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Pentellated 6 simplexes

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Pentellated 6-simplexes

In six-dimensional geometry, a pentellated 6-simplex is a convex uniform 6-polytope with 5th order truncations of the regular 6-simplex.

Contents

There are unique 10 degrees of pentellations of the 6-simplex with permutations of truncations, cantellations, runcinations, and sterications. The simple pentellated 6-simplex is also called an expanded 6-simplex, constructed by an expansion operation applied to the regular 6-simplex. The highest form, the pentisteriruncicantitruncated 6-simplex, is called an omnitruncated 6-simplex with all of the nodes ringed.

Alternate names

  • Expanded 6-simplex
  • Small terated tetradecapeton (Acronym: staf) (Jonathan Bowers)

    • Coordinates

    The vertices of the pentellated 6-simplex can be positioned in 7-space as permutations of (0,1,1,1,1,1,2). This construction is based on facets of the pentellated 7-orthoplex.

    A second construction in 7-space, from the center of a rectified 7-orthoplex is given by coordinate permutations of:

    (1,-1,0,0,0,0,0)

    Root vectors

    Its 42 vertices represent the root vectors of the simple Lie group A6. It is the vertex figure of the 6-simplex honeycomb.

    Images

    Note: (*) Symmetry doubled for Ak graphs with even k due to symmetrically-ringed Coxeter-Dynkin diagram.

    Alternate names

  • Teracellated heptapeton (Acronym: tocal) (Jonathan Bowers)

    • Coordinates

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

    Alternate names

  • Teriprismated heptapeton (Acronym: topal) (Jonathan Bowers)

    • Coordinates

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

    Alternate names

  • Terigreatorhombated heptapeton (Acronym: togral) (Jonathan Bowers)

    • Coordinates

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

    Alternate names

  • Tericellirhombated heptapeton (Acronym: tocral) (Jonathan Bowers)

    • Coordinates

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

    Alternate names

  • Teriprismatorhombated tetradecapeton (Acronym: taporf) (Jonathan Bowers)

    • Coordinates

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

    Images

    Note: (*) Symmetry doubled for Ak graphs with even k due to symmetrically-ringed Coxeter-Dynkin diagram.

    Alternate names

  • Terigreatoprismated heptapeton (Acronym: tagopal) (Jonathan Bowers)

    • Coordinates

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

    Alternate names

  • Tericellitruncated tetradecapeton (Acronym: tactaf) (Jonathan Bowers)

    • Coordinates

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

    Images

    Note: (*) Symmetry doubled for Ak graphs with even k due to symmetrically-ringed Coxeter-Dynkin diagram.

    Alternate names

  • Great teracellirhombated heptapeton (Acronym: gatocral) (Jonathan Bowers)

    • Coordinates

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

    Omnitruncated 6-simplex

    The omnitruncated 6-simplex has 5040 vertices, 15120 edges,16800 faces (4200 hexagons and 1260 squares), 8400 cells, 1806 4-faces, and 126 5-faces. With 5040 vertices, it is the largest of 35 uniform 6-polytopes generated from the regular 6-simplex.

    Alternate names

  • Pentisteriruncicantitruncated 6-simplex (Johnson's omnitruncation for 6-polytopes)
  • Omnitruncated heptapeton
  • Great terated tetradecapeton (Acronym: gotaf) (Jonathan Bowers)

    • The omnitruncated 6-simplex is the permutohedron of order 7. The omnitruncated 6-simplex is a zonotope, the Minkowski sum of seven line segments parallel to the seven lines through the origin and the seven vertices of the 6-simplex.

    Like all uniform omnitruncated n-simplices, the omnitruncated 6-simplex can tessellate space by itself, in this case 6-dimensional space with three facets around each hypercell. It has Coxeter-Dynkin diagram of .

    Coordinates

    The vertices of the omnitruncated 6-simplex can be most simply positioned in 7-space as permutations of (0,1,2,3,4,5,6). This construction is based on facets of the pentisteriruncicantitruncated 7-orthoplex, t0,1,2,3,4,5{35,4}, .

    Images

    Note: (*) Symmetry doubled for Ak graphs with even k due to symmetrically-ringed Coxeter-Dynkin diagram.

    The pentellated 6-simplex is one of 35 uniform 6-polytopes based on the [3,3,3,3,3] Coxeter group, all shown here in A6 Coxeter plane orthographic projections.

    References

    Pentellated 6-simplexes Wikipedia
    (Text) CC BY-SA


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