Cantellated 6-orthoplexes

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File:6-cube t5.svg
6-orthoplex
File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel node.png 		File:6-cube t35.svg
Cantellated 6-orthoplex
File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel node.png 		File:6-cube t24.svg
Bicantellated 6-orthoplex
File:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel node.png
File:6-cube t0.svg
6-cube
File:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel node 1.png 		File:6-cube t02.svg
Cantellated 6-cube
File:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel node 1.png 		File:6-cube t13.svg
Bicantellated 6-cube
File:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 4.pngFile:CDel node.png
File:6-cube t345.svg
Cantitruncated 6-orthoplex
File:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel node.png 		File:6-cube t234.svg
Bicantitruncated 6-orthoplex
File:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel node.png 		File:6-cube t123.svg
Bicantitruncated 6-cube
File:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 4.pngFile:CDel node.png 		File:6-cube t0123.svg
Cantitruncated 6-cube
File:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 4.pngFile:CDel node 1.png
Orthogonal projections in B6 Coxeter plane

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 File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel node.png

File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel split1.pngFile:CDel nodes.png = File:CDel nodes 10r.pngFile:CDel 3ab.pngFile:CDel nodes.pngFile:CDel split5c.pngFile:CDel nodes 01l.png

5-faces 136
4-faces 1656
Cells 5040
Faces 6400
Edges 3360
Vertices 480
Vertex figure
Coxeter groups B6, [3,3,3,3,4]
D6, [33,1,1]
Properties convex

Alternate names

  • Cantellated hexacross
  • Small rhombated hexacontatetrapeton (acronym: srog) (Jonathan Bowers)[1]

Construction

There are two Coxeter groups associated with the cantellated 6-orthoplex, one with the B6 or [4,3,3,3,3] Coxeter group, and a lower symmetry with the D6 or [33,1,1] 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

orthographic projections
Coxeter plane B6 B5 B4
Graph File:6-cube t35.svg File:6-cube t35 B5.svg File:6-cube t35 B4.svg
Dihedral symmetry [12] [10] [8]
Coxeter plane B3 B2
Graph File:6-cube t35 B3.svg File:6-cube t35 B2.svg
Dihedral symmetry [6] [4]
Coxeter plane A5 A3
Graph File:6-cube t35 A5.svg File:6-cube t35 A3.svg
Dihedral symmetry [6] [4]

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 File:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel node.png

File:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel split1.pngFile:CDel nodes.png

5-faces
4-faces
Cells
Faces
Edges 8640
Vertices 1440
Vertex figure
Coxeter groups B6, [3,3,3,3,4]
D6, [33,1,1]
Properties convex

Alternate names

  • Bicantellated hexacross, bicantellated hexacontatetrapeton
  • Small birhombated hexacontatetrapeton (acronym: siborg) (Jonathan Bowers)[2]

Construction

There are two Coxeter groups associated with the bicantellated 6-orthoplex, one with the B6 or [4,3,3,3,3] Coxeter group, and a lower symmetry with the D6 or [33,1,1] 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

orthographic projections
Coxeter plane B6 B5 B4
Graph File:6-cube t24.svg File:6-cube t24 B5.svg File:6-cube t24 B4.svg
Dihedral symmetry [12] [10] [8]
Coxeter plane B3 B2
Graph File:6-cube t24 B3.svg File:6-cube t24 B2.svg
Dihedral symmetry [6] [4]
Coxeter plane A5 A3
Graph File:6-cube t24 A5.svg File:6-cube t24 A3.svg
Dihedral symmetry [6] [4]

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 File:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel node.png

File:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel split1.pngFile:CDel nodes.png

5-faces
4-faces
Cells
Faces
Edges 3840
Vertices 960
Vertex figure
Coxeter groups B6, [3,3,3,3,4]
D6, [33,1,1]
Properties convex

Alternate names

  • Cantitruncated hexacross, cantitruncated hexacontatetrapeton
  • Great rhombihexacontatetrapeton (acronym: grog) (Jonathan Bowers)[3]

Construction

There are two Coxeter groups associated with the cantitruncated 6-orthoplex, one with the B6 or [4,3,3,3,3] Coxeter group, and a lower symmetry with the D6 or [33,1,1] 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

orthographic projections
Coxeter plane B6 B5 B4
Graph File:6-cube t345.svg File:6-cube t345 B5.svg File:6-cube t345 B4.svg
Dihedral symmetry [12] [10] [8]
Coxeter plane B3 B2
Graph File:6-cube t345 B3.svg File:6-cube t345 B2.svg
Dihedral symmetry [6] [4]
Coxeter plane A5 A3
Graph File:6-cube t345 A5.svg File:6-cube t345 A3.svg
Dihedral symmetry [6] [4]

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 File:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel node.png

File:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel split1.pngFile:CDel nodes.png

5-faces
4-faces
Cells
Faces
Edges 10080
Vertices 2880
Vertex figure
Coxeter groups B6, [3,3,3,3,4]
D6, [33,1,1]
Properties convex

Alternate names

  • Bicantitruncated hexacross, bicantitruncated hexacontatetrapeton
  • Great birhombihexacontatetrapeton (acronym: gaborg) (Jonathan Bowers)[4]

Construction

There are two Coxeter groups associated with the bicantitruncated 6-orthoplex, one with the B6 or [4,3,3,3,3] Coxeter group, and a lower symmetry with the D6 or [33,1,1] 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

orthographic projections
Coxeter plane B6 B5 B4
Graph File:6-cube t234.svg File:6-cube t234 B5.svg File:6-cube t234 B4.svg
Dihedral symmetry [12] [10] [8]
Coxeter plane B3 B2
Graph File:6-cube t234 B3.svg File:6-cube t234 B2.svg
Dihedral symmetry [6] [4]
Coxeter plane A5 A3
Graph File:6-cube t234 A5.svg File:6-cube t234 A3.svg
Dihedral symmetry [6] [4]

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.

B6 polytopes
File:6-cube t5.svg
β6 		File:6-cube t4.svg
t1β6 		File:6-cube t3.svg
t2β6 		File:6-cube t2.svg
t2γ6 		File:6-cube t1.svg
t1γ6 		File:6-cube t0.svg
γ6 		File:6-cube t45.svg
t0,1β6 		File:6-cube t35.svg
t0,2β6
File:6-cube t34.svg
t1,2β6 		File:6-cube t25.svg
t0,3β6 		File:6-cube t24.svg
t1,3β6 		File:6-cube t23.svg
t2,3γ6 		File:6-cube t15.svg
t0,4β6 		File:6-cube t14.svg
t1,4γ6 		File:6-cube t13.svg
t1,3γ6 		File:6-cube t12.svg
t1,2γ6
File:6-cube t05.svg
t0,5γ6 		File:6-cube t04.svg
t0,4γ6 		File:6-cube t03.svg
t0,3γ6 		File:6-cube t02.svg
t0,2γ6 		File:6-cube t01.svg
t0,1γ6 		File:6-cube t345.svg
t0,1,2β6 		File:6-cube t245.svg
t0,1,3β6 		File:6-cube t235.svg
t0,2,3β6
File:6-cube t234.svg
t1,2,3β6 		File:6-cube t145.svg
t0,1,4β6 		File:6-cube t135.svg
t0,2,4β6 		File:6-cube t134.svg
t1,2,4β6 		File:6-cube t125.svg
t0,3,4β6 		File:6-cube t124.svg
t1,2,4γ6 		File:6-cube t123.svg
t1,2,3γ6 		File:6-cube t045.svg
t0,1,5β6
File:6-cube t035.svg
t0,2,5β6 		File:6-cube t034.svg
t0,3,4γ6 		File:6-cube t025.svg
t0,2,5γ6 		File:6-cube t024.svg
t0,2,4γ6 		File:6-cube t023.svg
t0,2,3γ6 		File:6-cube t015.svg
t0,1,5γ6 		File:6-cube t014.svg
t0,1,4γ6 		File:6-cube t013.svg
t0,1,3γ6
File:6-cube t012.svg
t0,1,2γ6 		File:6-cube t2345.svg
t0,1,2,3β6 		File:6-cube t1345.svg
t0,1,2,4β6 		File:6-cube t1245.svg
t0,1,3,4β6 		File:6-cube t1235.svg
t0,2,3,4β6 		File:6-cube t1234.svg
t1,2,3,4γ6 		File:6-cube t0345.svg
t0,1,2,5β6 		File:6-cube t0245.svg
t0,1,3,5β6
File:6-cube t0235.svg
t0,2,3,5γ6 		File:6-cube t0234.svg
t0,2,3,4γ6 		File:6-cube t0145.svg
t0,1,4,5γ6 		File:6-cube t0135.svg
t0,1,3,5γ6 		File:6-cube t0134.svg
t0,1,3,4γ6 		File:6-cube t0125.svg
t0,1,2,5γ6 		File:6-cube t0124.svg
t0,1,2,4γ6 		File:6-cube t0123.svg
t0,1,2,3γ6
File:6-cube t12345.svg
t0,1,2,3,4β6 		File:6-cube t02345.svg
t0,1,2,3,5β6 		File:6-cube t01345.svg
t0,1,2,4,5β6 		File:6-cube t01245.svg
t0,1,2,4,5γ6 		File:6-cube t01235.svg
t0,1,2,3,5γ6 		File:6-cube t01234.svg
t0,1,2,3,4γ6 		File:6-cube t012345.svg
t0,1,2,3,4,5γ6

Notes

  1. ↑ Klitzing, (x3o3x3o3o4o - srog)
  2. ↑ Klitzing, (o3x3o3x3o4o - siborg)
  3. ↑ Klitzing, (x3x3x3o3o4o - grog)
  4. ↑ 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 [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. "6D uniform polytopes (polypeta)". x3o3x3o3o4o - srog, o3x3o3x3o4o - siborg, x3x3x3o3o4o - grog, o3x3x3x3o4o - gaborg

External links

Fundamental convex regular and uniform polytopes in dimensions 2–10
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
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