Runcic 6-cubes

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

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

Runcic 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

Alternate names

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

Cartesian coordinates

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.

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

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

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

Alternate names

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

Cartesian coordinates

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.

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:

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}