Cantellated 7-demicube
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Cantellated 7-demicube

Cantitruncated 7-demicube
Orthogonal projections in D7 Coxeter plane

In seven-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 7-demicube is a convex uniform 7-polytope, being a cantellation of the uniform 7-demicube. There are 2 unique cantellation for the 7-demicube including a truncation.

Cantellated 7-demicube

Cantellated 7-demicube
Type uniform polyexon
Schläfli symbol t0,2{3,34,1}
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 16800
Vertices 2240
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
D7, [34,1,1]
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 demihepteract centered at the origin are coordinate permutations:
(±1,±1,±1,±1,±1,±3,±3)

with an odd number of plus signs.

Cantitruncated 7-demicube

Cantitruncated 7-demicube
Type uniform polyexon
Schläfli symbol t0,1,2{3,34,1}
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 23520
Vertices 6720
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 demihepteract centered at the origin are coordinate permutations:
(±1,±1,±1,±1,±3,±5,±5)

with an odd number of plus signs.

Related polytopes

This polytope is based on the 7-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 95 uniform polytopes with D7 symmetry, 63 are shared by the BC6 symmetry, and 32 are unique:

External links

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