Cantellated 8-simplexes

Orthogonal projections in A₈ Coxeter plane
Cantellated 8-simplex	Bicantellated 8-simplex	Tricantellated 8-simplex
Cantitruncated 8-simplex	Bicantitruncated 8-simplex	Tricantitruncated 8-simplex

In eight-dimensional geometry, a cantellated 8-simplex is a convex uniform 8-polytope, being a cantellation of the regular 8-simplex.

There are six unique cantellations for the 8-simplex, including permutations of truncation.

Cantellated 8-simplex

Cantellated 8-simplex
Type	uniform 8-polytope
Schläfli symbol	rr{3,3,3,3,3,3,3}
Coxeter-Dynkin diagram
7-faces
6-faces
5-faces
4-faces
Cells
Faces
Edges	1764
Vertices	252
Vertex figure	6-simplex prism
Coxeter group	A₈, [3⁷], order 362880
Properties	convex

Alternate names

Small rhombated enneazetton (acronym: srene) (Jonathan Bowers)^[1]

Coordinates

The Cartesian coordinates of the vertices of the cantellated 8-simplex can be most simply positioned in 9-space as permutations of (0,0,0,0,0,0,1,1,2). This construction is based on facets of the cantellated 9-orthoplex.

Images

orthographic projections
A_k Coxeter plane	A₈	A₇	A₆	A₅
Graph
Dihedral symmetry	[9]	[8]	[7]	[6]
A_k Coxeter plane	A₄	A₃	A₂
Graph
Dihedral symmetry	[5]	[4]	[3]

Bicantellated 8-simplex

Bicantellated 8-simplex
Type	uniform 8-polytope
Schläfli symbol	r2r{3,3,3,3,3,3,3}
Coxeter-Dynkin diagram
7-faces
6-faces
5-faces
4-faces
Cells
Faces
Edges	5292
Vertices	756
Vertex figure
Coxeter group	A₈, [3⁷], order 362880
Properties	convex

Alternate names

Small birhombated enneazetton (acronym: sabrene) (Jonathan Bowers)^[2]

Coordinates

The Cartesian coordinates of the vertices of the bicantellated 8-simplex can be most simply positioned in 9-space as permutations of (0,0,0,0,0,1,1,2,2). This construction is based on facets of the bicantellated 9-orthoplex.

Images

orthographic projections
A_k Coxeter plane	A₈	A₇	A₆	A₅
Graph
Dihedral symmetry	[9]	[8]	[7]	[6]
A_k Coxeter plane	A₄	A₃	A₂
Graph
Dihedral symmetry	[5]	[4]	[3]

Tricantellated 8-simplex

tricantellated 8-simplex
Type	uniform 8-polytope
Schläfli symbol	r3r{3,3,3,3,3,3,3}
Coxeter-Dynkin diagram
7-faces
6-faces
5-faces
4-faces
Cells
Faces
Edges	8820
Vertices	1260
Vertex figure
Coxeter group	A₈, [3⁷], order 362880
Properties	convex

Alternate names

Small trirhombihexadecaexon (acronym: satrene) (Jonathan Bowers)^[3]

Coordinates

The Cartesian coordinates of the vertices of the tricantellated 8-simplex can be most simply positioned in 9-space as permutations of (0,0,0,0,0,1,1,2,2). This construction is based on facets of the tricantellated 9-orthoplex.

Images

orthographic projections
A_k Coxeter plane	A₈	A₇	A₆	A₅
Graph
Dihedral symmetry	[9]	[8]	[7]	[6]
A_k Coxeter plane	A₄	A₃	A₂
Graph
Dihedral symmetry	[5]	[4]	[3]

Cantitruncated 8-simplex

Cantitruncated 8-simplex
Type	uniform 8-polytope
Schläfli symbol	tr{3,3,3,3,3,3,3}
Coxeter-Dynkin diagram
7-faces
6-faces
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure
Coxeter group	A₈, [3⁷], order 362880
Properties	convex

Alternate names

Great rhombated enneazetton (acronym: grene) (Jonathan Bowers)^[4]

Coordinates

The Cartesian coordinates of the vertices of the cantitruncated 8-simplex can be most simply positioned in 9-space as permutations of (0,0,0,0,0,0,1,2,3). This construction is based on facets of the bicantitruncated 9-orthoplex.

Images

orthographic projections
A_k Coxeter plane	A₈	A₇	A₆	A₅
Graph
Dihedral symmetry	[9]	[8]	[7]	[6]
A_k Coxeter plane	A₄	A₃	A₂
Graph
Dihedral symmetry	[5]	[4]	[3]

Bicantitruncated 8-simplex

Bicantitruncated 8-simplex
Type	uniform 8-polytope
Schläfli symbol	t2r{3,3,3,3,3,3,3}
Coxeter-Dynkin diagram
7-faces
6-faces
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure
Coxeter group	A₈, [3⁷], order 362880
Properties	convex

Alternate names

Great birhombated enneazetton (acronym: gabrene) (Jonathan Bowers)^[5]

Coordinates

The Cartesian coordinates of the vertices of the bicantitruncated 8-simplex can be most simply positioned in 9-space as permutations of (0,0,0,0,0,1,2,3,3). This construction is based on facets of the bicantitruncated 9-orthoplex.

Images

orthographic projections
A_k Coxeter plane	A₈	A₇	A₆	A₅
Graph
Dihedral symmetry	[9]	[8]	[7]	[6]
A_k Coxeter plane	A₄	A₃	A₂
Graph
Dihedral symmetry	[5]	[4]	[3]

Tricantitruncated 8-simplex

Tricantitruncated 8-simplex
Type	uniform 8-polytope
Schläfli symbol	t3r{3,3,3,3,3,3,3}
Coxeter-Dynkin diagram
7-faces
6-faces
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure
Coxeter group	A₈, [3⁷], order 362880
Properties	convex

Great trirhombated enneazetton (acronym: gatrene) (Jonathan Bowers)^[6]

Coordinates

The Cartesian coordinates of the vertices of the tricantitruncated 8-simplex can be most simply positioned in 9-space as permutations of (0,0,0,0,1,2,3,3,3). This construction is based on facets of the bicantitruncated 9-orthoplex.

Images

orthographic projections
A_k Coxeter plane	A₈	A₇	A₆	A₅
Graph
Dihedral symmetry	[9]	[8]	[7]	[6]
A_k Coxeter plane	A₄	A₃	A₂
Graph
Dihedral symmetry	[5]	[4]	[3]

Related polytopes

The six presented polytopes are in the family of 135 uniform 8-polytopes with A₈ symmetry.

A8 polytopes
t₀	t₁	t₂	t₃	t₀₁	t₀₂	t₁₂	t₀₃	t₁₃	t₂₃	t₀₄	t₁₄	t₂₄	t₃₄	t₀₅
t₁₅	t₂₅	t₀₆	t₁₆	t₀₇	t₀₁₂	t₀₁₃	t₀₂₃	t₁₂₃	t₀₁₄	t₀₂₄	t₁₂₄	t₀₃₄	t₁₃₄	t₂₃₄
t₀₁₅	t₀₂₅	t₁₂₅	t₀₃₅	t₁₃₅	t₂₃₅	t₀₄₅	t₁₄₅	t₀₁₆	t₀₂₆	t₁₂₆	t₀₃₆	t₁₃₆	t₀₄₆	t₀₅₆
t₀₁₇	t₀₂₇	t₀₃₇	t₀₁₂₃	t₀₁₂₄	t₀₁₃₄	t₀₂₃₄	t₁₂₃₄	t₀₁₂₅	t₀₁₃₅	t₀₂₃₅	t₁₂₃₅	t₀₁₄₅	t₀₂₄₅	t₁₂₄₅
t₀₃₄₅	t₁₃₄₅	t₂₃₄₅	t₀₁₂₆	t₀₁₃₆	t₀₂₃₆	t₁₂₃₆	t₀₁₄₆	t₀₂₄₆	t₁₂₄₆	t₀₃₄₆	t₁₃₄₆	t₀₁₅₆	t₀₂₅₆	t₁₂₅₆
t₀₃₅₆	t₀₄₅₆	t₀₁₂₇	t₀₁₃₇	t₀₂₃₇	t₀₁₄₇	t₀₂₄₇	t₀₃₄₇	t₀₁₅₇	t₀₂₅₇	t₀₁₆₇	t₀₁₂₃₄	t₀₁₂₃₅	t₀₁₂₄₅	t₀₁₃₄₅
t₀₂₃₄₅	t₁₂₃₄₅	t₀₁₂₃₆	t₀₁₂₄₆	t₀₁₃₄₆	t₀₂₃₄₆	t₁₂₃₄₆	t₀₁₂₅₆	t₀₁₃₅₆	t₀₂₃₅₆	t₁₂₃₅₆	t₀₁₄₅₆	t₀₂₄₅₆	t₀₃₄₅₆	t₀₁₂₃₇
t₀₁₂₄₇	t₀₁₃₄₇	t₀₂₃₄₇	t₀₁₂₅₇	t₀₁₃₅₇	t₀₂₃₅₇	t₀₁₄₅₇	t₀₁₂₆₇	t₀₁₃₆₇	t₀₁₂₃₄₅	t₀₁₂₃₄₆	t₀₁₂₃₅₆	t₀₁₂₄₅₆	t₀₁₃₄₅₆	t₀₂₃₄₅₆
t₁₂₃₄₅₆	t₀₁₂₃₄₇	t₀₁₂₃₅₇	t₀₁₂₄₅₇	t₀₁₃₄₅₇	t₀₂₃₄₅₇	t₀₁₂₃₆₇	t₀₁₂₄₆₇	t₀₁₃₄₆₇	t₀₁₂₅₆₇	t_0123456	t_0123457	t_0123467	t_0123567	t_01234567

Notes

^ Klitizing, (x3o3x3o3o3o3o3o - srene)
^ Klitizing, (o3x3o3x3o3o3o3o - sabrene)
^ Klitizing, (o3o3x3o3x3o3o3o - satrene)
^ Klitizing, (x3x3x3o3o3o3o3o - grene)
^ Klitizing, (o3x3x3x3o3o3o3o - gabrene)
^ Klitizing, (o3o3x3x3x3o3o3o - gatrene)

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, wiley.com, 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. "8D uniform polytopes (polyzetta)". x3o3x3o3o3o3o3o - srene, o3x3o3x3o3o3o3o - sabrene, o3o3x3o3x3o3o3o - satrene, x3x3x3o3o3o3o3o - grene, o3x3x3x3o3o3o3o - gabrene, o3o3x3x3x3o3o3o - gatrene

External links

v t e Fundamental convex regular and uniform polytopes in dimensions 2–10
Family	$A n$	$B n$	$I 2 (p)$ / $D n$	$E 6$ / $E 7$ / $E 8$ / $F 4$ / $G 2$	$H n$
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	1_k2 • 2_k1 • k₂₁	n-pentagonal polytope
Topics: Polytope families • Regular polytope • List of regular polytopes and compounds • Polytope operations

[1] Klitizing, (x3o3x3o3o3o3o3o - srene)

[2] Klitizing, (o3x3o3x3o3o3o3o - sabrene)

[3] Klitizing, (o3o3x3o3x3o3o3o - satrene)

[4] Klitizing, (x3x3x3o3o3o3o3o - grene)

[5] Klitizing, (o3x3x3x3o3o3o3o - gabrene)

[6] Klitizing, (o3o3x3x3x3o3o3o - gatrene)

[1]

[2]

[3]

[4]

[5]

[6]