Runcinated 8-simplexes

In eight-dimensional geometry, a runcinated 8-simplex is a convex uniform 8-polytope with 3rd order truncations (runcination) of the regular 8-simplex.

Orthogonal projections in A₈ Coxeter plane
8-simplex	Runcinated 8-simplex	Biruncinated 8-simplex	Triruncinated 8-simplex
Runcitruncated 8-simplex	Biruncitruncated 8-simplex	Triruncitruncated 8-simplex	Runcicantellated 8-simplex
Biruncicantellated 8-simplex	Runcicantitruncated 8-simplex	Biruncicantitruncated 8-simplex	Triruncicantitruncated 8-simplex

There are eleven unique runcinations of the 8-simplex, including permutations of truncation and cantellation. The triruncinated 8-simplex and triruncicantitruncated 8-simplex have a doubled symmetry, showing [18] order reflectional symmetry in the A₈ Coxeter plane.

Runcinated 8-simplex

Runcinated 8-simplex
Type	uniform 8-polytope
Schläfli symbol	t_0,3{3,3,3,3,3,3,3}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges	4536
Vertices	504
Vertex figure
Coxeter group	A₈, [3⁷], order 362880
Properties	convex

Alternate names

Runcinated enneazetton
Small prismated enneazetton (Acronym: spene) (Jonathan Bowers)[1]

Coordinates

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

Biruncinated 8-simplex

Biruncinated 8-simplex
Type	uniform 8-polytope
Schläfli symbol	t_1,4{3,3,3,3,3,3,3}
Coxeter-Dynkin diagram
7-faces
6-faces
5-faces
4-faces
Cells
Faces
Edges	11340
Vertices	1260
Vertex figure
Coxeter group	A₈, [3⁷], order 362880
Properties	convex

Alternate names

Biruncinated enneazetton
Small biprismated enneazetton (Acronym: sabpene) (Jonathan Bowers)[2]

Coordinates

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

Triruncinated 8-simplex

Triruncinated 8-simplex
Type	uniform 8-polytope
Schläfli symbol	t_2,5{3,3,3,3,3,3,3}
Coxeter-Dynkin diagrams
7-faces
6-faces
5-faces
4-faces
Cells
Faces
Edges	15120
Vertices	1680
Vertex figure
Coxeter group	A₈×2, [[3⁷]], order 725760
Properties	convex

Alternate names

Triruncinated enneazetton
Small triprismated enneazetton (Acronym: satpeb) (Jonathan Bowers)[3]

Coordinates

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

Images

orthographic projections
A_k Coxeter plane	A₈	A₇	A₆	A₅
Graph
Dihedral symmetry	[[9]] = [18]	[8]	[[7]] = [14]	[6]
A_k Coxeter plane	A₄	A₃	A₂
Graph
Dihedral symmetry	[[5]] = [10]	[4]	[[3]] = [6]

Runcitruncated 8-simplex

Images

orthographic projections
A_k Coxeter plane	A₈	A₇	A₆	A₅
Graph
Dihedral symmetry	[[9]] = [18]	[8]	[[7]] = [14]	[6]
A_k Coxeter plane	A₄	A₃	A₂
Graph
Dihedral symmetry	[[5]] = [10]	[4]	[[3]] = [6]

Biruncitruncated 8-simplex

Images

orthographic projections
A_k Coxeter plane	A₈	A₇	A₆	A₅
Graph
Dihedral symmetry	[[9]] = [18]	[8]	[[7]] = [14]	[6]
A_k Coxeter plane	A₄	A₃	A₂
Graph
Dihedral symmetry	[[5]] = [10]	[4]	[[3]] = [6]

Triruncitruncated 8-simplex

Images

orthographic projections
A_k Coxeter plane	A₈	A₇	A₆	A₅
Graph
Dihedral symmetry	[[9]] = [18]	[8]	[[7]] = [14]	[6]
A_k Coxeter plane	A₄	A₃	A₂
Graph
Dihedral symmetry	[[5]] = [10]	[4]	[[3]] = [6]

Runcicantellated 8-simplex

Images

orthographic projections
A_k Coxeter plane	A₈	A₇	A₆	A₅
Graph
Dihedral symmetry	[[9]] = [18]	[8]	[[7]] = [14]	[6]
A_k Coxeter plane	A₄	A₃	A₂
Graph
Dihedral symmetry	[[5]] = [10]	[4]	[[3]] = [6]

Biruncicantellated 8-simplex

Images

orthographic projections
A_k Coxeter plane	A₈	A₇	A₆	A₅
Graph
Dihedral symmetry	[[9]] = [18]	[8]	[[7]] = [14]	[6]
A_k Coxeter plane	A₄	A₃	A₂
Graph
Dihedral symmetry	[[5]] = [10]	[4]	[[3]] = [6]

Runcicantitruncated 8-simplex

Images

orthographic projections
A_k Coxeter plane	A₈	A₇	A₆	A₅
Graph
Dihedral symmetry	[[9]] = [18]	[8]	[[7]] = [14]	[6]
A_k Coxeter plane	A₄	A₃	A₂
Graph
Dihedral symmetry	[[5]] = [10]	[4]	[[3]] = [6]

Biruncicantitruncated 8-simplex

Images

orthographic projections
A_k Coxeter plane	A₈	A₇	A₆	A₅
Graph
Dihedral symmetry	[[9]] = [18]	[8]	[[7]] = [14]	[6]
A_k Coxeter plane	A₄	A₃	A₂
Graph
Dihedral symmetry	[[5]] = [10]	[4]	[[3]] = [6]

Triruncicantitruncated 8-simplex

Images

orthographic projections
A_k Coxeter plane	A₈	A₇	A₆	A₅
Graph
Dihedral symmetry	[[9]] = [18]	[8]	[[7]] = [14]	[6]
A_k Coxeter plane	A₄	A₃	A₂
Graph
Dihedral symmetry	[[5]] = [10]	[4]	[[3]] = [6]

Related polytopes

This polytope is one 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

Klitzing (x3o3o3x3o3o3o3o - spene)
Klitzing (o3x3o3o3x3o3o3o - sabpene)
Klitzing (o3o3x3o3o3x3o3o - satpeb)

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. "8D uniform polytopes (polyzetta)". x3o3o3x3o3o3o3o - spene, o3x3o3o3x3o3o3o - sabpene, o3o3x3o3o3x3o3o - satpeb

External links

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 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

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[1] Klitzing (x3o3o3x3o3o3o3o - spene)

[2] Klitzing (o3x3o3o3x3o3o3o - sabpene)

[3] Klitzing (o3o3x3o3o3x3o3o - satpeb)