|
Uniform 6-polytope
In six-dimensional geometry, a uniform 6-polytope is a six-dimensional uniform polytope. A uniform polypeton is vertex-transitive, and all facets are uniform 5-polytopes. The complete set of convex uniform 6-polytopes has not been determined, but most can be made as Wythoff constructions from a small set of symmetry groups. These construction operations are represented by the permutations of rings of the Coxeter-Dynkin diagrams. Each combination of at least one ring on every connected group of nodes in the diagram produces a uniform 6-polytope. The simplest uniform polypeta are regular polytopes: the 6-simplex , the 6-cube (hexeract) , and the 6-orthoplex (hexacross) . History of discovery * Regular polytopes: (convex faces) ** 1852: Ludwig Schläfli proved in his manuscript ''Theorie der vielfachen Kontinuität'' that there are exactly 3 regular polytopes in 5 or more dimensions. * Convex semiregular polytopes: (Various definitions before Coxeter's uniform category) ** ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
List Of Regular Polytopes
This article lists the regular polytopes in Euclidean, spherical and hyperbolic spaces. Overview This table shows a summary of regular polytope counts by rank. There are no Euclidean regular star tessellations in any number of dimensions. 1-polytopes There is only one polytope of rank 1 (1-polytope), the closed line segment bounded by its two endpoints. Every realization of this 1-polytope is regular. It has the Schläfli symbol , or a Coxeter diagram with a single ringed node, . Norman Johnson calls it a ''dion'' and gives it the Schläfli symbol . Although trivial as a polytope, it appears as the edges of polygons and other higher dimensional polytopes. It is used in the definition of uniform prisms like Schläfli symbol ×, or Coxeter diagram as a Cartesian product of a line segment and a regular polygon. 2-polytopes (polygons) The polytopes of rank 2 (2-polytopes) are called polygons. Regular polygons are equilateral and cyclic. A -gonal regular polygon is repre ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
6-orthoplex
In geometry, a 6-orthoplex, or 6-cross polytope, is a regular 6-polytope with 12 Vertex (geometry), vertices, 60 Edge (geometry), edges, 160 triangle Face (geometry), faces, 240 tetrahedron Cell (mathematics), cells, 192 5-cell ''4-faces'', and 64 ''5-faces''. It has two constructed forms, the first being regular with Schläfli symbol , and the second with alternately labeled (checkerboarded) facets, with Schläfli symbol or Coxeter symbol 311. It is a part of an infinite family of polytopes, called cross-polytopes or ''orthoplexes''. The dual polytope is the 6-hypercube, or hexeract. Alternate names *Hexacross, derived from combining the family name cross polytope with ''hex'' for six (dimensions) in Greek language, Greek. * Hexacontitetrapeton as a 64-Facet (geometry), facetted 6-polytope. As a configuration This Regular 4-polytope#As configurations, configuration matrix represents the 6-orthoplex. The rows and columns correspond to vertices, edges, faces, cells, 4-faces an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
6-cube T03
In geometry, a 6-cube is a six-dimensional hypercube with 64 vertices, 192 edges, 240 square faces, 160 cubic cells, 60 tesseract 4-faces, and 12 5-cube 5-faces. It has Schläfli symbol , being composed of 3 5-cubes around each 4-face. It can be called a hexeract, a portmanteau of tesseract (the ''4-cube'') with ''hex'' for six (dimensions) in Greek. It can also be called a regular dodeca-6-tope or dodecapeton, being a 6-dimensional polytope constructed from 12 regular facets. Related polytopes It is a part of an infinite family of polytopes, called hypercubes. The dual of a 6-cube can be called a 6-orthoplex, and is a part of the infinite family of cross-polytopes. It is composed of various 5-cubes, at perpendicular angles on the u-axis, forming coordinates (x,y,z,w,v,u). Applying an '' alternation'' operation, deleting alternating vertices of the 6-cube, creates another uniform polytope, called a 6-demicube, (part of an infinite family called demihypercubes), which has ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cantellated 6-cube
In six-dimensional geometry, a cantellated 6-cube is a convex uniform 6-polytope, being a cantellation of the regular 6-cube. There are 8 cantellations for the 6-cube, including truncations. Half of them are more easily constructed from the dual 6-orthoplex. Cantellated 6-cube Alternate names * Cantellated hexeract * Small rhombated hexeract (acronym: srox) (Jonathan Bowers) Images Bicantellated 6-cube Alternate names * Bicantellated hexeract * Small birhombated hexeract (acronym: saborx) (Jonathan Bowers) Images Cantitruncated 6-cube Alternate names * Cantitruncated hexeract * Great rhombihexeract (acronym: grox) (Jonathan Bowers) Images It is fourth in a series of cantitruncated hypercubes: Bicantitruncated 6-cube Alternate names * Bicantitruncated hexeract * Great birhombihexeract (acronym: gaborx) (Jonathan Bowers) Images Related polytopes These polytopes are part of a set of 63 uniform 6-polytopes generated from the B6 Coxeter ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
6-cube T02
In geometry, a 6-cube is a six-dimensional hypercube with 64 vertices, 192 edges, 240 square faces, 160 cubic cells, 60 tesseract 4-faces, and 12 5-cube 5-faces. It has Schläfli symbol , being composed of 3 5-cubes around each 4-face. It can be called a hexeract, a portmanteau of tesseract (the ''4-cube'') with ''hex'' for six (dimensions) in Greek. It can also be called a regular dodeca-6-tope or dodecapeton, being a 6-dimensional polytope constructed from 12 regular facets. Related polytopes It is a part of an infinite family of polytopes, called hypercubes. The dual of a 6-cube can be called a 6-orthoplex, and is a part of the infinite family of cross-polytopes. It is composed of various 5-cubes, at perpendicular angles on the u-axis, forming coordinates (x,y,z,w,v,u). Applying an '' alternation'' operation, deleting alternating vertices of the 6-cube, creates another uniform polytope, called a 6-demicube, (part of an infinite family called demihypercubes), which has ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stericated 6-orthoplex
In six-dimensional geometry, a stericated 6-orthoplex is a convex uniform 6-polytope, constructed as a sterication (4th order truncation) of the regular 6-orthoplex. There are 16 unique sterications for the 6-orthoplex with permutations of truncations, cantellations, and runcinations. Eight are better represented from the stericated 6-cubes. Stericated 6-orthoplex Alternate names * Small cellated hexacontatetrapeton (Acronym: scag) (Jonathan Bowers) Images Steritruncated 6-orthoplex Alternate names * Cellitruncated hexacontatetrapeton (Acronym: catog) (Jonathan Bowers) Images Stericantellated 6-orthoplex Alternate names * Cellirhombated hexacontatetrapeton (Acronym: crag) (Jonathan Bowers) Images Stericantitruncated 6-orthoplex Alternate names * Celligreatorhombated hexacontatetrapeton (Acronym: cagorg) (Jonathan Bowers) Images Steriruncinated 6-orthoplex Alternate names * Celliprismated hexacontatetrapeton (Acronym: copog) (Jo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
6-cube T15
In geometry, a 6-cube is a six-dimensional hypercube with 64 vertices, 192 edges, 240 square faces, 160 cubic cells, 60 tesseract 4-faces, and 12 5-cube 5-faces. It has Schläfli symbol , being composed of 3 5-cubes around each 4-face. It can be called a hexeract, a portmanteau of tesseract (the ''4-cube'') with ''hex'' for six (dimensions) in Greek. It can also be called a regular dodeca-6-tope or dodecapeton, being a 6-dimensional polytope constructed from 12 regular facets. Related polytopes It is a part of an infinite family of polytopes, called hypercubes. The dual of a 6-cube can be called a 6-orthoplex, and is a part of the infinite family of cross-polytopes. It is composed of various 5-cubes, at perpendicular angles on the u-axis, forming coordinates (x,y,z,w,v,u). Applying an '' alternation'' operation, deleting alternating vertices of the 6-cube, creates another uniform polytope, called a 6-demicube, (part of an infinite family called demihypercubes), which has ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Runcinated 6-orthoplex
In six-dimensional geometry, a runcinated 6-orthplex is a convex uniform 6-polytope with 3rd order Truncation (geometry), truncations (runcination) of the regular 6-orthoplex. There are 12 unique runcinations of the 6-orthoplex with permutations of truncations, and cantellations. 7 are expressed relative to the dual 6-cube. Runcinated 6-orthoplex Alternate names * Small prismatohexacontatetrapeton (spog) (Jonathan Bowers) Images Runcicantellated 6-orthoplex Alternate names * Prismatorhombated hexacontatetrapeton (prog) (Jonathan Bowers) Images Biruncitruncated 6-orthoplex Alternate names * Biprismatotruncated hexacontatetrapeton (boprax) (Jonathan Bowers) Images Runcitruncated 6-orthoplex Alternate names * Prismatotruncated hexacontatetrapeton (potag) (Jonathan Bowers) Images Runcicantitruncated 6-orthoplex Alternate names * Great prismated hexacontatetrapeton (gopog) (Jonathan Bowers) Images Related polytopes These polytopes a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
6-cube T25
In geometry, a 6-cube is a six-dimensional hypercube with 64 Vertex (geometry), vertices, 192 Edge (geometry), edges, 240 square Face (geometry), faces, 160 cubic Cell (mathematics), cells, 60 tesseract 4-faces, and 12 5-cube 5-faces. It has Schläfli symbol , being composed of 3 5-cubes around each 4-face. It can be called a hexeract, a portmanteau of tesseract (the ''4-cube'') with ''hex'' for six (dimensions) in Greek language, Greek. It can also be called a regular dodeca-6-tope or dodecapeton, being a 6-polytope, 6-dimensional polytope constructed from 12 regular Facet (geometry), facets. Related polytopes It is a part of an infinite family of polytopes, called hypercubes. The Dual polytope, dual of a 6-cube can be called a 6-orthoplex, and is a part of the infinite family of cross-polytopes. It is composed of various 5-cubes, at perpendicular angles on the u-axis, forming coordinates (x,y,z,w,v,u). Applying an ''Alternation (geometry), alternation'' operation, deleting ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cantellated 6-orthoplex
In six-dimensional geometry, a cantellated 6-orthoplex is a convex uniform 6-polytope, being a cantellation of the regular 6-orthoplex. There are 8 cantellation for the 6-orthoplex including truncations. Half of them are more easily constructed from the dual 6-cube Cantellated 6-orthoplex Alternate names * Cantellated hexacross * Small rhombated hexacontatetrapeton (acronym: srog) (Jonathan Bowers) Construction There are two Coxeter groups associated with the ''cantellated 6-orthoplex'', one with the B6 or ,3,3,3,3Coxeter group, and a lower symmetry with the D6 or 3,1,1Coxeter group. Coordinates Cartesian coordinates for the 480 vertices of a cantellated 6-orthoplex, centered at the origin, are all the sign and coordinate permutations of : (2,1,1,0,0,0) Images Bicantellated 6-orthoplex Alternate names * Bicantellated hexacross, bicantellated hexacontatetrapeton * Small birhombated hexacontatetrapeton (acronym: siborg) (Jonathan Bowers) Construction ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
6-cube T35
In geometry, a 6-cube is a six-dimensional hypercube with 64 vertices, 192 edges, 240 square faces, 160 cubic cells, 60 tesseract 4-faces, and 12 5-cube 5-faces. It has Schläfli symbol , being composed of 3 5-cubes around each 4-face. It can be called a hexeract, a portmanteau of tesseract (the ''4-cube'') with ''hex'' for six (dimensions) in Greek. It can also be called a regular dodeca-6-tope or dodecapeton, being a 6-dimensional polytope constructed from 12 regular facets. Related polytopes It is a part of an infinite family of polytopes, called hypercubes. The dual of a 6-cube can be called a 6-orthoplex, and is a part of the infinite family of cross-polytopes. It is composed of various 5-cubes, at perpendicular angles on the u-axis, forming coordinates (x,y,z,w,v,u). Applying an '' alternation'' operation, deleting alternating vertices of the 6-cube, creates another uniform polytope, called a 6-demicube, (part of an infinite family called demihypercubes), which has ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rectified 6-orthoplex
In six-dimensional geometry, a rectified 6-orthoplex is a convex uniform 6-polytope, being a rectification of the regular 6-orthoplex. There are unique 6 degrees of rectifications, the zeroth being the 6-orthoplex, and the 6th and last being the 6-cube. Vertices of the rectified 6-orthoplex are located at the edge-centers of the 6-orthoplex. Vertices of the birectified 6-orthoplex are located in the triangular face centers of the 6-orthoplex. Rectified 6-orthoplex The ''rectified 6-orthoplex'' is the vertex figure for the demihexeractic honeycomb. : or Alternate names * rectified hexacross * rectified hexacontitetrapeton (acronym: rag) (Jonathan Bowers) Construction There are two Coxeter groups associated with the ''rectified hexacross'', one with the C6 or ,3,3,3,3Coxeter group, and a lower symmetry with two copies of pentacross facets, alternating, with the D6 or 3,1,1Coxeter group. Cartesian coordinates Cartesian coordinates for the vertices of a rectified he ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
