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Rectified 8 orthoplexes

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In eight-dimensional geometry, a rectified 8-orthoplex is a convex uniform 8-polytope, being a rectification of the regular 8-orthoplex.

Contents

There are unique 8 degrees of rectifications, the zeroth being the 8-orthoplex, and the 7th and last being the 8-cube. Vertices of the rectified 8-orthoplex are located at the edge-centers of the 8-orthoplex. Vertices of the birectified 8-orthoplex are located in the triangular face centers of the 8-orthoplex. Vertices of the trirectified 8-orthoplex are located in the tetrahedral cell centers of the 8-orthoplex.

Rectified 8-orthoplex

The rectified 8-orthoplex has 112 vertices. These represent the root vectors of the simple Lie group D8. The vertices can be seen in 3 hyperplanes, with the 28 vertices rectified 7-simplexs cells on opposite sides, and 56 vertices of an expanded 7-simplex passing through the center. When combined with the 16 vertices of the 8-orthoplex, these vertices represent the 128 root vectors of the B8 and C8 simple Lie groups.

The rectified 8-orthoplex is the vertex figure for the demiocteractic honeycomb.

or

Alternate names

  • rectified octacross
  • rectified diacosipentacontahexazetton (Acronym: rek) (Jonathan Bowers)

    • Construction

    There are two Coxeter groups associated with the rectified 8-orthoplex, one with the C8 or [4,36] Coxeter group, and a lower symmetry with two copies of heptcross facets, alternating, with the D8 or [35,1,1] Coxeter group.

    Cartesian coordinates

    Cartesian coordinates for the vertices of a rectified 8-orthoplex, centered at the origin, edge length 2 are all permutations of:

    (±1,±1,0,0,0,0,0,0)

    Alternate names

  • birectified octacross
  • birectified diacosipentacontahexazetton (Acronym: bark) (Jonathan Bowers)

    • Cartesian coordinates

    Cartesian coordinates for the vertices of a birectified 8-orthoplex, centered at the origin, edge length 2 are all permutations of:

    (±1,±1,±1,0,0,0,0,0)

    Trirectified 8-orthoplex

    The trirectified 8-orthoplex can tessellate space in the quadrirectified 8-cubic honeycomb.

    Alternate names

  • trirectified octacross
  • trirectified diacosipentacontahexazetton (acronym: tark) (Jonathan Bowers)

    • Cartesian coordinates

    Cartesian coordinates for the vertices of a trirectified 8-orthoplex, centered at the origin, edge length 2 are all permutations of:

    (±1,±1,±1,±1,0,0,0,0)

    References

    Rectified 8-orthoplexes Wikipedia
    (Text) CC BY-SA