# 8-demicube

Demiocteract
(8-demicube)

Petrie polygon projection
Type Uniform 8-polytope
Family demihypercube
Coxeter symbol 151
Schläfli symbol {3,35,1} = h{4,36}
s{27}
Coxeter diagram =
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, [37,1,1] = [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 truncated. It is part of a dimensionally infinite family of uniform polytopes called demihypercubes.

Coxeter named this polytope as 151 from its Coxeter-Dynkin diagram, with a ring on one of the 1-length branches.

## 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:

## Images

orthographic projections
Coxeter plane B8 D8 D7 D6 D5
Graph
Dihedral symmetry [16/2] [14] [12] [10] [8]
Coxeter plane D4 D3 A7 A5 A3
Graph
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-Strass, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 (Chapter 26. pp. 409: Hemicubes: 1n1)