Hexicated 7-simplexes


In seven-dimensional geometry, a hexicated 7-simplex is a convex uniform 7-polytope, including 6th-order truncations 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 an 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 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

    Coordinates

The vertices of the hexicated 7-simplex can be most simply positioned in 8-space as permutations of. 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:

Images

Hexitruncated 7-simplex

Alternate names

  • Petitruncated octaexon

    Coordinates

The vertices of the hexitruncated 7-simplex can be most simply positioned in 8-space as permutations of. This construction is based on facets of the hexitruncated 8-orthoplex,.

Images

Hexicantellated 7-simplex

Alternate names

  • Petirhombated octaexon

    Coordinates

The vertices of the hexicantellated 7-simplex can be most simply positioned in 8-space as permutations of. This construction is based on facets of the hexicantellated 8-orthoplex,.

Images

Hexiruncinated 7-simplex

Alternate names

  • Petaprismated hexadecaexon

    Coordinates

The vertices of the hexiruncinated 7-simplex can be most simply positioned in 8-space as permutations of. This construction is based on facets of the hexiruncinated 8-orthoplex,.

Images

Hexicantitruncated 7-simplex

Alternate names

  • Petigreatorhombated octaexon

    Coordinates

The vertices of the hexicantitruncated 7-simplex can be most simply positioned in 8-space as permutations of. This construction is based on facets of the hexicantitruncated 8-orthoplex,.

Images

Hexiruncitruncated 7-simplex

Alternate names

  • Petiprismatotruncated octaexon

    Coordinates

The vertices of the hexiruncitruncated 7-simplex can be most simply positioned in 8-space as permutations of. 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

    Coordinates

The vertices of the hexiruncicantellated 7-simplex can be most simply positioned in 8-space as permutations of. This construction is based on facets of the hexiruncicantellated 8-orthoplex,.

Images

Hexisteritruncated 7-simplex

Alternate names

  • Peticellitruncated octaexon

    Coordinates

The vertices of the hexisteritruncated 7-simplex can be most simply positioned in 8-space as permutations of. This construction is based on facets of the hexisteritruncated 8-orthoplex,.

Images

Hexistericantellated 7-simplex

Alternate names

  • Peticellirhombihexadecaexon

    Coordinates

The vertices of the hexistericantellated 7-simplex can be most simply positioned in 8-space as permutations of. This construction is based on facets of the hexistericantellated 8-orthoplex,.

Images

Hexipentitruncated 7-simplex

Alternate names

  • Petiteritruncated hexadecaexon

    Coordinates

The vertices of the hexipentitruncated 7-simplex can be most simply positioned in 8-space as permutations of. This construction is based on facets of the hexipentitruncated 8-orthoplex,.

Images

Hexiruncicantitruncated 7-simplex

Alternate names

  • Petigreatoprismated octaexon

    Coordinates

The vertices of the hexiruncicantitruncated 7-simplex can be most simply positioned in 8-space as permutations of. This construction is based on facets of the hexiruncicantitruncated 8-orthoplex,.

Images

Hexistericantitruncated 7-simplex

Alternate names

  • Peticelligreatorhombated octaexon

    Coordinates

The vertices of the hexistericantitruncated 7-simplex can be most simply positioned in 8-space as permutations of. This construction is based on facets of the hexistericantitruncated 8-orthoplex,.

Images

Hexisteriruncitruncated 7-simplex

Alternate names

  • Peticelliprismatotruncated octaexon

    Coordinates

The vertices of the hexisteriruncitruncated 7-simplex can be most simply positioned in 8-space as permutations of. This construction is based on facets of the hexisteriruncitruncated 8-orthoplex,.

Images

Hexisteriruncicantellated 7-simplex

Alternate names

  • Peticelliprismatorhombihexadecaexon

    Coordinates

The vertices of the hexisteriruncitruncated 7-simplex can be most simply positioned in 8-space as permutations of. This construction is based on facets of the hexisteriruncitruncated 8-orthoplex,.

Images

Hexipenticantitruncated 7-simplex

Alternate names

  • Petiterigreatorhombated octaexon

    Coordinates

The vertices of the hexipenticantitruncated 7-simplex can be most simply positioned in 8-space as permutations of. This construction is based on facets of the hexipenticantitruncated 8-orthoplex,.

Images

Hexipentiruncitruncated 7-simplex

Alternate names

  • Petiteriprismatotruncated hexadecaexon

    Coordinates

The vertices of the hexipentiruncitruncated 7-simplex can be most simply positioned in 8-space as permutations of. This construction is based on facets of the hexipentiruncitruncated 8-orthoplex,.

Images

Hexisteriruncicantitruncated 7-simplex

Alternate names

  • Petigreatocellated octaexon

    Coordinates

The vertices of the hexisteriruncicantitruncated 7-simplex can be most simply positioned in 8-space as permutations of. This construction is based on facets of the hexisteriruncicantitruncated 8-orthoplex,.

Images

Hexipentiruncicantitruncated 7-simplex

Alternate names

  • Petiterigreatoprismated octaexon

    Coordinates

The vertices of the hexipentiruncicantitruncated 7-simplex can be most simply positioned in 8-space as permutations of. This construction is based on facets of the hexipentiruncicantitruncated 8-orthoplex,.

Images

Hexipentistericantitruncated 7-simplex

Alternate names

  • Petitericelligreatorhombihexadecaexon

    Coordinates

The vertices of the hexipentistericantitruncated 7-simplex can be most simply positioned in 8-space as permutations of. This construction is based on facets of the hexipentistericantitruncated 8-orthoplex,.

Images

Omnitruncated 7-simplex

The omnitruncated 7-simplex is composed of 40320 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

    Coordinates

The vertices of the omnitruncated 7-simplex can be most simply positioned in 8-space as permutations of. This construction is based on facets of the hexipentisteriruncicantitruncated 8-orthoplex, t0,1,2,3,4,5,6,.

Images

Related polytopes

The 20 polytopes presented in this article are a part of 71 uniform 7-polytopes with A7 symmetry shown in the table below.