In seven-dimensional

geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...

, 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 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

, 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 expansion operation. Ammann-Beenker_tiling_example

Root vectors

Its 56 vertices represent the root vectors of the

simple Lie group In mathematics, a simple Lie group is a connected non-abelian Lie group ''G'' which does not have nontrivial connected normal subgroups. The list of simple Lie groups can be used to read off the list of simple Lie algebras and Riemannian symme ...

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 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

, 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

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) ...

) 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 In geometry, the Minkowski sum (also known as dilation) of two sets of position vectors ''A'' and ''B'' in Euclidean space is formed by adding each vector in ''A'' to each vector in ''B'', i.e., the set : A + B = \. Analogously, the Minkowsk ...

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