# Cantellated 5-orthoplex

 Orthogonal projections in BC5Coxeter plane 5-orthoplex Cantellated 5-orthoplex Bicantellated 5-cube Cantellated 5-cube 5-cube Cantitruncated 5-orthoplex Bicantitruncated 5-cube Cantitruncated 5-cube

In five-dimensional geometry, a cantellated 5-orthoplex is a convex uniform 5-polytope, being a cantellation of the regular 5-orthoplex.

There are 6 cantellation for the 5-orthoplex, including truncations. Some of them are more easily constructed from the dual 5-cube.

## Cantellated 5-orthoplex

 Cantellated 5-orthoplex Type Uniform 5-polytope Schläfli symbol t0,2{3,3,3,4} t0,2{3,3,31,1} Coxeter-Dynkin diagram 4-faces 122 Cells 680 Faces 1520 Edges 1280 Vertices 320 Vertex figure Coxeter group BC5 [4,3,3,3] D5 [32,1,1] Properties convex

### Alternate names

• Cantellated 5-orthoplex
• Bicantellated 5-demicube
• Small rhombated triacontiditeron (Acronym: sart) (Jonathan Bowers)[1]

### Coordinates

The vertices of the can be made in 5-space, as permutations and sign combinations of:

(0,0,1,1,2)

### Images

The cantellated 5-orthoplex is constructed by a cantellation operation applied to the 5-orthoplex.

orthographic projections
Coxeter plane B5 B4 / D5 B3 / D4 / A2
Graph
Dihedral symmetry [10] [8] [6]
Coxeter plane B2 A3
Graph
Dihedral symmetry [4] [4]
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## Cantitruncated 5-orthoplex

Cantitruncated 5-orthoplex
Type uniform polyteron
Schläfli symbol t0,1,2{3,3,3,4}
t0,1,2{3,31,1}
Coxeter-Dynkin diagrams
4-faces 122
Cells 680
Faces 1520
Edges 1600
Vertices 640
Vertex figure
Coxeter groups BC5, [3,3,3,4]
D5, [32,1,1]
Properties convex

### Alternate names

• Cantitruncated pentacross
• Cantitruncated triacontiditeron (Acronym: gart) (Jonathan Bowers)[2]

### Coordinates

Cartesian coordinates for the vertices of a cantitruncated 5-orthoplex, centered at the origin, are all sign and coordinate permutations of

(±3,±2,±1,0,0)

### Images

orthographic projections
Coxeter plane B5 B4 / D5 B3 / D4 / A2
Graph
Dihedral symmetry [10] [8] [6]
Coxeter plane B2 A3
Graph
Dihedral symmetry [4] [4]
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## Related polytopes

These polytopes are from a set of 31 uniform polytera generated from the regular 5-cube or 5-orthoplex.

 β5 t1β5 t2γ5 t1γ5 γ5 t0,1β5 t0,2β5 t1,2β5 t0,3β5 t1,3γ5 t1,2γ5 t0,4γ5 t0,3γ5 t0,2γ5 t0,1γ5 t0,1,2β5 t0,1,3β5 t0,2,3β5 t1,2,3γ5 t0,1,4β5 t0,2,4γ5 t0,2,3γ5 t0,1,4γ5 t0,1,3γ5 t0,1,2γ5 t0,1,2,3β5 t0,1,2,4β5 t0,1,3,4γ5 t0,1,2,4γ5 t0,1,2,3γ5 t0,1,2,3,4γ5
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## Notes

1. ^ Klitizing, (x3o3x3o4o - sart)
2. ^ Klitizing, (x3x3x3o4o - gart)
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## 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, editied 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.
• Richard Klitzing, 5D, uniform polytopes (polytera) x3o3x3o4o - sart, x3x3x3o4o - gart
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