Pentellated 8-simplex

Orthogonal projections in A₈ Coxeter plane
8-simplex	Pentellated 8-simplex	Bipentitruncated 8-simplex

In eight-dimensional geometry, a pentellated 8-simplex is a convex uniform 8-polytope with 5th order truncations of the regular 8-simplex.

There are two unique pentellations of the 8-simplex. Including truncations, canetellations, runcinations, and sterications, there are 32 more pentallations. These polytopes are a part of a family 135 uniform 8-polytopes with A₈ symmetry. A₈, [3⁸] has order 9 factorial symmetry, or 362880. The bipentalled form is symmetrically ringed, doubling the symmetry order to 725760, and is represented the double-bracketed group [[3⁸]]. The A₈ Coxeter plane projection shows order [9] symmetry for the pentellated 8-simplex, while the bipentellated 8-simple is doubled to [18] symmetry.

Contents

1 Pentellated 8-simplex
- 1.1 Coordinates
- 1.2 Images
2 Bipentellated 8-simplex
- 2.1 Coordinates
- 2.2 Images
3 Related polytopes
4 Notes
5 References
6 External links

Pentellated 8-simplex

Pentellated 8-simplex
Type	uniform polyzetton
Schläfli symbol	t_0,5{3,3,3,3,3,3,3}
Coxeter-Dynkin diagrams
7-faces
6-faces
5-faces
4-faces
Cells
Faces
Edges	5040
Vertices	504
Vertex figure
Coxeter group	A₈, [3⁷], order 362880
Properties	convex

Coordinates

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

Bipentellated 8-simplex

Bipentellated 8-simplex
Type	uniform polyzetton
Schläfli symbol	t_1,6{3,3,3,3,3,3,3}
Coxeter-Dynkin diagrams
7-faces	t_0,5{3,3,3,3,3,3}
6-faces
5-faces
4-faces
Cells
Faces
Edges	7560
Vertices	756
Vertex figure
Coxeter group	A₈, [[3⁷]], order 725760
Properties	convex, facet-transitive

Coordinates

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

This polytope is one of 135 uniform 8-polytopes with A₈ symmetry.

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

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, editied by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6 [1]
  - (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, 8D, uniform polytopes (polyzetta) x3o3o3o3o3x3o3o, o3x3o3o3o3o3x3o

External links

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


Family	A_n	BC_n	D_n	E₆ / E₇ / E₈ / F₄ / G₂	H_n
Regular polygon	Triangle	Square		Hexagon	Pentagon
Uniform polyhedron	Tetrahedron	Octahedron • Cube	Demicube		Dodecahedron • Icosahedron
Uniform polychoron	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
n-polytopes	n-simplex	n-orthoplex • n-cube	n-demicube	1_k2 • 2_k1 • k₂₁	pentagonal polytope
Topics: Polytope families • Regular polytope • List of regular polytopes