|
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 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 distinguis ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
List Of Regular Polytopes
This article lists the regular polytopes and regular polytope compounds in Euclidean geometry, Euclidean, spherical geometry, spherical and hyperbolic geometry, hyperbolic spaces. The Schläfli symbol describes every regular tessellation of an ''n''-sphere, Euclidean and hyperbolic spaces. A Schläfli symbol describing an ''n''-polytope equivalently describes a tessellation of an (''n'' − 1)-sphere. In addition, the symmetry of a regular polytope or tessellation is expressed as a Coxeter group, which Coxeter expressed identically to the Schläfli symbol, except delimiting by square brackets, a notation that is called Coxeter notation. Another related symbol is the Coxeter-Dynkin diagram which represents a symmetry group with no rings, and the represents regular polytope or tessellation with a ring on the first node. For example, the cube has Schläfli symbol , and with its octahedral symmetry, [4,3] or , it is represented by Coxeter diagram . The regular polytopes are ... [...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] |
7-cube T16
In geometry, a 7-cube is a seven-dimensional hypercube with 128 vertices, 448 edges, 672 square faces, 560 cubic cells, 280 tesseract 4-faces, 84 penteract 5-faces, and 14 hexeract 6-faces. It can be named by its Schläfli symbol , being composed of 3 6-cubes around each 5-face. It can be called a hepteract, a portmanteau of tesseract (the ''4-cube'') and ''hepta'' for seven (dimensions) in Greek. It can also be called a regular tetradeca-7-tope or tetradecaexon, being a 7 dimensional polytope constructed from 14 regular facets. Related polytopes The ''7-cube'' is 7th in a series of hypercube: The dual of a 7-cube is called a 7-orthoplex, and is a part of the infinite family of cross-polytopes. Applying an '' alternation'' operation, deleting alternating vertices of the hepteract, creates another uniform polytope, called a demihepteract, (part of an infinite family called demihypercubes), which has 14 demihexeractic and 64 6-simplex 6-faces. As a configurat ... [...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] |
7-cube T26
In geometry, a 7-cube is a seven-dimensional hypercube with 128 vertices, 448 edges, 672 square faces, 560 cubic cells, 280 tesseract 4-faces, 84 penteract 5-faces, and 14 hexeract 6-faces. It can be named by its Schläfli symbol , being composed of 3 6-cubes around each 5-face. It can be called a hepteract, a portmanteau of tesseract (the ''4-cube'') and ''hepta'' for seven (dimensions) in Greek. It can also be called a regular tetradeca-7-tope or tetradecaexon, being a 7 dimensional polytope constructed from 14 regular facets. Related polytopes The ''7-cube'' is 7th in a series of hypercube: The dual of a 7-cube is called a 7-orthoplex, and is a part of the infinite family of cross-polytopes. Applying an '' alternation'' operation, deleting alternating vertices of the hepteract, creates another uniform polytope, called a demihepteract, (part of an infinite family called demihypercubes), which has 14 demihexeractic and 64 6-simplex 6-faces. As a configurat ... [...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] |
7-cube T36
In geometry, a 7-cube is a seven-dimensional hypercube with 128 vertices, 448 edges, 672 square faces, 560 cubic cells, 280 tesseract 4-faces, 84 penteract 5-faces, and 14 hexeract 6-faces. It can be named by its Schläfli symbol , being composed of 3 6-cubes around each 5-face. It can be called a hepteract, a portmanteau of tesseract (the ''4-cube'') and ''hepta'' for seven (dimensions) in Greek. It can also be called a regular tetradeca-7-tope or tetradecaexon, being a 7 dimensional polytope constructed from 14 regular facets. Related polytopes The ''7-cube'' is 7th in a series of hypercube: The dual of a 7-cube is called a 7-orthoplex, and is a part of the infinite family of cross-polytopes. Applying an '' alternation'' operation, deleting alternating vertices of the hepteract, creates another uniform polytope, called a demihepteract, (part of an infinite family called demihypercubes), which has 14 demihexeractic and 64 6-simplex 6-faces. As a configurat ... [...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] |
7-cube T46
In geometry, a 7-cube is a seven-dimensional hypercube with 128 vertices, 448 edges, 672 square faces, 560 cubic cells, 280 tesseract 4-faces, 84 penteract 5-faces, and 14 hexeract 6-faces. It can be named by its Schläfli symbol , being composed of 3 6-cubes around each 5-face. It can be called a hepteract, a portmanteau of tesseract (the ''4-cube'') and ''hepta'' for seven (dimensions) in Greek. It can also be called a regular tetradeca-7-tope or tetradecaexon, being a 7 dimensional polytope constructed from 14 regular facets. Related polytopes The ''7-cube'' is 7th in a series of hypercube: The dual of a 7-cube is called a 7-orthoplex In geometry, a 7-orthoplex, or 7-cross polytope, is a regular 7-polytope with 14 vertices, 84 edges, 280 triangle faces, 560 tetrahedron cells, 672 5-cells ''4-faces'', 448 ''5-faces'', and 128 ''6-faces''. It has two constructed forms, the fi ..., and is a part of the infinite family of cross-polytopes. Applying an ''Al ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rectified 7-orthoplex
In seven-dimensional geometry, a rectified 7-orthoplex is a convex uniform 7-polytope, being a rectification of the regular 7-orthoplex. There are unique 7 degrees of rectifications, the zeroth being the 7-orthoplex, and the 6th and last being the 7-cube. Vertices of the rectified 7-orthoplex are located at the edge-centers of the 7-orthoplex. Vertices of the birectified 7-orthoplex are located in the triangular face centers of the 7-orthoplex. Vertices of the trirectified 7-orthoplex are located in the tetrahedral cell centers of the 7-orthoplex. Rectified 7-orthoplex The ''rectified 7-orthoplex'' is the vertex figure for the demihepteractic honeycomb. The rectified 7-orthoplex's 84 vertices represent the kissing number of a sphere-packing constructed from this honeycomb. : or Alternate names * rectified heptacross * rectified hecatonicosoctaexon (Acronym rez) (Jonathan Bowers) - rectified 128-faceted polyexon Images Construction There are two Coxeter groups asso ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
7-cube T5
In geometry, a 7-cube is a seven-dimensional hypercube with 128 vertices, 448 edges, 672 square faces, 560 cubic cells, 280 tesseract 4-faces, 84 penteract 5-faces, and 14 hexeract 6-faces. It can be named by its Schläfli symbol , being composed of 3 6-cubes around each 5-face. It can be called a hepteract, a portmanteau of tesseract (the ''4-cube'') and ''hepta'' for seven (dimensions) in Greek. It can also be called a regular tetradeca-7-tope or tetradecaexon, being a 7 dimensional polytope constructed from 14 regular facets. Related polytopes The ''7-cube'' is 7th in a series of hypercube: The dual of a 7-cube is called a 7-orthoplex, and is a part of the infinite family of cross-polytopes. Applying an '' alternation'' operation, deleting alternating vertices of the hepteract, creates another uniform polytope, called a demihepteract, (part of an infinite family called demihypercubes), which has 14 demihexeractic and 64 6-simplex 6-faces. As a configurat ... [...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 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 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 with the C7 or [4,35] Coxeter group, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
