8-demicube
In geometry, a demiocteract or 8-demicube is a uniform 8-polytope, constructed from the 8-hypercube, octeract, with alternated vertices removed. It is part of a dimensionally infinite family of uniform polytopes called demihypercubes. E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as HM8 for an 8-dimensional half measure polytope. Coxeter named this polytope as 151 from its Coxeter diagram, with a ring on one of the 1-length branches, File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel split1.pngFile:CDel nodes.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.png and Schläfli symbol or {3,35,1}.
Cartesian coordinates
Cartesian coordinates for the vertices of an 8-demicube centered at the origin are alternate halves of the 8-cube:
- (±1,±1,±1,±1,±1,±1,±1,±1)
with an odd number of plus signs.
Related polytopes and honeycombs
This polytope is the vertex figure for the uniform tessellation, 251 with Coxeter-Dynkin diagram:
- File:CDel nodea 1.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel branch.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.png
Images
| Coxeter plane | B8 | D8 | D7 | D6 | D5 |
|---|---|---|---|---|---|
| Graph | File:8-demicube t0 B8.svg | File:8-demicube t0 D8.svg | File:8-demicube t0 D7.svg | File:8-demicube t0 D6.svg | File:8-demicube t0 D5.svg |
| Dihedral symmetry | [16/2] | [14] | [12] | [10] | [8] |
| Coxeter plane | D4 | D3 | A7 | A5 | A3 |
| Graph | File:8-demicube t0 D4.svg | File:8-demicube t0 D3.svg | File:8-demicube t0 A7.svg | File:8-demicube t0 A5.svg | File:8-demicube t0 A3.svg |
| Dihedral symmetry | [6] | [4] | [8] | [6] | [4] |
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-Strauss, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 (Chapter 26. pp. 409: Hemicubes: 1n1)
External links
- Olshevsky, George. "Demiocteract". Glossary for Hyperspace. Archived from the original on 4 February 2007.
- Multi-dimensional Glossary