 6demicube

Demihexeract
(6demicube)
Petrie polygon projectionType Uniform 6polytope Family demihypercube Schläfli symbol {3,3^{3,1}}
h{4,3,3,3,3}
s{2,2,2,2,2}CoxeterDynkin diagram
Coxeter symbol 1_{31} 5faces 44 12 {3^{1,2,1}}
32 {3^{4}}4faces 252 60 {3^{1,1,1}}
192 {3^{3}}Cells 640 160 {3^{1,0,1}}
480 {3,3}Faces 640 {3} Edges 240 Vertices 32 Vertex figure Rectified 5simplex
Symmetry group D_{6}, [3^{5,1,1}] = [1^{+},4,3^{4}]
[2^{5}]^{+}Petrie polygon decagon Properties convex In geometry, a 6demicube or demihexteract is a uniform 6polytope, constructed from a 6cube (hexeract) with alternate vertices deleted. It is part of a dimensionally infinite family of uniform polytopes called demihypercubes.
Coxeter named this polytope as 1_{31} from its CoxeterDynkin diagram, with a ring on one of the 1length CoxeterDynkin diagram branches. It can named similarly by a 3dimensional exponential Schläfli symbol, {3,3^{3,1}}.
Contents
Cartesian coordinates
Cartesian coordinates for the vertices of a demihexeract centered at the origin are alternate halves of the hexeract:
 (±1,±1,±1,±1,±1,±1)
with an odd number of plus signs.
Images
orthographic projections Coxeter plane B_{6} Graph Dihedral symmetry [12/2] Coxeter plane D_{6} D_{5} Graph Dihedral symmetry [10] [8] Coxeter plane D_{4} D_{3} Graph Dihedral symmetry [6] [4] Coxeter plane A_{5} A_{3} Graph Dihedral symmetry [6] [4] Related polytopes
There are 47 uniform polytopes with D_{6} symmetry, 31 are shared by the B_{6} symmetry, and 16 are unique:
References
 H.S.M. Coxeter:
 Coxeter, Regular Polytopes, (3rd edition, 1973), Dover edition, ISBN 0486614808, p.296, Table I (iii): Regular Polytopes, three regular polytopes in ndimensions (n≥5)
 H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973, p.296, Table I (iii): Regular Polytopes, three regular polytopes in ndimensions (n≥5)
 Kaleidoscopes: Selected Writings of H.S.M. Coxeter, editied by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, WileyInterscience Publication, 1995, ISBN 9780471010036 [1]
 (Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380407, MR 2,10]
 (Paper 23) H.S.M. Coxeter, Regular and SemiRegular Polytopes II, [Math. Zeit. 188 (1985) 559591]
 (Paper 24) H.S.M. Coxeter, Regular and SemiRegular Polytopes III, [Math. Zeit. 200 (1988) 345]
 John H. Conway, Heidi Burgiel, Chaim GoodmanStrass, The Symmetries of Things 2008, ISBN 9781568812205 (Chapter 26. pp. 409: Hemicubes: 1_{n1})
 Richard Klitzing, 6D uniform polytopes (polypeta), x3o3o *b3o3o3o – hax
External links
 Olshevsky, George, Demihexeract at Glossary for Hyperspace.
 Multidimensional Glossary
Fundamental convex regular and uniform polytopes in dimensions 2–10 Family A_{n} BC_{n} D_{n} E_{6} / E_{7} / E_{8} / F_{4} / G_{2} H_{n} Regular polygon Triangle Square Hexagon Pentagon Uniform polyhedron Tetrahedron Octahedron • Cube Demicube Dodecahedron • Icosahedron Uniform polychoron 5cell 16cell • Tesseract Demitesseract 24cell 120cell • 600cell Uniform 5polytope 5simplex 5orthoplex • 5cube 5demicube Uniform 6polytope 6simplex 6orthoplex • 6cube 6demicube 1_{22} • 2_{21} Uniform 7polytope 7simplex 7orthoplex • 7cube 7demicube 1_{32} • 2_{31} • 3_{21} Uniform 8polytope 8simplex 8orthoplex • 8cube 8demicube 1_{42} • 2_{41} • 4_{21} Uniform 9polytope 9simplex 9orthoplex • 9cube 9demicube Uniform 10polytope 10simplex 10orthoplex • 10cube 10demicube npolytopes nsimplex northoplex • ncube ndemicube 1_{k2} • 2_{k1} • k_{21} pentagonal polytope Topics: Polytope families • Regular polytope • List of regular polytopes Categories: 6polytopes
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