7-demicube Hexic 7-cube Hexicantic 7-cube Hexiruncic 7-cubeHexiruncicantic 7-cube Hexisteric 7-cube Hexistericantic 7-cube Hexisteriruncic 7-cube Hexisteriruncicantic 7-cube Hexipentic 7-cube Hexipenticantic 7-cube Hexipentiruncic 7-cube Hexipentiruncicantic 7-cube Hexipentisteric 7-cube Hexipentistericantic 7-cube Hexipentisteriruncic 7-cube Hexipentisteriruncicantic 7-cube Orthogonal projections in D7 Coxeter plane
In seven-dimensional geometry, a hexic 7-cube is a convex uniform 7-polytope, constructed from the uniform 7-demicube. There are 16 unique forms.
Hexic 7-cube Hexic 7-cubeTypeuniform 7-polytopeSchläfli symbol t0,5{3,34,1}h6{4,35}Coxeter-Dynkin diagram6-faces5-faces4-facesCellsFacesEdges4704Vertices448Vertex figureCoxeter groupsD7, [34,1,1]Propertiesconvex
Alternate names Small terated demihepteract (acronym: suthesa)Klitzing, (x3o3o *b3o3o3o3x - suthesa)
Cartesian coordinates The Cartesian coordinates for the vertices of a hexic 7-cube centered at the origin are coordinate permutations: (±1,±1,±1,±1,±1,±1,±3) with an odd number of plus signs.
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Hexicantic 7-cube
Alternate names Teritruncated demihepteract (acronym: tuthesa)Klitzing, (x3x3o *b3o3o3o3x - tuthesa)
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Hexiruncic 7-cube
Alternate names Terirhombated demihepteract (acronym: turhesa)Klitzing, (x3o3o *b3x3o3o3x - turhesa)
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Hexisteric 7-cube
Alternate names Teriprismated demihepteract (acronym: tuphesa)Klitzing, (x3o3o *b3o3x3o3x - tuphesa)
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Hexipentic 7-cube
Alternate names Tericellated demihepteract (acronym: tuchesa)Klitzing, (x3o3o *b3o3o3x3x - tuchesa)
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Hexiruncicantic 7-cube
Alternate names Terigreatorhombated demihepteract (acronym: tugrohesa)Klitzing, (x3x3o *b3x3o3o3x - tugrohesa)
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Hexistericantic 7-cube
Alternate names Teriprismatotruncated demihepteract (acronym: tupthesa)Klitzing, (x3x3o *b3o3x3o3x - tupthesa)
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Hexipenticantic 7-cube
Alternate names Tericellitruncated demihepteract (acronym: tucothesa)Klitzing, (x3x3o *b3o3o3x3x - tucothesa)
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Hexisteriruncic 7-cube
Alternate names Teriprismatorhombated demihepteract (acronym: tuprohesa)Klitzing, (x3o3o *b3x3x3o3x - tuprohesa)
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Hexipentiruncic 7-cube
Alternate names Tericellirhombated demihepteract (acronym: tucrohesa)Klitzing, (x3o3o *b3x3o3x3x - tucrohesa)
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Hexipentisteric 7-cube
Alternate names Tericelliprismated demihepteract (acronym: tucophesa)Klitzing, (x3o3o *b3o3x3x3x - tucophesa)
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Hexisteriruncicantic 7-cube
Alternate names Terigreatoprismated demihepteract (acronym: tugphesa)Klitzing, (x3x3o *b3x3x3o3x - tugphesa)
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Hexipentiruncicantic 7-cube
Alternate names Tericelligreatorhombated demihepteract (acronym: tucagrohesa)Klitzing, (x3x3o *b3x3o3x3x - tucagrohesa)
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Hexipentisteriruncic 7-cube
Alternate names Tericelliprismatorhombated demihepteract (acronym: tucprohesa)Klitzing, (x3o3o *b3x3x3x3x - tucprohesa)
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Hexipentistericantic 7-cube
Alternate names Tericelliprismatotruncated demihepteract (acronym: tucpathesa)Klitzing, (x3x3o *b3o3x3x3x - tucpathesa)
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Hexipentisteriruncicantic 7-cube
Alternate names Great terated demihepteract (acronym: guthesa)Klitzing, (x3x3o *b3x3x3x3x - guthesa)
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Related polytopes These polytopes are based on the 7-demicube, a member of a dimensional family of uniform polytopes called demihypercubes for being alternation of the hypercube family.
There are 95 uniform polytopes with D7 symmetry, 63 are shared by the BC7 symmetry, and 32 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.
External links
Polytopes of Various Dimensions Multi-dimensional Glossary