# 10-demicube

Demidekeract
(10-demicube)

Petrie polygon projection
Type Uniform 10-polytope
Family demihypercube
Coxeter symbol 171
Schläfli symbol {31,7,1}
h{4,38}
s{21,1,1,1,1,1,1,1,1}
Coxeter diagram =
9-faces 532 20 {31,6,1}
512 {38}
8-faces 5300 180 {31,5,1}
5120 {37}
7-faces 24000 960 {31,4,1}
23040 {36}
6-faces 64800 3360 {31,3,1}
61440 {35}
5-faces 115584 8064 {31,2,1}
107520 {34}
4-faces 142464 13440 {31,1,1}
129024 {33}
Cells 122880 15360 {31,0,1}
107520 {3,3}
Faces 61440 {3}
Edges 11520
Vertices 512
Vertex figure Rectified 9-simplex
Symmetry group D10, [37,1,1] = [1+,4,38]
[29]+
Dual ?
Properties convex

In geometry, a demidekeract or 10-demicube is a uniform 10-polytope, constructed from the 10-cube with alternated vertices truncated. It is part of a dimensionally infinite family of uniform polytopes called demihypercubes.

Coxeter named this polytope as 171 from its Coxeter diagram, with a ring on one of the 1-length branches, .

## Cartesian coordinates

Cartesian coordinates for the vertices of a demidekeract centered at the origin are alternate halves of the dekeract:

(±1,±1,±1,±1,±1,±1,±1,±1,±1,±1)

with an odd number of plus signs.

## Images

 B10 coxeter plane D10 coxeter plane (Vertices are colored by multiplicity: red, orange, yellow, green = 1,2,4,8)

## 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)
• Norman Johnson Uniform Polytopes, Manuscript (1991)
• N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. (1966)
• Richard Klitzing, 10D uniform polytopes (polyxenna), x3o3o *b3o3o3o3o3o3o3o - hede