Steric 6-cubes

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

6-demicube =	Steric 6-cube =	Stericantic 6-cube =
	Steriruncic 6-cube =	Steriruncicantic 6-cube =
Orthogonal projections in D5 Coxeter plane

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
• Runcinated 6-demicube
• Small prismated hemihexeract (Acronym: sophax) (Jonathan Bowers)^[1]

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.

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

Dimensional family of steric n-cubes
n	5	6	7	8
[1+,4,3n-2] = [3,3n-3,1]	[1+,4,33] = [3,32,1]	[1+,4,34] = [3,33,1]	[1+,4,35] = [3,34,1]	[1+,4,36] = [3,35,1]
Steric figure
Coxeter	=	=	=	=
Schläfli	h4{4,33}	h4{4,34}	h4{4,35}	h4{4,36}

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
• Runcitruncated 6-demicube
• Prismatotruncated hemihexeract (Acronym: pithax) (Jonathan Bowers)^[2]

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.

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]

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
• Runcicantellated 6-demicube
• Prismatorhombated hemihexeract (Acronym: prohax) (Jonathan Bowers)^[3]

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.

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]

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
• Runcicantitruncated 6-demicube
• Great prismated hemihexeract (Acronym: gophax) (Jonathan Bowers)^[4]

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.

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:

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}