- 7-demicube
-
Demihepteract
(7-demicube)
Petrie polygon projectionType Uniform 7-polytope Family demihypercube Coxeter symbol 141 Schläfli symbol {31,4,1}
h{4,35}
s{26}Coxeter-Dynkin diagram 
6-faces 78 14 {31,3,1} 
64 {35}
5-faces 532 84 {31,2,1} 
448 {34}
4-faces 1624 280 {31,1,1} 
1344 {33}
Cells 2800 560 {31,0,1} 
2240 {3,3}Faces 2240 {3} 
Edges 672 Vertices 64 Vertex figure Rectified 6-simplex
Symmetry group D7, [36,1,1] = [1+,4,35]
[26]+Dual ? Properties convex In geometry, a demihepteract or 7-demicube is a uniform 7-polytope, constructed from the 7-hypercube (hepteract) with alternated vertices deleted. It is part of a dimensionally infinite family of uniform polytopes called demihypercubes.
Coxeter named this polytope as 141 from its Coxeter-Dynkin diagram, with a ring on one of the 1-length Coxeter-Dynkin diagram branches.
Contents
Cartesian coordinates
Cartesian coordinates for the vertices of a demihepteract centered at the origin are alternate halves of the hepteract:
- (±1,±1,±1,±1,±1,±1,±1)
with an odd number of plus signs.
Images
orthographic projections Coxeter plane B7 D7 D6 Graph 
Dihedral symmetry [14/2] [12] [10] Coxeter plane D5 D4 D3 Graph 
Dihedral symmetry [8] [6] [4] Coxeter plane A5 A3 Graph 
Dihedral symmetry [6] [4] Related polytopes
There are 95 uniform polytopes with D6 symmetry, 63 are shared by the B6 symmetry, and 32 are unique:
t0(141)
t0,1(141)
t0,2(141)
t0,1,2(141)
edit] References- H.S.M. Coxeter:
- Coxeter, Regular Polytopes, (3rd edition, 1973), Dover edition, ISBN 0-486-61480-8, p.296, Table I (iii): Regular Polytopes, three regular polytopes in n-dimensions (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 n-dimensions (n≥5)
- Kaleidoscopes: Selected Writings of H.S.M. Coxeter, editied by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6 [1]
- (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]
- John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 (Chapter 26. pp. 409: Hemicubes: 1n1)
- Richard Klitzing, 7D uniform polytopes (polyexa), x3o3o *b3o3o3o3o - hesa
External links
- Olshevsky, George, Demihepteract at Glossary for Hyperspace.
- Multi-dimensional Glossary
Fundamental convex regular and uniform polytopes in dimensions 2–10 Family An BCn Dn E6 / E7 / E8 / F4 / G2 Hn Regular polygon Triangle Square Hexagon Pentagon Uniform polyhedron Tetrahedron Octahedron • Cube Demicube Dodecahedron • Icosahedron Uniform polychoron 5-cell 16-cell • Tesseract Demitesseract 24-cell 120-cell • 600-cell Uniform 5-polytope 5-simplex 5-orthoplex • 5-cube 5-demicube Uniform 6-polytope 6-simplex 6-orthoplex • 6-cube 6-demicube 122 • 221 Uniform 7-polytope 7-simplex 7-orthoplex • 7-cube 7-demicube 132 • 231 • 321 Uniform 8-polytope 8-simplex 8-orthoplex • 8-cube 8-demicube 142 • 241 • 421 Uniform 9-polytope 9-simplex 9-orthoplex • 9-cube 9-demicube Uniform 10-polytope 10-simplex 10-orthoplex • 10-cube 10-demicube n-polytopes n-simplex n-orthoplex • n-cube n-demicube 1k2 • 2k1 • k21 pentagonal polytope Topics: Polytope families • Regular polytope • List of regular polytopes 
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