6-cube	Cantellated 6-demicube	Cantitruncated 6-demicube
Orthogonal projections in D6 Coxeter plane

In six-dimensional geometry

Geometry

Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....

, a cantellated 6-demicube is a convex uniform 6-polytope, being a cantellation of the uniform 6-demicube. There are 2 unique cantellation for the 6-demicube including a truncation.

Cantellated 6-demicube

Cantellated 6-demicube
Type	uniform polypeton
Schläfli symbol	t0,2{3,33,1}
Coxeter-Dynkin diagram Coxeter-Dynkin diagram In geometry, a Coxeter–Dynkin diagram is a graph with numerically labeled edges representing the spatial relations between a collection of mirrors...
5-faces
4-faces
Cells
Faces
Edges	3840
Vertices	640
Vertex figure Vertex figure In geometry a vertex figure is, broadly speaking, the figure exposed when a corner of a polyhedron or polytope is sliced off.-Definitions - theme and variations:...
Coxeter group Coxeter group In mathematics, a Coxeter group, named after H.S.M. Coxeter, is an abstract group that admits a formal description in terms of mirror symmetries. Indeed, the finite Coxeter groups are precisely the finite Euclidean reflection groups; the symmetry groups of regular polyhedra are an example... s	D6, [33,1,1]
Properties	convex Convex polytope A convex polytope is a special case of a polytope, having the additional property that it is also a convex set of points in the n-dimensional space Rn...

Cartesian coordinates

The Cartesian coordinates for the vertices of a cantellated demihexeract centered at the origin are coordinate permutations:

(±1,±1,±1,±1,±3,±3)

with an odd number of plus signs.

Cantitruncated 6-demicube

Cantitruncated 6-demicube
Type	uniform polypeton
Schläfli symbol	t0,1,2{3,33,1}
Coxeter-Dynkin diagram Coxeter-Dynkin diagram In geometry, a Coxeter–Dynkin diagram is a graph with numerically labeled edges representing the spatial relations between a collection of mirrors...
5-faces
4-faces
Cells
Faces
Edges	5760
Vertices	1920
Vertex figure Vertex figure In geometry a vertex figure is, broadly speaking, the figure exposed when a corner of a polyhedron or polytope is sliced off.-Definitions - theme and variations:...
Coxeter group Coxeter group In mathematics, a Coxeter group, named after H.S.M. Coxeter, is an abstract group that admits a formal description in terms of mirror symmetries. Indeed, the finite Coxeter groups are precisely the finite Euclidean reflection groups; the symmetry groups of regular polyhedra are an example... s	D6, [33,1,1]
Properties	convex Convex polytope A convex polytope is a special case of a polytope, having the additional property that it is also a convex set of points in the n-dimensional space Rn...

Cartesian coordinates

The Cartesian coordinates for the vertices of a cantitruncated demihexeract centered at the origin are coordinate permutations:

(±1,±1,±1,±3,±5,±5)

with an odd number of plus signs.

Related polytopes

This polytope is based on the 6-demicube, a part of a dimensional family of uniform polytope

Uniform polytope

A uniform polytope is a vertex-transitive polytope made from uniform polytope facets of a lower dimension. Uniform polytopes of 2 dimensions are the regular polygons....

s called demihypercubes for being alternation of the hypercube

Hypercube

In geometry, a hypercube is an n-dimensional analogue of a square and a cube . It is a closed, compact, convex figure whose 1-skeleton consists of groups of opposite parallel line segments aligned in each of the space's dimensions, perpendicular to each other and of the same length.An...

family.

There are 47 uniform polytopes with D₆ symmetry, 31 are shared by the B₆ symmetry, and 16 are unique:

External links

• Polytopes of Various Dimensions
• Multi-dimensional Glossary