Heptellated 8-simplexes

8-simplex	Heptellated 8-simplex	Heptihexipentisteriruncicantitruncated 8-simplex (Omnitruncated 8-simplex)
Orthogonal projections in A8 Coxeter plane (A7 for omnitruncation)

In eight-dimensional geometry, a heptellated 8-simplex is a convex uniform 8-polytope, including 7th-order truncations (heptellation) from the regular 8-simplex.

There are 35 unique heptellations for the 8-simplex, including all permutations of truncations, cantellations, runcinations, sterications, pentellations, and hexications. The simplest heptellated 8-simplex is also called an expanded 8-simplex, with only the first and last nodes ringed, is constructed by an expansion operation applied to the regular 8-simplex. The highest form, the heptihexipentisteriruncicantitruncated 8-simplex is more simply called a omnitruncated 8-simplex with all of the nodes ringed.

Heptellated 8-simplex

Heptellated 8-simplex
Type	uniform 8-polytope
Schläfli symbol	t0,7{3,3,3,3,3,3,3}
Coxeter-Dynkin diagrams
7-faces
6-faces
5-faces
4-faces
Cells
Faces
Edges	504
Vertices	72
Vertex figure	6-simplex antiprism
Coxeter group	A8×2, [[37]], order 725760
Properties	convex

Alternate names

• Expanded 8-simplex
• Small exated enneazetton (soxeb) (Jonathan Bowers)

The vertices of the heptellated 8-simplex can bepositioned in 8-space as permutations of (0,1,1,1,1,1,1,1,2). This construction is based on facets of the heptellated 9-orthoplex.

A second construction in 9-space, from the center of a rectified 9-orthoplex is given by coordinate permutations of:

(1,-1,0,0,0,0,0,0,0)

Root vectors

Its 72 vertices represent the root vectors of the simple Lie group A₈.

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]

Omnitruncated 8-simplex

Omnitruncated 8-simplex
Type	uniform 8-polytope
Schläfli symbol	t0,1,2,3,4,5,6,7{37}
Coxeter-Dynkin diagrams
7-faces
6-faces
5-faces
4-faces
Cells
Faces
Edges	1451520
Vertices	362880
Vertex figure	irr. 7-simplex
Coxeter group	A8, [[37]], order 725760
Properties	convex

The symmetry order of an omnitruncated 8-simplex is 725760. The symmetry of a family of a uniform polytopes is equal to the number of vertices of the omnitruncation, being 362880 (9 factorial) in the case of the omnitruncated 8-simplex; but when the CD symbol is palindromic, the symmetry order is doubled, 725760 here, because the element corresponding to any element of the underlying 8-simplex can be exchanged with one of those corresponding to an element of its dual.

Alternate names

• Heptihexipentisteriruncicantitruncated 8-simplex
• Great exated enneazetton (goxeb) (Jonathan Bowers)

Coordinates

The Cartesian coordinates of the vertices of the omnitruncated 8-simplex can be most simply positioned in 9-space as permutations of (0,1,2,3,4,5,6,7,8). This construction is based on facets of the heptihexipentisteriruncicantitruncated 9-orthoplex, t_{0,1,2,3,4,5,6,7}{3⁷,4}

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]

Permutohedron and related tessellation

The omnitruncated 8-simplex is the permutohedron of order 9. The omnitruncated 8-simplex is a zonotope, the Minkowski sum of nine line segments parallel to the nine lines through the origin and the nine vertices of the 8-simplex.

Like all uniform omnitruncated n-simplices, the omnitruncated 8-simplex can tessellate space by itself, in this case 8-dimensional space with three facets around each ridge. It has Coxeter-Dynkin diagram of .

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)". x3o3o3o3o3o3o3x - soxeb, x3x3x3x3x3x3x3x - goxeb

External links

• Polytopes of Various Dimensions
• Multi-dimensional Glossary