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Truncated 5-orthoplexes

## Summary

 Orthogonal projections in B5 Coxeter plane 5-orthoplex Truncated 5-orthoplex Bitruncated 5-orthoplex 5-cube Truncated 5-cube Bitruncated 5-cube

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

There are 4 unique truncations of the 5-orthoplex. Vertices of the truncation 5-orthoplex are located as pairs on the edge of the 5-orthoplex. Vertices of the bitruncated 5-orthoplex are located on the triangular faces of the 5-orthoplex. The third and fourth truncations are more easily constructed as second and first truncations of the 5-cube.

## Truncated 5-orthoplex

Truncated 5-orthoplex
Type uniform 5-polytope
Schläfli symbol t{3,3,3,4}
t{3,31,1}
Coxeter-Dynkin diagrams

4-faces 42 10
32
Cells 240 160
80
Faces 400 320
80
Edges 280 240
40
Vertices 80
Vertex figure
( )v{3,4}
Coxeter groups B5, [3,3,3,4], order 3840
D5, [32,1,1], order 1920
Properties convex

### Alternate names

• Truncated pentacross
• Truncated triacontaditeron (Acronym: tot) (Jonathan Bowers)[1]

### Coordinates

Cartesian coordinates for the vertices of a truncated 5-orthoplex, centered at the origin, are all 80 vertices are sign (4) and coordinate (20) permutations of

(±2,±1,0,0,0)

### Images

The truncated 5-orthoplex is constructed by a truncation operation applied to the 5-orthoplex. All edges are shortened, and two new vertices are added on each original edge.

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]

## Bitruncated 5-orthoplex

Bitruncated 5-orthoplex
Type uniform 5-polytope
Schläfli symbol 2t{3,3,3,4}
2t{3,31,1}
Coxeter-Dynkin diagrams

4-faces 42 10
32
Cells 280 40
160
80
Faces 720 320
320
80
Edges 720 480
240
Vertices 240
Vertex figure
{ }v{4}
Coxeter groups B5, [3,3,3,4], order 3840
D5, [32,1,1], order 1920
Properties convex

The bitruncated 5-orthoplex can tessellate space in the tritruncated 5-cubic honeycomb.

### Alternate names

• Bitruncated pentacross
• Bitruncated triacontiditeron (acronym: bittit) (Jonathan Bowers)[2]

### Coordinates

Cartesian coordinates for the vertices of a truncated 5-orthoplex, centered at the origin, are all 80 vertices are sign and coordinate permutations of

(±2,±2,±1,0,0)

### Images

The bitrunacted 5-orthoplex is constructed by a bitruncation 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]

This polytope is one of 31 uniform 5-polytopes generated from the regular 5-cube or 5-orthoplex.

B5 polytopes

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

## Notes

1. ^ Klitzing, (x3x3o3o4o - tot)
2. ^ Klitzing, (o3x3x3o4o - bittit)

## 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)". x3x3o3o4o - tot, o3x3x3o4o - bittit