# Runcic 7-cubes

(Redirected from Cantellated 7-demicubes)
 Orthogonal projections in D7 Coxeter plane 7-demicube Runcic 7-cube Runcicantic 7-cube

In seven-dimensional geometry, a runcic 7-cube is a convex uniform 7-polytope, related to the uniform 7-demicube. There are 2 unique forms.

## Runcic 7-cube

Runcic 7-cube
Type uniform 7-polytope
Schläfli symbol t0,2{3,34,1}
h3{4,35}
Coxeter-Dynkin diagram
5-faces
4-faces
Cells
Faces
Edges 16800
Vertices 2240
Vertex figure
Coxeter groups D7, [34,1,1]
Properties convex

A runcic 7-cube, h3{4,35}, has half the vertices of a runcinated 7-cube, t0,3{4,35}.

### Alternate names

• Small rhombated hemihepteract (Acronym sirhesa) (Jonathan Bowers)[1]

### Cartesian coordinates

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

(±1,±1,±1,±3,±3,±3,±3)

with an odd number of plus signs.

### Images

orthographic projections
Coxeter
plane
B7 D7 D6
Graph
Dihedral
symmetry
[14/2] [12] [10]
Coxeter plane D5 D4 D3
Graph
Dihedral
symmetry
[8] [6] [4]
Coxeter
plane
A5 A3
Graph
Dihedral
symmetry
[6] [4]

## Runcicantic 7-cube

Runcicantic 7-cube
Type uniform 7-polytope
Schläfli symbol t0,1,2{3,34,1}
h2,3{4,35}
Coxeter-Dynkin diagram
5-faces
4-faces
Cells
Faces
Edges 23520
Vertices 6720
Vertex figure
Coxeter groups D6, [33,1,1]
Properties convex

A runcicantic 7-cube, h2,3{4,35}, has half the vertices of a runcicantellated 7-cube, t0,1,3{4,35}.

### Alternate names

• Great rhombated hemihepteract (Acronym girhesa) (Jonathan Bowers)[2]

### Cartesian coordinates

The Cartesian coordinates for the vertices of a runcicantic 7-cube centered at the origin are coordinate permutations:

(±1,±1,±1,±1,±3,±5,±5)

with an odd number of plus signs.

### Images

orthographic projections
Coxeter
plane
B7 D7 D6
Graph
Dihedral
symmetry
[14/2] [12] [10]
Coxeter plane D5 D4 D3
Graph
Dihedral
symmetry
[8] [6] [4]
Coxeter
plane
A5 A3
Graph
Dihedral
symmetry
[6] [4]

## Related polytopes

This polytope is based on the 7-demicube, a part of a dimensional family of uniform polytopes called demihypercubes for being alternation of the hypercube family.

There are 95 uniform polytopes with D7 symmetry, 63 are shared by the BC6 symmetry, and 32 are unique:

## Notes

1. ^ Klitzing, (x3o3o *b3x3o3o3o - sirhesa)
2. ^ Klitzing, (x3x3o *b3x3o3o3o - girhesa)

## References

• H.S.M. Coxeter:
• H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973
• Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6 [1]
• (Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380-407, MR 2,10]
• (Paper 23) H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) 559-591]
• (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
• Norman Johnson Uniform Polytopes, Manuscript (1991)
• N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D.
• Klitzing, Richard. "7D uniform polytopes (polyexa)". x3o3o *b3x3o3o3o - sirhesa, x3x3o *b3x3o3o3o - girhesa