Cantellated 6-demicube

6-cube	Cantellated 6-demicube	Cantitruncated 6-demicube
Orthogonal projections in D6Coxeter plane

In six-dimensional geometry, a cantellated 6-demicube is a convex uniform 6-polytope, being a cantellation of the uniform 6-demicube. There are 2 unique cantellation for the 6-demicube including a truncation.

Cantellated 6-demicube

Cantellated 6-demicube
Type	uniform polypeton
Schläfli symbol	t0,2{3,33,1}
Coxeter-Dynkin diagram
5-faces
4-faces
Cells
Faces
Edges	3840
Vertices	640
Vertex figure
Coxeter groups	D6, [33,1,1]
Properties	convex

Alternate names

• Small rhombated hemihexeract (Acronym sirhax) (Jonathan Bowers)

Cartesian coordinates

The Cartesian coordinates for the vertices of a cantellated demihexeract centered at the origin are coordinate permutations:

(±1,±1,±1,±1,±3,±3)

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]

Cantitruncated 6-demicube

Cantitruncated 6-demicube
Type	uniform polypeton
Schläfli symbol	t0,1,2{3,33,1}
Coxeter-Dynkin diagram
5-faces
4-faces
Cells
Faces
Edges	5760
Vertices	1920
Vertex figure
Coxeter groups	D6, [33,1,1]
Properties	convex

Alternate names

• Great rhombated hemihexeract (Acronym girhax) (Jonathan Bowers)

Cartesian coordinates

The Cartesian coordinates for the vertices of a cantitruncated demihexeract centered at the origin are coordinate permutations:

(±1,±1,±1,±3,±5,±5)

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

This polytope is based on the 6-demicube, a part of a dimensional family of uniform polytopes called demihypercubes for being alternation of the hypercube family.

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:
  • H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973
  • 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]
• Norman Johnson Uniform Polytopes, Manuscript (1991)
  • N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D.
• Richard Klitzing, 6D, uniform polytopes (polypeta) x3o3o *b3x3o3o, x3x3o *b3x3o3o

External links

• Weisstein, Eric W., "Hypercube", MathWorld.
• Olshevsky, George, Measure polytope at Glossary for Hyperspace.
• Polytopes of Various Dimensions
• Multi-dimensional Glossary