Truncated 7-orthoplexes
7-orthoplex |
Truncated 7-orthoplex |
Bitruncated 7-orthoplex |
Tritruncated 7-orthoplex |
7-cube |
Truncated 7-cube |
Bitruncated 7-cube |
Tritruncated 7-cube |
Orthogonal projections in B7 Coxeter plane |
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In seven-dimensional geometry, a truncated 7-orthoplex is a convex uniform 7-polytope, being a truncation of the regular 7-orthoplex.
There are 6 truncations of the 7-orthoplex. Vertices of the truncation 7-orthoplex are located as pairs on the edge of the 7-orthoplex. Vertices of the bitruncated 7-orthoplex are located on the triangular faces of the 7-orthoplex. Vertices of the tritruncated 7-orthoplex are located inside the tetrahedral cells of the 7-orthoplex. The final three truncations are best expressed relative to the 7-cube.
Truncated 7-orthoplex
Truncated 7-orthoplex | |
---|---|
Type | uniform 7-polytope |
Schläfli symbol | t{35,4} |
Coxeter-Dynkin diagrams |
|
6-faces | |
5-faces | |
4-faces | |
Cells | 3920 |
Faces | 2520 |
Edges | 924 |
Vertices | 168 |
Vertex figure | ( )v{3,3,4} |
Coxeter groups | B7, [35,4] D7, [34,1,1] |
Properties | convex |
Alternate names
- Truncated heptacross
- Truncated hecatonicosoctaexon (Jonathan Bowers)[1]
Coordinates
Cartesian coordinates for the vertices of a truncated 7-orthoplex, centered at the origin, are all 168 vertices are sign (4) and coordinate (42) permutations of
- (±2,±1,0,0,0,0,0)
Images
Coxeter plane | B7 / A6 | B6 / D7 | B5 / D6 / A4 |
---|---|---|---|
Graph | |||
Dihedral symmetry | [14] | [12] | [10] |
Coxeter plane | B4 / D5 | B3 / D4 / A2 | B2 / D3 |
Graph | |||
Dihedral symmetry | [8] | [6] | [4] |
Coxeter plane | A5 | A3 | |
Graph | |||
Dihedral symmetry | [6] | [4] |
Construction
There are two Coxeter groups associated with the truncated 7-orthoplex, one with the C7 or [4,35] Coxeter group, and a lower symmetry with the D7 or [34,1,1] Coxeter group.
Bitruncated 7-orthoplex
Bitruncated 7-orthoplex | |
---|---|
Type | uniform 7-polytope |
Schläfli symbol | 2t{35,4} |
Coxeter-Dynkin diagrams |
|
6-faces | |
5-faces | |
4-faces | |
Cells | |
Faces | |
Edges | 4200 |
Vertices | 840 |
Vertex figure | { }v{3,3,4} |
Coxeter groups | B7, [35,4] D7, [34,1,1] |
Properties | convex |
Alternate names
- Bitruncated heptacross
- Bitruncated hecatonicosoctaexon (Jonathan Bowers)[2]
Coordinates
Cartesian coordinates for the vertices of a bitruncated 7-orthoplex, centered at the origin, are all sign and coordinate permutations of
- (±2,±2,±1,0,0,0,0)
Images
Coxeter plane | B7 / A6 | B6 / D7 | B5 / D6 / A4 |
---|---|---|---|
Graph | |||
Dihedral symmetry | [14] | [12] | [10] |
Coxeter plane | B4 / D5 | B3 / D4 / A2 | B2 / D3 |
Graph | |||
Dihedral symmetry | [8] | [6] | [4] |
Coxeter plane | A5 | A3 | |
Graph | |||
Dihedral symmetry | [6] | [4] |
Tritruncated 7-orthoplex
The tritruncated 7-orthoplex can tessellation space in the quadritruncated 7-cubic honeycomb.
Tritruncated 7-orthoplex | |
---|---|
Type | uniform 7-polytope |
Schläfli symbol | 3t{35,4} |
Coxeter-Dynkin diagrams |
|
6-faces | |
5-faces | |
4-faces | |
Cells | |
Faces | |
Edges | 10080 |
Vertices | 2240 |
Vertex figure | {3}v{3,4} |
Coxeter groups | B7, [35,4] D7, [34,1,1] |
Properties | convex |
Alternate names
- Tritruncated heptacross
- Tritruncated hecatonicosoctaexon (Jonathan Bowers)[3]
Coordinates
Cartesian coordinates for the vertices of a tritruncated 7-orthoplex, centered at the origin, are all sign and coordinate permutations of
- (±2,±2,±2,±1,0,0,0)
Images
Coxeter plane | B7 / A6 | B6 / D7 | B5 / D6 / A4 |
---|---|---|---|
Graph | |||
Dihedral symmetry | [14] | [12] | [10] |
Coxeter plane | B4 / D5 | B3 / D4 / A2 | B2 / D3 |
Graph | |||
Dihedral symmetry | [8] | [6] | [4] |
Coxeter plane | A5 | A3 | |
Graph | |||
Dihedral symmetry | [6] | [4] |
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 [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. "7D uniform polytopes (polyexa)". https://bendwavy.org/klitzing/dimensions/polyexa.htm. x3x3o3o3o3o4o - tez, o3x3x3o3o3o4o - botaz, o3o3x3x3o3o4o - totaz
External links
Fundamental convex regular and uniform polytopes in dimensions 2–10
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Family | An | Bn | I2(p) / Dn | E6 / E7 / E8 / 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 |
Original source: https://en.wikipedia.org/wiki/Truncated 7-orthoplexes.
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