Truncated 5-cube
5-cube |
Truncated 5-cube |
Bitruncated 5-cube |
|
5-orthoplex |
Truncated 5-orthoplex |
Bitruncated 5-orthoplex |
|
| Orthogonal projections in BC5 Coxeter plane | |||
|---|---|---|---|
In five-dimensional geometry, a truncated 5-cube is a convex uniform 5-polytope, being a truncation of the regular 5-cube.
There are four unique truncations of the 5-cube. Vertices of the truncated 5-cube are located as pairs on the edge of the 5-cube. Vertices of the bitruncated 5-cube are located on the square faces of the 5-cube. The third and fourth truncations are more easily constructed as second and first truncations of the 5-orthoplex.
|
Truncated 5-cube
| Truncated 5-cube | |
|---|---|
| Type | uniform polyteron |
| Schläfli symbol | t0,1{4,3,3,3} |
| Coxeter-Dynkin diagrams | |
| 4-faces | 42 |
| Cells | 200 |
| Faces | 400 |
| Edges | 400 |
| Vertices | 160 |
| Vertex figure | Elongated tetrahedral pyramid |
| Coxeter groups | BC5, [3,3,3,4] |
| Properties | convex |
Alternate names
- Truncated penteract (Acronym: tan) (Jonathan Bowers)
Construction and coordinates
The truncated 5-cube may be constructed by truncating the vertices of the 5-cube at 
of the edge length. A regular 5-cell is formed at each truncated vertex.
The Cartesian coordinates of the vertices of a truncated 5-cube having edge length 2 are all permutations of:
Images
The truncated 5-cube is constructed by a truncation applied to the 5-cube. All edges are shortened, and two new vertices are added on each original edge.
| 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-cube
| Bitruncated 5-cube | |
|---|---|
| Type | uniform polyteron |
| Schläfli symbol | t1,2{4,3,3,3} |
| Coxeter-Dynkin diagrams | |
| 4-faces | 42 |
| Cells | 280 |
| Faces | 720 |
| Edges | 800 |
| Vertices | 320 |
| Vertex figure | |
| Coxeter groups | BC5, [3,3,3,4] |
| Properties | convex |
Alternate names
- Bitruncated penteract (Acronym: bittin) (Jonathan Bowers)
Construction and coordinates
The bitruncated 5-cube may be constructed by bitruncating the vertices of the 5-cube at 
of the edge length.
The Cartesian coordinates of the vertices of a bitruncated 5-cube having edge length 2 are all permutations of:
Images
| 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 polytera generated from the regular 5-cube or 5-orthoplex.
Notes
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, editied 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.
- Richard Klitzing, 5D, uniform polytopes (polytera) o3o3o3x4x - tan, o3o3x3x4o - bittin
External links
- Weisstein, Eric W., "Hypercube" from MathWorld.
- Olshevsky, George, Measure polytope at Glossary for Hyperspace.
- Polytopes of Various Dimensions
- Multi-dimensional Glossary