Cantellated 6-simplexes

Orthogonal projections in A₆ Coxeter plane
6-simplex	Cantellated 6-simplex	Bicantellated 6-simplex
Birectified 6-simplex	Cantitruncated 6-simplex	Bicantitruncated 6-simplex

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

There are unique 4 degrees of cantellation for the 6-simplex, including truncations.

Cantellated 6-simplex

Cantellated 6-simplex
Type	uniform 6-polytope
Schläfli symbol	rr{3,3,3,3,3} or $r\left\{{\begin{array}{l}3,3,3,3\\3\end{array}}\right\}$
Coxeter-Dynkin diagrams
5-faces	35
4-faces	210
Cells	560
Faces	805
Edges	525
Vertices	105
Vertex figure	5-cell prism
Coxeter group	A₆, [3⁵], order 5040
Properties	convex

Alternate names

Small rhombated heptapeton (Acronym: sril) (Jonathan Bowers)^[1]

Coordinates

The vertices of the cantellated 6-simplex can be most simply positioned in 7-space as permutations of (0,0,0,0,1,1,2). This construction is based on facets of the cantellated 7-orthoplex.

Images

orthographic projections
A_k Coxeter plane	A₆	A₅	A₄
Graph
Dihedral symmetry	[7]	[6]	[5]
A_k Coxeter plane	A₃	A₂
Graph
Dihedral symmetry	[4]	[3]

^[2]

Bicantellated 6-simplex

Bicantellated 6-simplex
Type	uniform 6-polytope
Schläfli symbol	2rr{3,3,3,3,3} or $r\left\{{\begin{array}{l}3,3,3\\3,3\end{array}}\right\}$
Coxeter-Dynkin diagrams
5-faces	49
4-faces	329
Cells	980
Faces	1540
Edges	1050
Vertices	210
Vertex figure
Coxeter group	A₆, [3⁵], order 5040
Properties	convex

Alternate names

Small prismated heptapeton (Acronym: sabril) (Jonathan Bowers)^[3]

Coordinates

The vertices of the bicantellated 6-simplex can be most simply positioned in 7-space as permutations of (0,0,0,1,1,2,2). This construction is based on facets of the bicantellated 7-orthoplex.

Images

orthographic projections
A_k Coxeter plane	A₆	A₅	A₄
Graph
Dihedral symmetry	[7]	[6]	[5]
A_k Coxeter plane	A₃	A₂
Graph
Dihedral symmetry	[4]	[3]

Cantitruncated 6-simplex

cantitruncated 6-simplex
Type	uniform 6-polytope
Schläfli symbol	tr{3,3,3,3,3} or $t\left\{{\begin{array}{l}3,3,3,3\\3\end{array}}\right\}$
Coxeter-Dynkin diagrams
5-faces	35
4-faces	210
Cells	560
Faces	805
Edges	630
Vertices	210
Vertex figure
Coxeter group	A₆, [3⁵], order 5040
Properties	convex

Alternate names

Great rhombated heptapeton (Acronym: gril) (Jonathan Bowers)^[4]

Coordinates

The vertices of the cantitruncated 6-simplex can be most simply positioned in 7-space as permutations of (0,0,0,0,1,2,3). This construction is based on facets of the cantitruncated 7-orthoplex.

Images

orthographic projections
A_k Coxeter plane	A₆	A₅	A₄
Graph
Dihedral symmetry	[7]	[6]	[5]
A_k Coxeter plane	A₃	A₂
Graph
Dihedral symmetry	[4]	[3]

Bicantitruncated 6-simplex

bicantitruncated 6-simplex
Type	uniform 6-polytope
Schläfli symbol	2tr{3,3,3,3,3} or $t\left\{{\begin{array}{l}3,3,3\\3,3\end{array}}\right\}$
Coxeter-Dynkin diagrams
5-faces	49
4-faces	329
Cells	980
Faces	1540
Edges	1260
Vertices	420
Vertex figure
Coxeter group	A₆, [3⁵], order 5040
Properties	convex

Alternate names

Great birhombated heptapeton (Acronym: gabril) (Jonathan Bowers)^[5]

Coordinates

The vertices of the bicantitruncated 6-simplex can be most simply positioned in 7-space as permutations of (0,0,0,1,2,3,3). This construction is based on facets of the bicantitruncated 7-orthoplex.

Images

orthographic projections
A_k Coxeter plane	A₆	A₅	A₄
Graph
Dihedral symmetry	[7]	[6]	[5]
A_k Coxeter plane	A₃	A₂
Graph
Dihedral symmetry	[4]	[3]

Related uniform 6-polytopes

The truncated 6-simplex is one of 35 uniform 6-polytopes based on the [3,3,3,3,3] Coxeter group, all shown here in A₆ Coxeter plane orthographic projections.

A6 polytopes
t₀	t₁	t₂	t_0,1	t_0,2	t_1,2	t_0,3	t_1,3	t_2,3
t_0,4	t_1,4	t_0,5	t_0,1,2	t_0,1,3	t_0,2,3	t_1,2,3	t_0,1,4	t_0,2,4
t_1,2,4	t_0,3,4	t_0,1,5	t_0,2,5	t_0,1,2,3	t_0,1,2,4	t_0,1,3,4	t_0,2,3,4	t_1,2,3,4
t_0,1,2,5	t_0,1,3,5	t_0,2,3,5	t_0,1,4,5	t_0,1,2,3,4	t_0,1,2,3,5	t_0,1,2,4,5	t_0,1,2,3,4,5

Notes

↑ Klitizing, (x3o3x3o3o3o - sril)
↑ Klitzing, (x3o3x3o3o3o - sril)
↑ Klitzing, (o3x3o3x3o3o - sabril)
↑ Klitzing, (x3x3x3o3o3o - gril)
↑ Klitzing, (o3x3x3x3o3o - gabril)

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)". x3o3x3o3o3o - sril, o3x3o3x3o3o - sabril, x3x3x3o3o3o - gril, o3x3x3x3o3o - gabril

External links

Olshevsky, George. "Cross polytope". Glossary for Hyperspace. Archived from the original on 4 February 2007.
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 / E₉ / E₁₀ / 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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