Cantic 5-cube

In geometry of five dimensions or higher, a cantic 5-cube, cantihalf 5-cube, truncated 5-demicube is a uniform 5-polytope, being a truncation of the 5-demicube. It has half the vertices of a cantellated 5-cube.

Truncated 5-demicube Cantic 5-cube
D5 Coxeter plane projection
Type	uniform 5-polytope
Schläfli symbol	h2{4,3,3,3} t{3,32,1}
Coxeter-Dynkin diagram	=
4-faces	42 total: 16 r{3,3,3} 16 t{3,3,3} 10 t{3,3,4}
Cells	280 total: 80 {3,3} 120 t{3,3} 80 {3,4}
Faces	640 total: 480 {3} 160 {6}
Edges	560
Vertices	160
Vertex figure	( )v{ }×{3}
Coxeter groups	D5, [32,1,1]
Properties	convex

Cartesian coordinates

The Cartesian coordinates for the 160 vertices of a cantic 5-cube centered at the origin and edge length 6√2 are coordinate permutations:

(±1,±1,±3,±3,±3)

with an odd number of plus signs.

Alternate names

• Cantic penteract, truncated demipenteract
• Truncated hemipenteract (thin) (Jonathan Bowers)^[1]

Images

orthographic projections

Coxeter plane	B5
Graph
Dihedral symmetry	[10/2]
Coxeter plane	D5	D4
Graph
Dihedral symmetry	[8]	[6]
Coxeter plane	D3	A3
Graph
Dihedral symmetry	[4]	[4]

Related polytopes

It has half the vertices of the cantellated 5-cube, as compared here in the B5 Coxeter plane projections:

Cantic 5-cube

Cantellated 5-cube

This polytope is based on the 5-demicube, a part of a dimensional family of uniform polytopes called demihypercubes for being alternation of the hypercube family.

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 23 uniform 5-polytope that can be constructed from the D₅ symmetry of the 5-demicube, of which are unique to this family, and 15 are shared within the 5-cube family.

D5 polytopes
h{4,3,3,3}	h2{4,3,3,3}	h3{4,3,3,3}	h4{4,3,3,3}	h2,3{4,3,3,3}	h2,4{4,3,3,3}	h3,4{4,3,3,3}	h2,3,4{4,3,3,3}