# 6-demicube

Demihexeract
(6-demicube)

Petrie polygon projection
Type Uniform 6-polytope
Family demihypercube
Schläfli symbol {3,33,1} = h{4,34}
s{25}
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 alternated vertices truncated. 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:

 h{4,34} h2{4,34} h3{4,34} h4{4,34} h5{4,34} h2,3{4,34} h2,4{4,34} h2,5{4,34} h3,4{4,34} h3,5{4,34} h4,5{4,34} h2,3,4{4,34} h2,3,5{4,34} h2,4,5{4,34} h3,4,5{4,34} h2,3,4,5{4,34}

The 6-demicube, 131 is third in a dimensional series of uniform polytopes, expressed by Coxeter as k31 series. The fifth figure is a Euclidean honeycomb, 331, and the final is a noncompact hyperbolic honeycomb, 431. Each progressive uniform polytope is constructed from the previous as its vertex figure.

k31 dimensional figures
n 4 5 6 7 8 9
Coxeter
group
A3×A1 A5 D6 E7 ${\tilde{E}}_{7}$ = E7+ E7++
Coxeter
diagram
Symmetry
(order)
[3-1,3,1]
(48)
[30,3,1]
(720)
[31,3,1]
(23,040)
[32,3,1]
(2,903,040)
[33,3,1]
(∞)
[34,3,1]
(∞)
Graph
Name −131 031 131 231 331 431

It is also the second in a dimensional series of uniform polytopes and honeycombs, expressed by Coxeter as 13k series. The next figure is the Euclidean honeycomb 133 and the final is a noncompact hyperbolic honeycomb, 134.

13k dimensional figures
n 4 5 6 7 8 9
Coxeter
group
A3×A1 A5 D6 E7 ${\tilde{E}}_{7}$=E7+ E7++
Coxeter
diagram
Symmetry
(order)
[3-1,3,1]
(48)
[30,3,1]
(720)
[31,3,1]
(23,040)
[32,3,1]
(2,903,040)
[[33,3,1]]
(∞)
[34,3,1]
(∞)
Graph
Name 13,-1 130 131 132 133 134

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