Pentellated 8-simplexes

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

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, cantellations, runcinations, and sterications, there are 32 more pentellations. These polytopes are a part of a family 135 uniform 8-polytopes with A₈ symmetry. A₈, [3⁷] 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 [[3⁷]]. The A₈ Coxeter plane projection shows order [9] symmetry for the pentellated 8-simplex, while the bipentellated 8-simple is doubled to [18] symmetry.

Pentellated 8-simplex

Pentellated 8-simplex
Type	uniform 8-polytope
Schläfli symbol	t0,5{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

The Cartesian coordinates of the vertices of the pentellated 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 pentellated 9-orthoplex.

Images

orthographic projections

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

Bipentellated 8-simplex

Bipentellated 8-simplex
Type	uniform 8-polytope
Schläfli symbol	t1,6{3,3,3,3,3,3,3}
Coxeter-Dynkin diagrams
7-faces	t0,5{3,3,3,3,3,3}
6-faces
5-faces
4-faces
Cells
Faces
Edges	7560
Vertices	756
Vertex figure
Coxeter group	A8×2, [[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 9-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

Ak Coxeter plane	A8	A7	A6	A5
Graph
Dihedral symmetry	[[9]] = [18]	[8]	[[7]] = [14]	[6]
Ak 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.

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 [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.
• Klitzing, Richard. "8D uniform polytopes (polyzetta)". x3o3o3o3o3x3o3o, o3x3o3o3o3o3x3o

External links

• Polytopes of Various Dimensions
• Multi-dimensional Glossary