Steric 6-cubes

6-demicube =	Steric 6-cube =	Stericantic 6-cube =
	Steriruncic 6-cube =	Stericruncicantic 6-cube =
Orthogonal projections in D6 Coxeter plane

In six-dimensional geometry, a steric 6-cube is a convex uniform 6-polytope. There are unique 4 steric forms of the 6-cube.

Steric 6-cube

Steric 6-cube
Type	uniform 6-polytope
Schläfli symbol	t0,3{3,33,1} h4{4,34}
Coxeter-Dynkin diagram	=
5-faces
4-faces
Cells
Faces
Edges	3360
Vertices	480
Vertex figure
Coxeter groups	D6, [33,1,1]
Properties	convex

Alternate names

• Runcinated demihexeract/6-demicube
• Small prismated hemihexeract (Acronym sophax) (Jonathan Bowers)

Cartesian coordinates

The Cartesian coordinates for the 480 vertices of a steric 6-cube centered at the origin are coordinate permutations:

(±1,±1,±1,±1,±1,±3)

with an odd number of plus signs.

Stericantic 6-cube

Stericantic 6-cube
Type	uniform 6-polytope
Schläfli symbol	t0,1,3{3,33,1} h2,4{4,34}
Coxeter-Dynkin diagram	=
5-faces
4-faces
Cells
Faces
Edges	12960
Vertices	2880
Vertex figure
Coxeter groups	D6, [33,1,1]
Properties	convex

Alternate names

• Runcitruncated demihexeract/6-demicube
• Prismatotruncated hemihexeract (Acronym pithax) (Jonathan Bowers)

Cartesian coordinates

The Cartesian coordinates for the 2880 vertices of a stericantic 6-cube centered at the origin are coordinate permutations:

(±1,±1,±1,±3,±3,±5)

with an odd number of plus signs.

Steriruncic 6-cube

Steriruncic 6-cube
Type	uniform 6-polytope
Schläfli symbol	t0,2,3{3,33,1} h3,4{4,34}
Coxeter-Dynkin diagram	=
5-faces
4-faces
Cells
Faces
Edges	7680
Vertices	1920
Vertex figure
Coxeter groups	D6, [33,1,1]
Properties	convex

Alternate names

• Runcicantellated demihexeract/6-demicube
• Prismatorhombated hemihexeract (Acronym prohax) (Jonathan Bowers)

Cartesian coordinates

The Cartesian coordinates for the 1920 vertices of a steriruncic 6-cube centered at the origin are coordinate permutations:

(±1,±1,±1,±1,±3,±5)

with an odd number of plus signs.

Steriruncicantic 6-cube

Steriruncicantic 6-cube
Type	uniform 6-polytope
Schläfli symbol	t0,1,2,3{3,32,1} h2,3,4{4,34}
Coxeter-Dynkin diagram	=
5-faces
4-faces
Cells
Faces
Edges	17280
Vertices	5760
Vertex figure
Coxeter groups	D6, [33,1,1]
Properties	convex

Alternate names

• Runcicantitruncated demihexeract/6-demicube
• Great prismated hemihexeract (Acronym gophax) (Jonathan Bowers)

Cartesian coordinates

The Cartesian coordinates for the 5760 vertices of a steriruncicantic 6-cube centered at the origin are coordinate permutations:

(±1,±1,±1,±3,±5,±7)

with an odd number of plus signs.

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:
  • H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973
  • 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]
• Norman Johnson Uniform Polytopes, Manuscript (1991)
  • N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D.
• Klitzing, Richard. "6D uniform polytopes (polypeta)". https://bendwavy.org/klitzing/dimensions/polypeta.htm.  x3o3o *b3o3x3o - sophax, x3x3o *b3o3x3o - pithax, x3o3o *b3x3x3o - prohax, x3x3o *b3x3x3o - gophax

External links

• Weisstein, Eric W.. "Hypercube". http://mathworld.wolfram.com/Hypercube.html. 
• Polytopes of Various Dimensions
• Multi-dimensional Glossary