# Runcic 5-cubes

(Redirected from Cantitruncated 5-demicube)
 Orthogonal projections in B5 Coxeter plane 5-cube Runcic 5-cube = 5-demicube = Runcicantic 5-cube =

In six-dimensional geometry, a runcic 5-cube or (runcic 5-demicube, runcihalf 5-cube) is a convex uniform 5-polytope. There are 2 runcic forms for the 5-cube. Runcic 5-cubes have half the vertices of runcinated 5-cubes.

## Runcic 5-cube

Runcic 5-cube
Type uniform 5-polytope
Schläfli symbol h3{4,3,3,3}
Coxeter-Dynkin diagram
4-faces 42
Cells 360
Faces 880
Edges 720
Vertices 160
Vertex figure
Coxeter groups D5, [32,1,1]
Properties convex

### Alternate names

• Cantellated 5-demicube/demipenteract
• Small rhombated hemipenteract (sirhin) (Jonathan Bowers)[1]

### Cartesian coordinates

The Cartesian coordinates for the 960 vertices of a runcic 5-cubes centered at the origin are coordinate permutations:

(±1,±1,±1,±3,±3)

with an odd number of plus signs.

### 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 runcinated 5-cube, as compared here in the B5 Coxeter plane projections:

 Runcic 5-cube Runcinated 5-cube

## Runcicantic 5-cube

Runcicantic 5-cube
Type uniform 5-polytope
Schläfli symbol t0,1,2{3,32,1}
h3{4,33}
Coxeter-Dynkin diagram
4-faces 42
Cells 360
Faces 1040
Edges 1200
Vertices 480
Vertex figure
Coxeter groups D5, [32,1,1]
Properties convex

### Alternate names

• Cantitruncated 5-demicube/demipenteract
• Great rhombated hemipenteract (girhin) (Jonathan Bowers)[2]

### Cartesian coordinates

The Cartesian coordinates for the 480 vertices of a runcicantic 5-cube centered at the origin are coordinate permutations:

(±1,±1,±3,±5,±5)

with an odd number of plus signs.

### 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 runcicantellated 5-cube, as compared here in the B5 Coxeter plane projections:

 Runcicantic 5-cube Runcicantellated 5-cube

## Related polytopes

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.

There are 23 uniform 5-polytopes 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.

## Notes

1. ^ Klitzing, (x3o3o *b3x3o - sirhin)
2. ^ Klitzing, (x3x3o *b3x3o - girhin)

## 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. "5D uniform polytopes (polytera)". x3o3o *b3x3o - sirhin, x3x3o *b3x3o - girhin