Truncated 5-orthoplexes
5-orthoplex |
Truncated 5-orthoplex |
Bitruncated 5-orthoplex | |
5-cube |
Truncated 5-cube |
Bitruncated 5-cube | |
Orthogonal projections in BC5 Coxeter plane |
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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 | Elongated octahedral pyramid |
Coxeter groups | BC5, [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 trunacted 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.
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 | square-pyramidal pyramid |
Coxeter groups | BC5, [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.
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
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
- (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
External links
- Olshevsky, George. "Measure polytope". Glossary for Hyperspace. Archived from the original on 4 February 2007.
- Polytopes of Various Dimensions
- Multi-dimensional Glossary
Fundamental convex regular and uniform polytopes in dimensions 2–10 | ||||||||||||
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Family | An | Bn | I2(p) / Dn | E6 / E7 / E8 / E9 / E10 / F4 / G2 | Hn | |||||||
Regular polygon | Triangle | Square | p-gon | Hexagon | Pentagon | |||||||
Uniform polyhedron | Tetrahedron | Octahedron • Cube | Demicube | Dodecahedron • Icosahedron | ||||||||
Uniform 4-polytope | 5-cell | 16-cell • Tesseract | Demitesseract | 24-cell | 120-cell • 600-cell | |||||||
Uniform 5-polytope | 5-simplex | 5-orthoplex • 5-cube | 5-demicube | |||||||||
Uniform 6-polytope | 6-simplex | 6-orthoplex • 6-cube | 6-demicube | 122 • 221 | ||||||||
Uniform 7-polytope | 7-simplex | 7-orthoplex • 7-cube | 7-demicube | 132 • 231 • 321 | ||||||||
Uniform 8-polytope | 8-simplex | 8-orthoplex • 8-cube | 8-demicube | 142 • 241 • 421 | ||||||||
Uniform 9-polytope | 9-simplex | 9-orthoplex • 9-cube | 9-demicube | |||||||||
Uniform 10-polytope | 10-simplex | 10-orthoplex • 10-cube | 10-demicube | |||||||||
Uniform n-polytope | n-simplex | n-orthoplex • n-cube | n-demicube | 1k2 • 2k1 • k21 | n-pentagonal polytope | |||||||
Topics: Polytope families • Regular polytope • List of regular polytopes and compounds |