Truncated 5-simplexes
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In five-dimensional geometry, a truncated 5-simplex is a convex uniform 5-polytope, being a truncation of the regular 5-simplex.
 5-simplex    |
 Truncated 5-simplex |
 Bitruncated 5-simplex |
| Orthogonal projections in A5 Coxeter plane | ||
|---|---|---|
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 |  ( )v{3,3} | |
| 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
| 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 |  { }v{3} | |
| 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
| 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)
