6-demicube(half 6-cube) = Pentic 6-cube = Penticantic 6-cube = Pentiruncic 6-cube = Pentiruncicantic 6-cube = Pentisteric 6-cube = Pentistericantic 6-cube = Pentisteriruncic 6-cube = Pentisteriruncicantic 6-cube = Orthogonal projections in D5 Coxeter plane In six-dimensional geometry, a pentic 6-cube is a convex uniform 6-polytope.
There are 8 pentic forms of the 6-cube.
Pentic 6-cube Pentic 6-cubeTypeuniform 6-polytopeSchläfli symbol t0,4{3,34,1}h5{4,34}Coxeter-Dynkin diagram = 5-faces4-facesCellsFacesEdges1440Vertices192Vertex figureCoxeter groupsD6, [33,1,1]Propertiesconvex
The pentic 6-cube, , has half of the vertices of a pentellated 6-cube, .
Alternate names Stericated 6-demicube Stericated demihexeract Small cellated hemihexeract (Acronym: sochax) (Jonathan Bowers)
Cartesian coordinates The Cartesian coordinates for the vertices, centered at the origin are coordinate permutations: (±1,±1,±1,±1,±1,±3) with an odd number of plus signs.
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Penticantic 6-cube Penticantic 6-cubeTypeuniform 6-polytopeSchläfli symbol t0,1,4{3,34,1}h2,5{4,34}Coxeter-Dynkin diagram = 5-faces4-facesCellsFacesEdges9600Vertices1920Vertex figureCoxeter groupsD6, [33,1,1]Propertiesconvex The penticantic 6-cube, , has half of the vertices of a penticantellated 6-cube, .
Alternate names Steritruncated 6-demicube Steritruncated demihexeract Cellitruncated hemihexeract (Acronym: cathix) (Jonathan Bowers)
Cartesian coordinates The Cartesian coordinates for the vertices, centered at the origin are coordinate permutations: (±1,±1,±3,±3,±3,±5) with an odd number of plus signs.
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Pentiruncic 6-cube Pentiruncic 6-cubeTypeuniform 6-polytopeSchläfli symbol t0,2,4{3,34,1}h3,5{4,34}Coxeter-Dynkin diagram = 5-faces4-facesCellsFacesEdges10560Vertices1920Vertex figureCoxeter groupsD6, [33,1,1]Propertiesconvex The pentiruncic 6-cube, , has half of the vertices of a pentiruncinated 6-cube (penticantellated 6-orthoplex), .
Alternate names Stericantellated 6-demicube Stericantellated demihexeract Cellirhombated hemihexeract (Acronym: crohax) (Jonathan Bowers)
Cartesian coordinates The Cartesian coordinates for the vertices, centered at the origin are coordinate permutations: (±1,±1,±1,±3,±3,±5) with an odd number of plus signs.
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Pentiruncicantic 6-cube Pentiruncicantic 6-cubeTypeuniform 6-polytopeSchläfli symbol t0,1,2,4{3,32,1}h2,3,5{4,34}Coxeter-Dynkin diagram = 5-faces4-facesCellsFacesEdges20160Vertices5760Vertex figureCoxeter groupsD6, [33,1,1]Propertiesconvex The pentiruncicantic 6-cube, , has half of the vertices of a pentiruncicantellated 6-cube or (pentiruncicantellated 6-orthoplex),
Alternate names Stericantitruncated demihexeract Stericantitruncated 6-demicube Great cellated hemihexeract (Acronym: cagrohax) (Jonathan Bowers)Klitzing, (x3x3o *b3x3x3o3o - cagrohax)
Cartesian coordinates The Cartesian coordinates for the vertices, centered at the origin are coordinate permutations: (±1,±1,±3,±3,±5,±7) with an odd number of plus signs.
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Pentisteric 6-cube Pentisteric 6-cubeTypeuniform 6-polytopeSchläfli symbol t0,3,4{3,34,1}h4,5{4,34}Coxeter-Dynkin diagram = 5-faces4-facesCellsFacesEdges5280Vertices960Vertex figureCoxeter groupsD6, [33,1,1]Propertiesconvex The pentisteric 6-cube, , has half of the vertices of a pentistericated 6-cube (pentitruncated 6-orthoplex),
Alternate names Steriruncinated 6-demicube Steriruncinated demihexeract Small celliprismated hemihexeract (Acronym: cophix) (Jonathan Bowers)
Cartesian coordinates The Cartesian coordinates for the vertices, centered at the origin are coordinate permutations: (±1,±1,±1,±1,±3,±5) with an odd number of plus signs.
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Pentistericantic 6-cube Pentistericantic 6-cubeTypeuniform 6-polytopeSchläfli symbol t0,1,3,4{3,34,1}h2,4,5{4,34}Coxeter-Dynkin diagram = 5-faces4-facesCellsFacesEdges23040Vertices5760Vertex figureCoxeter groupsD6, [33,1,1]Propertiesconvex The pentistericantic 6-cube, , has half of the vertices of a pentistericantellated 6-cube (pentiruncitruncated 6-orthoplex), .
Alternate names Steriruncitruncated demihexeract Steriruncitruncated 6-demicube Cellitruncated hemihexeract (Acronym: capthix) (Jonathan Bowers)
Cartesian coordinates The Cartesian coordinates for the vertices, centered at the origin are coordinate permutations: (±1,±1,±3,±3,±5,±7) with an odd number of plus signs.
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Pentisteriruncic 6-cube Pentisteriruncic 6-cubeTypeuniform 6-polytopeSchläfli symbol t0,2,3,4{3,34,1}h3,4,5{4,34}Coxeter-Dynkin diagram = 5-faces4-facesCellsFacesEdges15360Vertices3840Vertex figureCoxeter groupsD6, [33,1,1]Propertiesconvex The pentisteriruncic 6-cube, , has half of the vertices of a pentisteriruncinated 6-cube (penticantitruncated 6-orthoplex), .
Alternate names Steriruncicantellated 6-demicube Steriruncicantellated demihexeract Celliprismatorhombated hemihexeract (Acronym: caprohax) (Jonathan Bowers)
Cartesian coordinates The Cartesian coordinates for the vertices, centered at the origin are coordinate permutations: (±1,±1,±1,±3,±5,±7) with an odd number of plus signs.
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Pentisteriruncicantic 6-cube Pentisteriruncicantic 6-cubeTypeuniform 6-polytopeSchläfli symbol t0,1,2,3,4{3,32,1}h2,3,4,5{4,34}Coxeter-Dynkin diagram = 5-faces4-facesCellsFacesEdges34560Vertices11520Vertex figureCoxeter groupsD6, [33,1,1]Propertiesconvex The pentisteriruncicantic 6-cube, , has half of the vertices of a pentisteriruncicantellated 6-cube (pentisteriruncicantitruncated 6-orthoplex), .
Alternate names Steriruncicantitruncated 6-demicube/demihexeract Great cellated hemihexeract (Acronym: gochax) (Jonathan Bowers)Klitzing, (x3x3o *b3x3x3x3o - gochax)
Cartesian coordinates The Cartesian coordinates for the vertices, centered at the origin are coordinate permutations: (±1,±1,±3,±3,±5,±7) with an odd number of plus signs.
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Related polytopes
There are 47 uniform polytopes with D6 symmetry, 31 are shared by the B6 symmetry, and 16 are unique:
Notes
References H.S.M. Coxeter: H.S.M. Coxeter, Regular Polytopes, 3rd edition, Dover, New York, 1973 Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivić Weiss, Wiley-Interscience Publication, 1995, wiley.com, (Paper 22) H.S.M. Coxeter, Regular and Semi-Regular Polytopes I, [Math. Zeit. 46 (1940) 380–407, MR 2,10] (Paper 23) H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) 559–591] (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3–45] Norman Johnson Uniform Polytopes, Manuscript (1991) N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. x3o3o *b3o3x3o3o - sochax, x3x3o *b3o3x3o3o - cathix, x3o3o *b3x3x3o3o - crohax, x3x3o *b3x3x3o3o - cagrohax, x3o3o *b3o3x3x3x - cophix, x3x3o *b3o3x3x3x - capthix, x3o3o *b3x3x3x3x - caprohax, x3x3o *b3x3x3x3o - gochax
External links
Polytopes of Various Dimensions Multi-dimensional Glossary