6-orthoplex |
Cantellated 6-orthoplex |
Bicantellated 6-orthoplex | |||||||||
6-cube |
Cantellated 6-cube |
Bicantellated 6-cube | |||||||||
Cantitruncated 6-orthoplex |
Bicantitruncated 6-orthoplex |
Bicantitruncated 6-cube |
Cantitruncated 6-cube | ||||||||
Orthogonal projections in B6 Coxeter plane |
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In six-dimensional geometry, a cantellated 6-orthoplex is a convex uniform 6-polytope, being a cantellation of the regular 6-orthoplex.
There are 8 cantellation for the 6-orthoplex including truncations. Half of them are more easily constructed from the dual 5-cube
Cantellated 6-orthoplex
Cantellated 6-orthoplex | |
---|---|
Type | uniform 6-polytope |
Schläfli symbol | t0,2{3,3,3,3,4} rr{3,3,3,3,4} |
Coxeter-Dynkin diagrams | = |
5-faces | 136 |
4-faces | 1656 |
Cells | 5040 |
Faces | 6400 |
Edges | 3360 |
Vertices | 480 |
Vertex figure | |
Coxeter groups | B6, D6, |
Properties | convex |
Alternate names
- Cantellated hexacross
- Small rhombated hexacontatetrapeton (acronym: srog) (Jonathan Bowers)
Construction
There are two Coxeter groups associated with the cantellated 6-orthoplex, one with the B6 or Coxeter group, and a lower symmetry with the D6 or Coxeter group.
Coordinates
Cartesian coordinates for the 480 vertices of a cantellated 6-orthoplex, centered at the origin, are all the sign and coordinate permutations of
- (2,1,1,0,0,0)
Images
Coxeter plane | B6 | B5 | B4 |
---|---|---|---|
Graph | |||
Dihedral symmetry | |||
Coxeter plane | B3 | B2 | |
Graph | |||
Dihedral symmetry | |||
Coxeter plane | A5 | A3 | |
Graph | |||
Dihedral symmetry |
Bicantellated 6-orthoplex
Bicantellated 6-orthoplex | |
---|---|
Type | uniform 6-polytope |
Schläfli symbol | t1,3{3,3,3,3,4} 2rr{3,3,3,3,4} |
Coxeter-Dynkin diagrams |
|
5-faces | |
4-faces | |
Cells | |
Faces | |
Edges | 8640 |
Vertices | 1440 |
Vertex figure | |
Coxeter groups | B6, D6, |
Properties | convex |
Alternate names
- Bicantellated hexacross, bicantellated hexacontatetrapeton
- Small birhombated hexacontatetrapeton (acronym: siborg) (Jonathan Bowers)
Construction
There are two Coxeter groups associated with the bicantellated 6-orthoplex, one with the B6 or Coxeter group, and a lower symmetry with the D6 or Coxeter group.
Coordinates
Cartesian coordinates for the 1440 vertices of a bicantellated 6-orthoplex, centered at the origin, are all the sign and coordinate permutations of
- (2,2,1,1,0,0)
Images
Coxeter plane | B6 | B5 | B4 |
---|---|---|---|
Graph | |||
Dihedral symmetry | |||
Coxeter plane | B3 | B2 | |
Graph | |||
Dihedral symmetry | |||
Coxeter plane | A5 | A3 | |
Graph | |||
Dihedral symmetry |
Cantitruncated 6-orthoplex
Cantitruncated 6-orthoplex | |
---|---|
Type | uniform 6-polytope |
Schläfli symbol | t0,1,2{3,3,3,3,4} tr{3,3,3,3,4} |
Coxeter-Dynkin diagrams |
|
5-faces | |
4-faces | |
Cells | |
Faces | |
Edges | 3840 |
Vertices | 960 |
Vertex figure | |
Coxeter groups | B6, D6, |
Properties | convex |
Alternate names
- Cantitruncated hexacross, cantitruncated hexacontatetrapeton
- Great rhombihexacontatetrapeton (acronym: grog) (Jonathan Bowers)
Construction
There are two Coxeter groups associated with the cantitruncated 6-orthoplex, one with the B6 or Coxeter group, and a lower symmetry with the D6 or Coxeter group.
Coordinates
Cartesian coordinates for the 960 vertices of a cantitruncated 6-orthoplex, centered at the origin, are all the sign and coordinate permutations of
- (3,2,1,0,0,0)
Images
Coxeter plane | B6 | B5 | B4 |
---|---|---|---|
Graph | |||
Dihedral symmetry | |||
Coxeter plane | B3 | B2 | |
Graph | |||
Dihedral symmetry | |||
Coxeter plane | A5 | A3 | |
Graph | |||
Dihedral symmetry |
Bicantitruncated 6-orthoplex
Bicantitruncated 6-orthoplex | |
---|---|
Type | uniform 6-polytope |
Schläfli symbol | t1,2,3{3,3,3,3,4} 2tr{3,3,3,3,4} |
Coxeter-Dynkin diagrams |
|
5-faces | |
4-faces | |
Cells | |
Faces | |
Edges | 10080 |
Vertices | 2880 |
Vertex figure | |
Coxeter groups | B6, D6, |
Properties | convex |
Alternate names
- Bicantitruncated hexacross, bicantitruncated hexacontatetrapeton
- Great birhombihexacontatetrapeton (acronym: gaborg) (Jonathan Bowers)
Construction
There are two Coxeter groups associated with the bicantitruncated 6-orthoplex, one with the B6 or Coxeter group, and a lower symmetry with the D6 or Coxeter group.
Coordinates
Cartesian coordinates for the 2880 vertices of a bicantitruncated 6-orthoplex, centered at the origin, are all the sign and coordinate permutations of
- (3,3,2,1,0,0)
Images
Coxeter plane | B6 | B5 | B4 |
---|---|---|---|
Graph | |||
Dihedral symmetry | |||
Coxeter plane | B3 | B2 | |
Graph | |||
Dihedral symmetry | |||
Coxeter plane | A5 | A3 | |
Graph | |||
Dihedral symmetry |
Related polytopes
These polytopes are part of a set of 63 uniform 6-polytopes generated from the B6 Coxeter plane, including the regular 6-cube or 6-orthoplex.
Notes
- Klitzing, (x3o3x3o3o4o - srog)
- Klitzing, (o3x3o3x3o4o - siborg)
- Klitzing, (x3x3x3o3o4o - grog)
- Klitzing, (o3x3x3x3o4o - gaborg)
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,
- (Paper 23) H.S.M. Coxeter, Regular and Semi-Regular Polytopes II,
- (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III,
- Norman Johnson Uniform Polytopes, Manuscript (1991)
- N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D.
- Klitzing, Richard. "6D uniform polytopes (polypeta)". x3o3x3o3o4o - srog, o3x3o3x3o4o - siborg, x3x3x3o3o4o - grog, o3x3x3x3o4o - gaborg
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 polychoron | Pentachoron | 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 |