7-demicube
Demihepteract (7-demicube) | ||
---|---|---|
Petrie polygon projection | ||
Type | Uniform 7-polytope | |
Family | demihypercube | |
Coxeter symbol | 141 | |
Schläfli symbol | {3,34,1} = h{4,35} s{26} | |
Coxeter 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 truncated. 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.
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
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:
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, edited 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)
- Klitzing, Richard. "7D uniform polytopes (polyexa) x3o3o *b3o3o3o3o - hesa".
External links
- Olshevsky, George. "Demihepteract". Glossary for Hyperspace. Archived from the original on 4 February 2007.
- Multi-dimensional Glossary