Cantellated 6-orthoplexes

In six-dimensional geometry, a cantellated 6-orthoplex is a convex uniform 6-polytope, being a cantellation of the regular 6-orthoplex.

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

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, [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 [3^3,1,1] 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
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, [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 [3^3,1,1] 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
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, [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 [3^3,1,1] 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
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, [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 [3^3,1,1] 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]

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
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

External links

• Polytopes of Various Dimensions
• Multi-dimensional Glossary

v t e 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