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

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
Typeuniform 6-polytope
Schläfli symbolt0,2{3,3,3,3,4}
rr{3,3,3,3,4}
Coxeter-Dynkin diagrams

=

5-faces136
4-faces1656
Cells5040
Faces6400
Edges3360
Vertices480
Vertex figure
Coxeter groupsB6, [3,3,3,3,4]
D6, [33,1,1]
Propertiesconvex

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
Dihedral symmetry [12] [10] [8]
Coxeter plane B3 B2
Graph
Dihedral symmetry [6] [4]
Coxeter plane A5 A3
Graph
Dihedral symmetry [6] [4]

Bicantellated 6-orthoplex

Bicantellated 6-orthoplex
Typeuniform 6-polytope
Schläfli symbolt1,3{3,3,3,3,4}
2rr{3,3,3,3,4}
Coxeter-Dynkin diagrams

5-faces
4-faces
Cells
Faces
Edges8640
Vertices1440
Vertex figure
Coxeter groupsB6, [3,3,3,3,4]
D6, [33,1,1]
Propertiesconvex

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
Dihedral symmetry [12] [10] [8]
Coxeter plane B3 B2
Graph
Dihedral symmetry [6] [4]
Coxeter plane A5 A3
Graph
Dihedral symmetry [6] [4]

Cantitruncated 6-orthoplex

Cantitruncated 6-orthoplex
Typeuniform 6-polytope
Schläfli symbolt0,1,2{3,3,3,3,4}
tr{3,3,3,3,4}
Coxeter-Dynkin diagrams

5-faces
4-faces
Cells
Faces
Edges3840
Vertices960
Vertex figure
Coxeter groupsB6, [3,3,3,3,4]
D6, [33,1,1]
Propertiesconvex

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
Dihedral symmetry [12] [10] [8]
Coxeter plane B3 B2
Graph
Dihedral symmetry [6] [4]
Coxeter plane A5 A3
Graph
Dihedral symmetry [6] [4]

Bicantitruncated 6-orthoplex

Bicantitruncated 6-orthoplex
Typeuniform 6-polytope
Schläfli symbolt1,2,3{3,3,3,3,4}
2tr{3,3,3,3,4}
Coxeter-Dynkin diagrams

5-faces
4-faces
Cells
Faces
Edges10080
Vertices2880
Vertex figure
Coxeter groupsB6, [3,3,3,3,4]
D6, [33,1,1]
Propertiesconvex

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
Dihedral symmetry [12] [10] [8]
Coxeter plane B3 B2
Graph
Dihedral symmetry [6] [4]
Coxeter plane A5 A3
Graph
Dihedral symmetry [6] [4]

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

β6
t1β6
t2β6
t2γ6
t1γ6
γ6
t0,1β6
t0,2β6

t1,2β6
t0,3β6
t1,3β6
t2,3γ6
t0,4β6
t1,4γ6
t1,3γ6
t1,2γ6

t0,5γ6
t0,4γ6
t0,3γ6
t0,2γ6
t0,1γ6
t0,1,2β6
t0,1,3β6
t0,2,3β6

t1,2,3β6
t0,1,4β6
t0,2,4β6
t1,2,4β6
t0,3,4β6
t1,2,4γ6
t1,2,3γ6
t0,1,5β6

t0,2,5β6
t0,3,4γ6
t0,2,5γ6
t0,2,4γ6
t0,2,3γ6
t0,1,5γ6
t0,1,4γ6
t0,1,3γ6

t0,1,2γ6
t0,1,2,3β6
t0,1,2,4β6
t0,1,3,4β6
t0,2,3,4β6
t1,2,3,4γ6
t0,1,2,5β6
t0,1,3,5β6

t0,2,3,5γ6
t0,2,3,4γ6
t0,1,4,5γ6
t0,1,3,5γ6
t0,1,3,4γ6
t0,1,2,5γ6
t0,1,2,4γ6
t0,1,2,3γ6

t0,1,2,3,4β6
t0,1,2,3,5β6
t0,1,2,4,5β6
t0,1,2,4,5γ6
t0,1,2,3,5γ6
t0,1,2,3,4γ6
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
      • (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
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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