Omnitruncated 8-simplex

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

Alternate names

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

Coordinates

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

Omnitruncated 8-simplex

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)Klitzing, (x3x3x3x3x3x3x3x - goxeb)

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}

Images

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.

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, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995,

*** (Paper 22) H.S.M. Coxeter, ''Regular and Semi Regular Polytopes I'', ath. Zeit. 46 (1940) 380-407, MR 2,10*** (Paper 23) H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes II'', ath. Zeit. 188 (1985) 559-591*** (Paper 24) H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes III'', ath. Zeit. 200 (1988) 3-45* Norman Johnson ''Uniform Polytopes'', Manuscript (1991)
** N.W. Johnson: ''The Theory of Uniform Polytopes and Honeycombs'', Ph.D.
* x3o3o3o3o3o3o3x - soxeb, x3x3x3x3x3x3x3x - goxeb

External links

Polytopes of Various Dimensions



{{Polytopes

8-polytopes