Contents

Runcic 6-cubes

6-demicube =	Runcic 6-cube =	Runcicantic 6-cube =
Orthogonal projections in D6 Coxeter plane

In six-dimensional geometry, a runcic 6-cube is a convex uniform 6-polytope. There are 2 unique runcic for the 6-cube.

Runcic 6-cube
Type	uniform 6-polytope
Schläfli symbol	t0,2{3,33,1} h3{4,34}
Coxeter-Dynkin diagram	=
5-faces
4-faces
Cells
Faces
Edges	3840
Vertices	640
Vertex figure
Coxeter groups	D6, [33,1,1]
Properties	convex

• Cantellated 6-demicube/demihexeract
• Small rhombated hemihexeract (Acronym sirhax) (Jonathan Bowers)^[1]

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

(±1,±1,±1,±3,±3,±3)

with an odd number of plus signs.

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]

Runcic n-cubes
n	4	5	6	7	8
[1+,4,3n-2] = [3,3n-3,1]	[1+,4,32] = [3,31,1]	[1+,4,33] = [3,32,1]	[1+,4,34] = [3,33,1]	[1+,4,35] = [3,34,1]	[1+,4,36] = [3,35,1]
Runcic figure
Coxeter	=	=	=	=	=
Schläfli	h3{4,32}	h3{4,33}	h3{4,34}	h3{4,35}	h3{4,36}

Runcicantic 6-cube
Type	uniform 6-polytope
Schläfli symbol	t0,1,2{3,33,1} h2,3{4,34}
Coxeter-Dynkin diagram	=
5-faces
4-faces
Cells
Faces
Edges	5760
Vertices	1920
Vertex figure
Coxeter groups	D6, [33,1,1]
Properties	convex

• Cantitruncated 6-demicube/demihexeract
• Great rhombated hemihexeract (Acronym girhax) (Jonathan Bowers)^[2]

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

(±1,±1,±3,±5,±5,±5)

with an odd number of plus signs.

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]

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:

D6 polytopes
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}

• 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)". x3o3o *b3x3o3o, x3x3o *b3x3o3o

• Weisstein, Eric W. "Hypercube". MathWorld.
• Polytopes of Various Dimensions
• Multi-dimensional Glossary