# 6-demicube

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6-demicube
Demihexeract
(6-demicube)

Petrie polygon projection
Type Uniform 6-polytope
Family demihypercube
Schläfli symbol {3,33,1}
h{4,3,3,3,3}
s{2,2,2,2,2}
Coxeter-Dynkin diagram

Coxeter symbol 131
5-faces 44 12 {31,2,1}
32 {34}
4-faces 252 60 {31,1,1}
192 {33}
Cells 640 160 {31,0,1}
480 {3,3}
Faces 640 {3}
Edges 240
Vertices 32
Vertex figure Rectified 5-simplex
Symmetry group D6, [35,1,1] = [1+,4,34]
[25]+
Petrie polygon decagon
Properties convex

In geometry, a 6-demicube or demihexteract is a uniform 6-polytope, constructed from a 6-cube (hexeract) with alternate vertices deleted. It is part of a dimensionally infinite family of uniform polytopes called demihypercubes.

Coxeter named this polytope as 131 from its Coxeter-Dynkin diagram, with a ring on one of the 1-length Coxeter-Dynkin diagram branches. It can named similarly by a 3-dimensional exponential Schläfli symbol, {3,33,1}.

## Cartesian coordinates

Cartesian coordinates for the vertices of a demihexeract centered at the origin are alternate halves of the hexeract:

(±1,±1,±1,±1,±1,±1)

with an odd number of plus signs.

## Images

orthographic projections
Coxeter plane B6
Graph
Dihedral symmetry [12/2]
Coxeter plane D6 D5
Graph
Dihedral symmetry [10] [8]
Coxeter plane D4 D3
Graph
Dihedral symmetry [6] [4]
Coxeter plane A5 A3
Graph
Dihedral symmetry [6] [4]

## Related polytopes

There are 47 uniform polytopes with D6 symmetry, 31 are shared by the B6 symmetry, and 16 are unique:

## 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, editied 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)
• Richard Klitzing, 6D uniform polytopes (polypeta), x3o3o *b3o3o3o – hax

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