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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.

More information Orthogonal projections in A8 Coxeter plane ...

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.

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Rectified 8-orthoplex

Summarize
Perspective
Rectified 8-orthoplex
Typeuniform 8-polytope
Schläfli symbolt1{3,3,3,3,3,3,4}
Coxeter-Dynkin diagrams
7-faces272
6-faces3072
5-faces8960
4-faces12544
Cells10080
Faces4928
Edges1344
Vertices112
Vertex figure6-orthoplex prism
Petrie polygonhexakaidecagon
Coxeter groupsC8, [4,36]
D8, [35,1,1]
Propertiesconvex

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)[1]

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 are all permutations of:

(±1,±1,0,0,0,0,0,0)

Images

More information B8, B7 ...
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Birectified 8-orthoplex

More information Birectified 8-orthoplex ...

Alternate names

  • birectified octacross
  • birectified diacosipentacontahexazetton (Acronym: bark) (Jonathan Bowers)[2]

Cartesian coordinates

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

(±1,±1,±1,0,0,0,0,0)

Images

More information B8, B7 ...
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Trirectified 8-orthoplex

More information 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)[3]

Cartesian coordinates

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

(±1,±1,±1,±1,0,0,0,0)

Images

More information B8, B7 ...

Notes

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References

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