8-demicube explained
| bgcolor=#e7dcc3 colspan=2 | Demiocteract
(8-demicube) |
|---|
| bgcolor=#ffffff align=center colspan=2 |
Petrie polygon projection |
| Type | Uniform 8-polytope |
| Family | demihypercube |
| Coxeter symbol | 151 |
| Schläfli symbols | = h
s |
| Coxeter diagrams | =
|
| 7-faces | 144:
16
128 |
| 6-faces | 112
1024 |
| 5-faces | 448
3584 |
| 4-faces | 1120
7168 |
| Cells | 10752:
1792
8960 |
| Faces | 7168 |
| Edges | 1792 |
| Vertices | 128 |
| Vertex figure | Rectified 7-simplex
|
| Symmetry group | D8, [3<sup>5,1,1</sup>] = [1<sup>+</sup>,4,3<sup>6</sup>]
A18, [2<sup>7</sup>]+ |
| Dual | ? |
| Properties | convex | |
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.
\left\{3\begin{array}{l}3,3,3,3,3\\3\end{array}\right\}
or .
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:
References
- H.S.M. Coxeter:
- Coxeter, Regular Polytopes, (3rd edition, 1973), Dover edition,, 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, http://www.wiley.com/WileyCDA/WileyTitle/productCd-0471010030.html
- (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, (Chapter 26. pp. 409: Hemicubes: 1n1)
External links
