Summary
| Demiocteract (8-demicube) | |
|---|---|
 Petrie polygon projection | |
| Type | Uniform 8-polytope |
| Family | demihypercube |
| Coxeter symbol | 151 |
| Schläfli symbols | {3,35,1} = h{4,36} s{21,1,1,1,1,1,1} |
| Coxeter diagrams |     =  
|
| 7-faces | 144: 16 {31,4,1}  128 {36} 
|
| 6-faces | 112 {31,3,1} 1024 {35} 
|
| 5-faces | 448 {31,2,1} 3584 {34} 
|
| 4-faces | 1120 {31,1,1} 7168 {3,3,3} 
|
| Cells | 10752: 1792 {31,0,1}  8960 {3,3}
|
| Faces | 7168 {3}
|
| Edges | 1792 |
| Vertices | 128 |
| Vertex figure | Rectified 7-simplex
|
| Symmetry group | D8, [35,1,1] = [1+,4,36] A18, [27]+ |
| 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.
Coxeter named this polytope as 151 from its Coxeter diagram, with a ring on
one of the 1-length branches, 
and Schläfli symbol or {3,35,1}.
Cartesian coordinates edit
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 edit
This polytope is the vertex figure for the uniform tessellation, 251 with Coxeter-Dynkin diagram:
Images edit
| Coxeter plane | B8 | D8 | D7 | D6 | D5 |
|---|---|---|---|---|---|
| Graph | 
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| Dihedral symmetry | [16/2] | [14] | [12] | [10] | [8] |
| Coxeter plane | D4 | D3 | A7 | A5 | A3 |
| Graph | 
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| Dihedral symmetry | [6] | [4] | [8] | [6] | [4] |
References edit
- 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 edit
- Olshevsky, George. "Demiocteract". Glossary for Hyperspace. Archived from the original on 4 February 2007.
- Multi-dimensional Glossary