Hexicated 7-simplexes

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

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

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,.

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,.

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,.

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,.

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,.

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,.

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,.

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,.

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,.

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,.

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,.

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,.

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,.

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,.

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,.

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,.

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,.

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,.

Omnitruncated 7-simplex
The omnitruncated 7-simplex is composed of 40320 (8 factorial) 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. It has Coxeter-Dynkin diagram of.

Alternate names

 * Great petated hexadecaexon (Acronym: guph) (Jonathan Bowers)

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, t0,1,2,3,4,5,6{36,4},.

Related polytopes
These polytope are a part of 71 uniform 7-polytopes with A7 symmetry.