Rectified 8-orthoplexes

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                

• rectified octacross
• rectified diacosipentacontahexazetton (Acronym: rek) (Jonathan Bowers)^[1]

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 [3^5,1,1] Coxeter group.

Cartesian coordinates for the vertices of a rectified 8-orthoplex, centered at the origin, edge length ${\displaystyle {\sqrt {2}}}$  are all permutations of:

(±1,±1,0,0,0,0,0,0)

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

Cartesian coordinates for the vertices of a birectified 8-orthoplex, centered at the origin, edge length ${\displaystyle {\sqrt {2}}}$  are all permutations of:

(±1,±1,±1,0,0,0,0,0)

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

• trirectified octacross
• trirectified diacosipentacontahexazetton (acronym: tark) (Jonathan Bowers)^[3]

Cartesian coordinates for the vertices of a trirectified 8-orthoplex, centered at the origin, edge length ${\displaystyle {\sqrt {2}}}$  are all permutations of:

(±1,±1,±1,±1,0,0,0,0)

• H.S.M. Coxeter:
  • H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973
  • Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6 [1]
    • (Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380-407, MR 2,10]
    • (Paper 23) H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) 559-591]
    • (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
• Norman Johnson Uniform Polytopes, Manuscript (1991)
  • N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D.
• Klitzing, Richard. "8D uniform polytopes (polyzetta)". o3x3o3o3o3o3o4o - rek, o3o3x3o3o3o3o4o - bark, o3o3o3x3o3o3o4o - tark

• Polytopes of Various Dimensions
• Multi-dimensional Glossary