Cantic 6-cube

In six-dimensional geometry, a cantic 6-cube (or a truncated 6-demicube) is a uniform 6-polytope.

Cantic 6-cube Truncated 6-demicube
D6 Coxeter plane projection
Type	uniform polypeton
Schläfli symbol	t0,1{3,33,1} h2{4,34}
Coxeter-Dynkin diagram	=
5-faces	76
4-faces	636
Cells	2080
Faces	3200
Edges	2160
Vertices	480
Vertex figure	( )v[{ }x{3,3}]
Coxeter groups	D6, [33,1,1]
Properties	convex

Alternate names

• Truncated 6-demicube
• Truncaced demihexeract
• Truncated hemihexeract (Acronym: thax) (Jonathan Bowers)^[1]

Cartesian coordinates

The Cartesian coordinates for the 480 vertices of a cantic 6-cube centered at the origin and edge length 6√2 are coordinate permutations:

(±1,±1,±3,±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

Dimensional family of cantic n-cubes

n	3	4	5	6	7	8
Symmetry [1+,4,3n-2]	[1+,4,3] = [3,3]	[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]
Cantic figure
Coxeter	=	=	=	=	=	=
Schläfli	h2{4,3}	h2{4,32}	h2{4,33}	h2{4,34}	h2{4,35}	h2{4,36}

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}