Heptellated 8-simplexes#Heptellated 8-simplex

• Expanded 8-simplex
• Small exated enneazetton (Acronym: soxeb) (Jonathan Bowers){{sfn|Klitzing|at=[https://bendwavy.org/klitzing/incmats/soxeb.htm (x3o3o3o3o3o3o3x - soxeb)]}}{{efn|Name of soxeb is different than that in the source, which begins with "Small exiated ...". It may seem to be incorrect, but it is the source that has a typo. The word "exiated" is inconsistent with the rule for creating names of this type. For instance:

Polypeton → pet-on → pet-ated. Suffix "on" is replaced by "ated", see e.g. [https://bendwavy.org/klitzing/explain/polytope-tree.htm Klitzing – Polytopes]}}

The vertices of the heptellated 8-simplex can be positioned 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)

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

{{8-simplex2 Coxeter plane graphs|t07|120}}

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.

• Heptihexipentisteriruncicantitruncated 8-simplex
• Great exated enneazetton (Acronym: goxeb) (Jonathan Bowers){{sfn|Klitzing|at=[https://bendwavy.org/klitzing/incmats/goxeb.htm (x3x3x3x3x3x3x3x - goxeb)]}}

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}

{{8-simplex2 Coxeter plane graphs|t01234567|120}}

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 {{CDD|branch_11|3ab|nodes_11|3ab|nodes_11|3ab|nodes_11|split2|node_1}}.

The two presented polytopes are selected from 135 uniform 8-polytopes with A₈ symmetry, shown in the table below.

{{Enneazetton family}}

{{reflist}}

{{notelist}}

• 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, [https://www.wiley.com/en-us/Kaleidoscopes-p-9780471010036 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.
• {{KlitzingPolytopes|polyzetta.htm|8D uniform polytopes (polyzetta) with acronyms}} x3o3o3o3o3o3o3x - soxeb, x3x3x3x3x3x3x3x - goxeb {{sfn whitelist|CITEREFKlitzing}}

• [https://web.archive.org/web/20070310205351/http://members.cox.net/hedrondude/topes.htm Polytopes of Various Dimensions]
• [http://tetraspace.alkaline.org/glossary.htm Multi-dimensional Glossary]

{{Polytopes}}

Category:8-polytopes