# Truncated 5-orthoplexes

(Redirected from Truncated 5-orthoplex)
 Orthogonal projections in B5 Coxeter plane 5-orthoplex Truncated 5-orthoplex Bitruncated 5-orthoplex 5-cube Truncated 5-cube Bitruncated 5-cube

In six-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
Cells 240
Faces 400
Edges 280
Vertices 80
Vertex figure
( )v{3,4}
Coxeter groups B5, [3,3,3,4]
D5, [32,1,1]
Properties convex

### Alternate names

• Truncated pentacross
• Truncated triacontiditeron (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
Cells 280
Faces 720
Edges 720
Vertices 240
Vertex figure
{ }v{4}
Coxeter groups B5, [3,3,3,4]
D5, [32,1,1]
Properties convex

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

### Alternate names

• Bitruncated pentacross
• Bitruncated triacontiditeron (acronym: gart) (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. 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]

## Related polytopes

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

## Notes

1. ^ Klitzing, (x3x3o3o4o - tot)
2. ^ Klitzing, (x3x3x3o4o - gart)

## 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, x3x3x3o4o - gart