In seven-dimensional geometry, a hexicated 7-simplex is a convex uniform 7-polytope, including 6th-order truncations (hexication) from the regular 7-simplex. There are 20 unique hexications for the 7-simplex, including all permutations of truncations, cantellations, runcinations, sterications, and pentellations. The simple hexicated 7-simplex is also called an expanded 7-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 7-simplex. The highest form, the ''hexipentisteriruncicantitruncated 7-simplex'' is more simply called a ''omnitruncated 7-simplex'' with all of the nodes ringed.

Hexicated 7-simplex

In seven-dimensional geometry, a hexicated 7-simplex is a convex uniform 7-polytope, a

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 ...

(6th order truncation) of the regular 7-simplex, or alternately can be seen as an

operation. Ammann-Beenker_tiling_example

Root vectors

Its 56 vertices represent the root vectors of the simple Lie group A₇.

Alternate names

* Expanded 7-simplex * Small petated hexadecaexon (acronym: suph) (Jonathan Bowers)

Coordinates

The vertices of the ''hexicated 7-simplex'' can be most simply positioned in 8-space as permutations of (0,1,1,1,1,1,1,2). This construction is based on facets of the hexicated 8-orthoplex, . A second construction in 8-space, from the center of a

rectified 8-orthoplex In eight-dimensional geometry, a rectified 8-orthoplex is a convex uniform 8-polytope, being a rectification of the regular 8-orthoplex. There are unique 8 degrees of rectifications, the zeroth being the 8-orthoplex, and the 7th and last being th ...

is given by coordinate permutations of: : (1,-1,0,0,0,0,0,0)

Images

Hexitruncated 7-simplex

Alternate names

* Petitruncated octaexon (acronym: puto) (Jonathan Bowers)

Coordinates

The vertices of the ''hexitruncated 7-simplex'' can be most simply positioned in 8-space as permutations of (0,1,1,1,1,1,2,3). This construction is based on facets of the hexitruncated 8-orthoplex, .

Images

Hexicantellated 7-simplex

Alternate names

* Petirhombated octaexon (acronym: puro) (Jonathan Bowers)

Coordinates

The vertices of the ''hexicantellated 7-simplex'' can be most simply positioned in 8-space as permutations of (0,1,1,1,1,2,2,3). This construction is based on facets of the hexicantellated 8-orthoplex, .

Images

Hexiruncinated 7-simplex

Alternate names

* Petiprismated hexadecaexon (acronym: puph) (Jonathan Bowers)

Coordinates

The vertices of the ''hexiruncinated 7-simplex'' can be most simply positioned in 8-space as permutations of (0,1,1,1,2,2,2,3). This construction is based on facets of the hexiruncinated 8-orthoplex, .

Images

Hexicantitruncated 7-simplex

Alternate names

* Petigreatorhombated octaexon (acronym: pugro) (Jonathan Bowers)

Coordinates

The vertices of the ''hexicantitruncated 7-simplex'' can be most simply positioned in 8-space as permutations of (0,1,1,1,1,2,3,4). This construction is based on facets of the hexicantitruncated 8-orthoplex, .

Images

Hexiruncitruncated 7-simplex

Alternate names

* Petiprismatotruncated octaexon (acronym: pupato) (Jonathan Bowers)

Coordinates

The vertices of the ''hexiruncitruncated 7-simplex'' can be most simply positioned in 8-space as permutations of (0,1,1,1,2,2,3,4). This construction is based on facets of the hexiruncitruncated 8-orthoplex, .

Images

Hexiruncicantellated 7-simplex

In seven-dimensional geometry, a hexiruncicantellated 7-simplex is a uniform 7-polytope.

Alternate names

* Petiprismatorhombated octaexon (acronym: pupro) (Jonathan Bowers)

Coordinates

The vertices of the ''hexiruncicantellated 7-simplex'' can be most simply positioned in 8-space as permutations of (0,1,1,1,2,3,3,4). This construction is based on facets of the hexiruncicantellated 8-orthoplex, .

Images

Hexisteritruncated 7-simplex

Alternate names

* Peticellitruncated octaexon (acronym: pucto) (Jonathan Bowers)

Coordinates

The vertices of the ''hexisteritruncated 7-simplex'' can be most simply positioned in 8-space as permutations of (0,1,1,2,2,2,3,4). This construction is based on facets of the hexisteritruncated 8-orthoplex, .

Images

Hexistericantellated 7-simplex

Alternate names

* Peticellirhombihexadecaexon (acronym: pucroh) (Jonathan Bowers)

Coordinates

The vertices of the ''hexistericantellated 7-simplex'' can be most simply positioned in 8-space as permutations of (0,1,1,2,2,3,3,4). This construction is based on facets of the hexistericantellated 8-orthoplex, .

Images

Hexipentitruncated 7-simplex

Alternate names

* Petiteritruncated hexadecaexon (acronym: putath) (Jonathan Bowers)

Coordinates

The vertices of the ''hexipentitruncated 7-simplex'' can be most simply positioned in 8-space as permutations of (0,1,2,2,2,2,3,4). This construction is based on facets of the hexipentitruncated 8-orthoplex, .

Images

Hexiruncicantitruncated 7-simplex

Alternate names

* Petigreatoprismated octaexon (acronym: pugopo) (Jonathan Bowers)

Coordinates

The vertices of the ''hexiruncicantitruncated 7-simplex'' can be most simply positioned in 8-space as permutations of (0,1,1,2,2,3,4,5). This construction is based on facets of the hexiruncicantitruncated 8-orthoplex, .

Images

Hexistericantitruncated 7-simplex

Alternate names

* Peticelligreatorhombated octaexon (acronym: pucagro) (Jonathan Bowers)

Coordinates

The vertices of the ''hexistericantitruncated 7-simplex'' can be most simply positioned in 8-space as permutations of (0,1,1,2,2,3,4,5). This construction is based on facets of the hexistericantitruncated 8-orthoplex, .

Images

Hexisteriruncitruncated 7-simplex

Alternate names

* Peticelliprismatotruncated octaexon (acronym: pucpato) (Jonathan Bowers)

Coordinates

The vertices of the ''hexisteriruncitruncated 7-simplex'' can be most simply positioned in 8-space as permutations of (0,1,1,2,3,3,4,5). This construction is based on facets of the hexisteriruncitruncated 8-orthoplex, .

Images

Hexisteriruncicantellated 7-simplex

Alternate names

* Peticelliprismatorhombihexadecaexon (acronym: pucproh) (Jonathan Bowers)

Coordinates

The vertices of the ''hexisteriruncitruncated 7-simplex'' can be most simply positioned in 8-space as permutations of (0,1,1,2,3,4,4,5). This construction is based on facets of the hexisteriruncitruncated 8-orthoplex, .

Images

Hexipenticantitruncated 7-simplex

Alternate names

* Petiterigreatorhombated octaexon (acronym: putagro) (Jonathan Bowers)

Coordinates

The vertices of the ''hexipenticantitruncated 7-simplex'' can be most simply positioned in 8-space as permutations of (0,1,2,2,2,3,4,5). This construction is based on facets of the hexipenticantitruncated 8-orthoplex, .

Images

Hexipentiruncitruncated 7-simplex

Alternate names

* Petiteriprismatotruncated hexadecaexon (acronym: putpath) (Jonathan Bowers)

Coordinates

The vertices of the ''hexipentiruncitruncated 7-simplex'' can be most simply positioned in 8-space as permutations of (0,1,2,2,3,4,4,5). This construction is based on facets of the hexipentiruncitruncated 8-orthoplex, .

Images

Hexisteriruncicantitruncated 7-simplex

Alternate names

* Petigreatocellated octaexon (acronym: pugaco) (Jonathan Bowers)

Coordinates

The vertices of the ''hexisteriruncicantitruncated 7-simplex'' can be most simply positioned in 8-space as permutations of (0,1,1,2,3,4,5,6). This construction is based on facets of the hexisteriruncicantitruncated 8-orthoplex, .

Images

Hexipentiruncicantitruncated 7-simplex

Alternate names

* Petiterigreatoprismated octaexon (acronym: putgapo) (Jonathan Bowers)

Coordinates

The vertices of the ''hexipentiruncicantitruncated 7-simplex'' can be most simply positioned in 8-space as permutations of (0,1,2,2,3,4,5,6). This construction is based on facets of the hexipentiruncicantitruncated 8-orthoplex, .

Images

Hexipentistericantitruncated 7-simplex

Alternate names

* Petitericelligreatorhombihexadecaexon (acronym: putcagroh) (Jonathan Bowers)

Coordinates

The vertices of the ''hexipentistericantitruncated 7-simplex'' can be most simply positioned in 8-space as permutations of (0,1,2,3,3,4,5,6). This construction is based on facets of the hexipentistericantitruncated 8-orthoplex, .

Images

Omnitruncated 7-simplex

The omnitruncated 7-simplex is composed of 40320 (8

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) vertices and is the largest uniform 7-polytope in the A₇ symmetry of the regular 7-simplex. It can also be called the ''hexipentisteriruncicantitruncated 7-simplex'' which is the long name for the omnitruncation for 7 dimensions, with all reflective mirrors active.

Permutohedron and related tessellation

The omnitruncated 7-simplex is the permutohedron of order 8. The omnitruncated 7-simplex is a zonotope, the Minkowski sum of eight line segments parallel to the eight lines through the origin and the eight vertices of the 7-simplex. Like all uniform omnitruncated n-simplices, the omnitruncated 7-simplex can tessellate space by itself, in this case 7-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 .

Alternate names

* Great petated hexadecaexon (Acronym: guph) (Jonathan Bowers)Klitzing, (x3x3x3x3x3x3x - guph)

Coordinates

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

Images

Related polytopes

These polytope are a part of 71 uniform 7-polytopes with A₇ 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,
wiley.com
*** (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'', PhD (1966) * x3o3o3o3o3o3x - suph, x3x3o3o3o3o3x- puto, x3o3x3o3o3o3x - puro, x3o3o3x3o3o3x - puph, x3o3o3o3x3o3x - pugro, x3x3x3o3o3o3x - pupato, x3o3x3x3o3o3x - pupro, x3x3o3o3x3o3x - pucto, x3o3x3o3x3o3x - pucroh, x3x3o3o3o3x3x - putath, x3x3x3x3o3o3x - pugopo, x3x3x3o3x3o3x - pucagro, x3x3o3x3x3o3x - pucpato, x3o3x3x3x3o3x - pucproh, x3x3x3o3o3x3x - putagro, x3x3x3x3o3x3x - putpath, x3x3x3x3x3o3x - pugaco, x3x3x3x3o3x3x - putgapo, x3x3x3o3x3x3x - putcagroh, x3x3x3x3x3x3x - guph

External links

Polytopes of Various Dimensions

{{Polytopes 7-polytopes