# Truncated 5-simplexes

(Redirected from Truncated 5-simplex)
 Orthogonal projections in A5 Coxeter plane 5-simplex Truncated 5-simplex Bitruncated 5-simplex

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

There are unique 2 degrees of truncation. Vertices of the truncation 5-simplex are located as pairs on the edge of the 5-simplex. Vertices of the bitruncation 5-simplex are located on the triangular faces of the 5-simplex.

## Truncated 5-simplex

 Truncated 5-simplex Type Uniform 5-polytope Schläfli symbol t{3,3,3,3} Coxeter-Dynkin diagram 4-faces 12 6 {3,3,3} 6 t{3,3,3} Cells 45 30 {3,3} 15 t{3,3} Faces 80 60 {3} 20 {6} Edges 75 Vertices 30 Vertex figure Tetra.pyr Coxeter group A5 [3,3,3,3], order 720 Properties convex

The truncated 5-simplex has 30 vertices, 75 edges, 80 triangular faces, 45 cells (15 tetrahedral, and 30 truncated tetrahedron), and 12 4-faces (6 5-cell and 6 truncated 5-cells).

### Alternate names

• Truncated hexateron (Acronym: tix) (Jonathan Bowers)[1]

### Coordinates

The vertices of the truncated 5-simplex can be most simply constructed on a hyperplane in 6-space as permutations of (0,0,0,0,1,2) or of (0,1,2,2,2,2). These coordinates come from facets of the truncated 6-orthoplex and bitruncated 6-cube respectively.

### Images

orthographic projections
Ak
Coxeter plane
A5 A4
Graph
Dihedral symmetry [6] [5]
Ak
Coxeter plane
A3 A2
Graph
Dihedral symmetry [4] [3]

## Bitruncated 5-simplex

 bitruncated 5-simplex Type Uniform 5-polytope Schläfli symbol 2t{3,3,3,3} Coxeter-Dynkin diagram 4-faces 12 6 2t{3,3,3} 6 t{3,3,3} Cells 60 45 {3,3} 15 t{3,3} Faces 140 80 {3} 60 {6} Edges 150 Vertices 60 Vertex figure Triangular-pyramidal pyramid Coxeter group A5 [3,3,3,3], order 720 Properties convex

### Alternate names

• Bitruncated hexateron (Acronym: bittix) (Jonathan Bowers)[2]

### Coordinates

The vertices of the bitruncated 5-simplex can be most simply constructed on a hyperplane in 6-space as permutations of (0,0,0,1,2,2) or of (0,0,1,2,2,2). These represent positive orthant facets of the bitruncated 6-orthoplex, and the tritruncated 6-cube respectively.

### Images

orthographic projections
Ak
Coxeter plane
A5 A4
Graph
Dihedral symmetry [6] [5]
Ak
Coxeter plane
A3 A2
Graph
Dihedral symmetry [4] [3]

## Related uniform 5-polytopes

The truncated 5-simplex is one of 19 uniform 5-polytopes based on the [3,3,3,3] Coxeter group, all shown here in A5 Coxeter plane orthographic projections. (Vertices are colored by projection overlap order, red, orange, yellow, green, cyan, blue, purple having progressively more vertices)

## Notes

1. ^ Klitizing, (x3x3o3o3o - tix)
2. ^ Klitizing, (o3x3x3o3o - bittix)

## 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)". x3x3o3o3o - tix, o3x3x3o3o - bittix