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 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, {3,3^3,1}.

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:

t0(131)	t0,1(131)	t0,2(131)	t0,3(131)	t0,4(131)	t0,1,2(131)	t0,1,3(131)	t0,1,4(131)
t0,2,3(131)	t0,2,4(131)	t0,3,4(131)	t0,1,2,3(131)	t0,1,2,4(131)	t0,1,3,4(131)	t0,2,3,4(131)	t0,1,2,3,4(131)

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: 1_n1)
• Richard Klitzing, 6D uniform polytopes (polypeta), x3o3o *b3o3o3o – hax

External links

• Olshevsky, George, Demihexeract at Glossary for Hyperspace.
• Multi-dimensional Glossary