In eight-dimensional geometry, a heptellated 8-simplex is a convex

uniform 8-polytope In Eight-dimensional space, eight-dimensional geometry, an eight-dimensional polytope or 8-polytope is a polytope contained by 7-polytope facets. Each 6-polytope Ridge (geometry), ridge being shared by exactly two 7-polytope Facet (mathematics), fa ...

, including 7th-order truncations (heptellation) from the regular

8-simplex In geometry, an 8-simplex is a self-dual regular 8-polytope. It has 9 vertices, 36 edges, 84 triangle faces, 126 tetrahedral cells, 126 5-cell 4-faces, 84 5-simplex 5-faces, 36 6-simplex 6-faces, and 9 7-simplex 7-faces. Its dihedral angle is co ...

. There are 35 unique heptellations for the 8-simplex, including all

permutation In mathematics, a permutation of a set is, loosely speaking, an arrangement of its members into a sequence or linear order, or if the set is already ordered, a rearrangement of its elements. The word "permutation" also refers to the act or proc ...

s of truncations, cantellations, runcinations, sterications, pentellations, and

hexication In geometry, a uniform polytope of dimension three or higher is a vertex-transitive polytope bounded by uniform facets. The uniform polytopes in two dimensions are the regular polygons (the definition is different in 2 dimensions to exclude vert ...

s. 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 Expansion may refer to: Arts, entertainment and media * ''L'Expansion'', a French monthly business magazine * ''Expansion'' (album), by American jazz pianist Dave Burrell, released in 2004 * ''Expansions'' (McCoy Tyner album), 1970 * ''Expansio ...

operation applied to the regular

. 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 In nine-dimensional geometry, a rectified 9-simplex is a convex uniform 9-polytope, being a rectification of the regular 9-orthoplex. There are 9 rectifications of the 9-orthoplex. Vertices of the rectified 9-orthoplex are located at the edge-cent ...

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 mathematics, the factorial of a non-negative denoted is the product of all positive integers less than or equal The factorial also equals the product of n with the next smaller factorial: \begin n! &= n \times (n-1) \times (n-2) \t ...

) 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 A ridge or a mountain ridge is a geographical feature consisting of a chain of mountains or hills that form a continuous elevated crest for an extended distance. The sides of the ridge slope away from the narrow top on either side. The line ...

. 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 Harold Scott MacDonald "Donald" Coxeter, (9 February 1907 – 31 March 2003) was a British and later also Canadian geometer. He is regarded as one of the greatest geometers of the 20th century. Biography Coxeter was born in Kensington t ...

: ** 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