Cantellated 6-orthoplexes

Orthogonal projections in B₆ Coxeter plane
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

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	t_0,2{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, [3^3,1,1]
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	t_1,3{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, [3^3,1,1]
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	t_0,1,2{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, [3^3,1,1]
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	t_1,2,3{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, [3^3,1,1]
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.

β₆	t₁β₆	t₂β₆	t₂γ₆	t₁γ₆	γ₆	t_0,1β₆	t_0,2β₆
t_1,2β₆	t_0,3β₆	t_1,3β₆	t_2,3γ₆	t_0,4β₆	t_1,4γ₆	t_1,3γ₆	t_1,2γ₆
t_0,5γ₆	t_0,4γ₆	t_0,3γ₆	t_0,2γ₆	t_0,1γ₆	t_0,1,2β₆	t_0,1,3β₆	t_0,2,3β₆
t_1,2,3β₆	t_0,1,4β₆	t_0,2,4β₆	t_1,2,4β₆	t_0,3,4β₆	t_1,2,4γ₆	t_1,2,3γ₆	t_0,1,5β₆
t_0,2,5β₆	t_0,3,4γ₆	t_0,2,5γ₆	t_0,2,4γ₆	t_0,2,3γ₆	t_0,1,5γ₆	t_0,1,4γ₆	t_0,1,3γ₆
t_0,1,2γ₆	t_0,1,2,3β₆	t_0,1,2,4β₆	t_0,1,3,4β₆	t_0,2,3,4β₆	t_1,2,3,4γ₆	t_0,1,2,5β₆	t_0,1,3,5β₆
t_0,2,3,5γ₆	t_0,2,3,4γ₆	t_0,1,4,5γ₆	t_0,1,3,5γ₆	t_0,1,3,4γ₆	t_0,1,2,5γ₆	t_0,1,2,4γ₆	t_0,1,2,3γ₆
t_0,1,2,3,4β₆	t_0,1,2,3,5β₆	t_0,1,2,4,5β₆	t_0,1,2,4,5γ₆	t_0,1,2,3,5γ₆	t_0,1,2,3,4γ₆	t_0,1,2,3,4,5γ₆

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, [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.
Richard Klitzing, 6D, uniform polytopes (polypeta) x3o3x3o3o4o - srog, o3x3o3x3o4o - siborg, x3x3x3o3o4o - grog, o3x3x3x3o4o - gaborg

External links

Olshevsky, George, Cross polytope at Glossary for Hyperspace.
Polytopes of Various Dimensions
Multi-dimensional Glossary

Fundamental convex regular and uniform polytopes in dimensions 2–10
Family	A_n	B_n	I₂(p) / D_n	E₆ / E₇ / E₈ / F₄ / G₂	H_n
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	1_k2 • 2_k1 • k₂₁	n-pentagonal polytope
Topics: Polytope families • Regular polytope • List of regular polytopes and compounds

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