex with permutations of truncations, cantellations, runcinations, and sterications. Pentellated 7-simplex Alternate names * Small terated octaexon (acronym: seto) (Jonathan Bowers) Coordinates The vertices of the ''pentellated 7-simplex'' can be most simply positioned in 8-space as permutations of (0,0,1,1,1,1,1,2). This construction is based on facets of the pentellated 8-orthoplex. Images Pentitruncated 7-simplex Alternate names * Teritruncated octaexon (acronym: teto) (Jonathan Bowers) Coordinates The vertices of the ''pentitruncated 7-simplex'' can be most simply positioned in 8-space as permutations of (0,0,1,1,1,1,2,3). This construction is based on facets of the pentitruncated 8-orthoplex. Images Penticantellated 7-simplex Alternate names * Terirhomb ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stericated 7-simplex
In seven-dimensional geometry, a stericated 7-simplex is a convex uniform 7-polytope with 4th order truncations (sterication) of the regular 7-simplex. There are 14 unique sterication for the 7-simplex with permutations of truncations, cantellations, and runcinations. Stericated 7-simplex Alternate names * Small cellated octaexon (acronym: sco) (Jonathan Bowers) Coordinates The vertices of the ''stericated 7-simplex'' can be most simply positioned in 8-space as permutations of (0,0,0,1,1,1,1,2). This construction is based on facets of the stericated 8-orthoplex. Images Bistericated 7-simplex Alternate names * Small bicellated hexadecaexon (acronym: sabach) (Jonathan Bowers) Coordinates The vertices of the ''bistericated 7-simplex'' can be most simply positioned in 8-space as permutations of (0,0,1,1,1,1,2,2). This construction is based on facets of the bistericated 8-orthoplex. Images Steritruncated 7-simplex Alternate names * Cellitruncated octaexo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stericated 7-orthoplex
In seven-dimensional geometry, a stericated 7-orthoplex is a convex uniform 7-polytope with 4th order truncations (sterication) of the regular 7-orthoplex. There are 24 unique sterication for the 7-orthoplex with permutations of truncations, cantellations, and runcinations. 14 are more simply constructed from the 7-cube. This polytope is one of 127 uniform 7-polytope 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 whose symmetry group is transitive on vertices and whose ...s with B7 symmetry. Stericated 7-orthoplex Alternate names * Small cellated hecatonicosoctaexon (acronym: ) (Jonathan Bowers) Images Steritruncated 7-orthoplex Alternate names * Cellitruncated hecatonicosoctaexon (acronym: ) (Jonathan Bowers) Images Bisteritruncated 7-orthoplex Alternate names * Bicellitruncated hecatonicosoctaexon (acronym: ) ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Runcinated 7-orthoplex
In seven-dimensional geometry, a runcinated 7-orthoplex is a convex uniform 7-polytope with 3rd order truncations (runcination) of the regular 7-orthoplex. There are 16 unique runcinations of the 7-orthoplex with permutations of truncations, and cantellations. 8 are more simply constructed from the 7-cube. These polytopes are among 127 uniform 7-polytope 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 whose symmetry group is transitive on vertices and whose ...s with B7 symmetry. Runcinated 7-orthoplex Alternate names * Small prismated hecatonicosoctaexon (acronym: spaz) (Jonathan Bowers) Images Biruncinated 7-orthoplex Alternate names * Small biprismated hecatonicosoctaexon (Acronym sibpaz) (Jonathan Bowers) Images Runcitruncated 7-orthoplex Alternate names * Prismatotruncated hecatonicosoctaexon (acronym: potaz) (J ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Runcinated 7-simplex
In seven-dimensional geometry, a runcinated 7-simplex is a convex uniform 7-polytope with 3rd order truncations (runcination) of the regular 7-simplex. There are 8 unique runcinations of the 7-simplex with permutations of truncations, and cantellations. Runcinated 7-simplex Alternate names * Small prismated octaexon (acronym: spo) (Jonathan Bowers) Coordinates The vertices of the ''runcinated 7-simplex'' can be most simply positioned in 8-space as permutations of (0,0,0,0,1,1,1,2). This construction is based on facets of the runcinated 8-orthoplex. Images Biruncinated 7-simplex Alternate names * Small biprismated octaexon (sibpo) (Jonathan Bowers) Coordinates The vertices of the ''biruncinated 7-simplex'' can be most simply positioned in 8-space as permutations of (0,0,0,1,1,1,2,2). This construction is based on facets of the biruncinated 8-orthoplex. Images Runcitruncated 7-simplex Alternate names * Prismatotruncated octaexon (acronym: patto) (Jonathan Bow ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cantellated 7-simplex
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. Cantellated 7-simplex Alternate names * Small rhombated octaexon (acronym: saro) (Jonathan Bowers) Coordinates The vertices of the ''cantellated 7-simplex'' can be most simply positioned in 8-space as permutations of (0,0,0,0,0,1,1,2). This construction is based on facets of the cantellated 8-orthoplex. Images Bicantellated 7-simplex Alternate names * Small birhombated octaexon (acronym: sabro) (Jonathan Bowers) Coordinates The vertices of the ''bicantellated 7-simplex'' can be most simply positioned in 8-space as permutations of (0,0,0,0,1,1,2,2). This construction is based on facets of the bicantellated 8-orthoplex. Images Tricantellated 7-simplex Alternate names * Small trirhombihexadecaexon (stiroh) (Jonathan Bowers) Coordina ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cantellated 7-orthoplex
In seven-dimensional geometry, a cantellated 7-orthoplex is a convex uniform 7-polytope, being a cantellation of the regular 7-orthoplex. There are ten degrees of cantellation for the 7-orthoplex, including truncations. Six are most simply constructible from the dual 7-cube. Cantellated 7-orthoplex Alternate names * Small rhombated hecatonicosoctaexon (acronym: sarz) (Jonathan Bowers) Images Bicantellated 7-orthoplex Alternate names * Small birhombated hecatonicosoctaexon (acronym: sebraz) (Jonathan Bowers) Images Cantitruncated 7-orthoplex Alternate names * Great rhombated hecatonicosoctaexon (acronym: garz) (Jonathan Bowers) Images Bicantitruncated 7-orthoplex Alternate names * Great birhombated hecatonicosoctaexon (acronym: gebraz) (Jonathan Bowers)Klitizing, (o3o3o3x3x3x4o - gebraz) Images Related polytopes These polytopes are from a family of 127 uniform 7-polytope In seven-dimensional geometry, a 7-polytope is a polytope cont ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rectified 7-simplex
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 unique degrees of rectifications, including the zeroth, the 7-simplex itself. 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 tetrahedral cell centers of the ''7-simplex''. Rectified 7-simplex The rectified 7-simplex is the edge figure of the 251 honeycomb. It is called 05,1 for its branching Coxeter-Dynkin diagram, shown as . E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as S. Alternate names * Rectified octaexon (Acronym: roc) (Jonathan Bowers) Coordinates The vertices of the ''rectified 7-simplex'' can be most simply positioned in 8-space as permutations of (0,0,0,0,0,0,1,1). This c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Truncated 7-simplex
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 triangular faces of the 7-simplex. Vertices of the tritruncated 7-simplex are located inside the tetrahedral cells of the 7-simplex. Truncated 7-simplex In seven-dimensional geometry, a truncated 7-simplex is a convex uniform 7-polytope, being a truncation of the regular 7-simplex. Alternate names * Truncated octaexon (Acronym: toc) (Jonathan Bowers) Coordinates The vertices of the ''truncated 7-simplex'' can be most simply positioned in 8-space as permutations of (0,0,0,0,0,0,1,2). This construction is based on facets of the truncated 8-orthoplex. Images Bitruncated 7-simplex Alternate names * Bitruncated octaexon (acronym: bittoc) (Jonathan Bowers ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Truncated 7-orthoplex
In seven-dimensional geometry, a truncated 7-orthoplex is a convex uniform 7-polytope, being a Truncation (geometry), truncation of the regular 7-orthoplex. There are 6 truncations of the 7-orthoplex. Vertices of the truncation 7-orthoplex are located as pairs on the edge of the 7-orthoplex. Vertices of the bitruncated 7-orthoplex are located on the triangular faces of the 7-orthoplex. Vertices of the tritruncated 7-orthoplex are located inside the tetrahedron, tetrahedral cells of the 7-orthoplex. The final three truncations are best expressed relative to the 7-cube. Truncated 7-orthoplex Alternate names * Truncated heptacross * Truncated hecatonicosoctaexon (Jonathan Bowers) Coordinates Cartesian coordinates for the vertices of a truncated 7-orthoplex, centered at the origin, are all 168 vertices are sign (4) and coordinate (42) permutations of : (±2,±1,0,0,0,0,0) Images Construction There are two Coxeter groups associated with the ''truncated 7-orthoplex'', one ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
