Stericated 8-simplexes

Orthogonal projections in A₈ Coxeter plane
8-simplex	Stericated 8-simplex	Bistericated 8-simplex
Steritruncated 8-simplex	Bisteritruncated 8-simplex	Stericantellated 8-simplex	Bisteri-cantellated 8-simplex
Stericanti-truncated 8-simplex	Bistericanti-truncated 8-simplex	Steri-runcinated 8-simplex	Bisteri-runcinated 8-simplex
Sterirunci-truncated 8-simplex	Bisterirunci-truncated 8-simplex	Sterirunci-cantellated 8-simplex	Bisterirunci-cantellated 8-simplex
Steriruncicanti-truncated 8-simplex	Bisteriruncicanti-truncated 8-simplex

In eight-dimensional geometry, a stericated 8-simplex is a convex uniform 8-polytope with 4th order truncations (sterication) of the regular 8-simplex. There are 16 unique sterications for the 8-simplex including permutations of truncation, cantellation, and runcination.

Stericated 8-simplex

Stericated 8-simplex
Type	uniform 8-polytope
Schläfli symbol	t_0,4{3,3,3,3,3,3,3}
Coxeter-Dynkin diagrams
7-faces
6-faces
5-faces
4-faces
Cells
Faces
Edges	6300
Vertices	630
Vertex figure
Coxeter group	A₈, [3⁷], order 362880
Properties	convex

Acronym: secane (Jonathan Bowers)^[1]

Coordinates

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

Bistericated 8-simplex

Bistericated 8-simplex
Type	uniform 8-polytope
Schläfli symbol	t_1,5{3,3,3,3,3,3,3}
Coxeter-Dynkin diagrams
7-faces
6-faces
5-faces
4-faces
Cells
Faces
Edges	12600
Vertices	1260
Vertex figure
Coxeter group	A₈, [3⁷], order 362880
Properties	convex

Acronym: sobcane (Jonathan Bowers)^[2]

Coordinates

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

Steritruncated 8-simplex

Steritruncated 8-simplex
Type	uniform 8-polytope
Schläfli symbol	t_0,1,4{3,3,3,3,3,3,3}
Coxeter-Dynkin diagrams
7-faces
6-faces
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure
Coxeter group	A₈, [3⁷], order 362880
Properties	convex

Acronym: catene (Jonathan Bowers)^[3]

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]

Bisteritruncated 8-simplex

Bisteritruncated 8-simplex
Type	uniform 8-polytope
Schläfli symbol	t_1,2,5{3,3,3,3,3,3,3}
Coxeter-Dynkin diagrams
7-faces
6-faces
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure
Coxeter group	A₈, [3⁷], order 362880
Properties	convex

Acronym: bictane (Jonathan Bowers)^[4]

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]

Stericantellated 8-simplex

Acronym: crane (Jonathan Bowers)^[5]

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]

Bistericantellated 8-simplex

Acronym: bocrane (Jonathan Bowers)^[6]

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]

Stericantitruncated 8-simplex

Acronym: cograne (Jonathan Bowers)^[7]

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]

Bistericantitruncated 8-simplex

Acronym: bocagrane (Jonathan Bowers)^[8]

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]

Steriruncinated 8-simplex

Acronym: capene (Jonathan Bowers)^[9]

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]

Bisteriruncinated 8-simplex

Acronym: bacpane (Jonathan Bowers)^[10]

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]

Steriruncitruncated 8-simplex

Acronym: coptane (Jonathan Bowers)^[11]

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]

Bisteriruncitruncated 8-simplex

Acronym: bicpotane (Jonathan Bowers)^[12]

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]

Steriruncicantellated 8-simplex

Acronym: coprene (Jonathan Bowers)^[13]

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]

Bisteriruncicantellated 8-simplex

Acronym: bicprene (Jonathan Bowers)^[14]

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]

Steriruncicantitruncated 8-simplex

Acronym: gacene (Jonathan Bowers)^[15]

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]

Bisteriruncicantitruncated 8-simplex

Acronym: gobcane (Jonathan Bowers)^[16]

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 16 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

^ Klitzing, (x3o3o3o3x3o3o3o - secane)
^ Klitzing, (o3x3o3o3o3x3o3o - sobcane)
^ Klitzing, (x3x3o3o3x3o3o3o - catene)
^ Klitzing, (o3x3x3o3o3x3o3o - bictane)
^ Klitzing, (x3o3x3o3x3o3o3o - crane)
^ Klitzing, (o3x3o3x3o3x3o3o - bocrane)
^ Klitzing, (x3x3x3o3x3o3o3o - cograne)
^ Klitzing, (o3x3x3x3ox3o3o3 - bocagrane)
^ Klitzing, (x3o3o3x3x3o3o3o - capene)
^ Klitzing, (o3x3o3o3x3x3o3o - bacpane)
^ Klitzing, (x3x3o3x3x3o3o3o - coptane)
^ Klitzing, (o3x3x3o3x3x3o3o - bicpotane)
^ Klitzing, (x3o3x3x3x3o3o3o - coprene)
^ Klitzing, (o3x3o3x3x3x3o3o - bicprene)
^ Klitzing, (x3x3x3x3x3o3o3o - gacene)
^ Klitzing, (o3x3x3x3x3x3o3o - gobcane)

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)". x3o3o3o3x3o3o3o - secane, o3x3o3o3o3x3o3o - sobcane, x3x3o3o3x3o3o3o - catene, o3x3x3o3o3x3o3o - bictane, x3o3x3o3x3o3o3o - crane, o3x3o3x3o3x3o3o - bocrane, x3x3x3o3x3o3o3o - cograne, o3x3x3x3ox3o3o3 - bocagrane, x3o3o3x3x3o3o3o - capene, o3x3o3o3x3x3o3o - bacpane, x3x3o3x3x3o3o3o - coptane, o3x3x3o3x3x3o3o - bicpotane, x3o3x3x3x3o3o3o - coprene, o3x3o3x3x3x3o3o - bicprene, x3x3x3x3x3o3o3o - gacene, o3x3x3x3x3x3o3o - gobcane

External links

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] Klitzing, (x3o3o3o3x3o3o3o - secane)

[2] Klitzing, (o3x3o3o3o3x3o3o - sobcane)

[3] Klitzing, (x3x3o3o3x3o3o3o - catene)

[4] Klitzing, (o3x3x3o3o3x3o3o - bictane)

[5] Klitzing, (x3o3x3o3x3o3o3o - crane)

[6] Klitzing, (o3x3o3x3o3x3o3o - bocrane)

[7] Klitzing, (x3x3x3o3x3o3o3o - cograne)

[8] Klitzing, (o3x3x3x3ox3o3o3 - bocagrane)

[9] Klitzing, (x3o3o3x3x3o3o3o - capene)

[10] Klitzing, (o3x3o3o3x3x3o3o - bacpane)

[11] Klitzing, (x3x3o3x3x3o3o3o - coptane)

[12] Klitzing, (o3x3x3o3x3x3o3o - bicpotane)

[13] Klitzing, (x3o3x3x3x3o3o3o - coprene)

[14] Klitzing, (o3x3o3x3x3x3o3o - bicprene)

[15] Klitzing, (x3x3x3x3x3o3o3o - gacene)

[16] Klitzing, (o3x3x3x3x3x3o3o - gobcane)

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