Truncated 5-simplexes
5-simplex |
Truncated 5-simplex |
Bitruncated 5-simplex |
| Orthogonal projections in A5 Coxeter plane | ||
|---|---|---|
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.
01Truncated 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)
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] |
02Bitruncated 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)
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] |
Sources and credits
This article is adapted from the Wikipedia article “Truncated 5-simplexes”, written by its contributors and licensed under CC BY-SA 4.0. Fathomly has changed the layout, removed citation markers, navigation and maintenance notices, and adjusted punctuation. This adapted version is shared under the same license. For references, see the original article.
Images, from Wikimedia Commons:
- 5-simplex t0.svg by Tomruen, Public domain
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