8-orthoplex

Alternate names

• Octacross, derived from combining the family name cross polytope with oct for eight (dimensions) in Greek
• Diacosipentacontahexazetton as a 256-facetted 8-polytope (polyzetton), acronym: ek

As a configuration

This configuration matrix represents the 8-orthoplex. The rows and columns correspond to vertices, edges, faces, cells, 4-faces, 5-faces, 6-faces and 7-faces. The diagonal numbers say how many of each element occur in the whole 8-orthoplex. The nondiagonal numbers say how many of the column's element occur in or at the row's element.

\begin{bmatrix}\begin{matrix} 16 & 14 & 84 & 280 & 560 & 672 & 448 & 128 \ 2 & 112 & 12 & 60 & 160 & 240 & 192 & 64 \ 3 & 3 & 448 & 10 & 40 & 80 & 80 & 32 \ 4 & 6 & 4 & 1120 & 8 & 24 & 32 & 16 \ 5 & 10 & 10 & 5 & 1792 & 6 & 12 & 8 \ 6 & 15 & 20 & 15 & 6 & 1792 & 4 & 4 \ 7 & 21 & 35 & 35 & 21 & 7 & 1024 & 2 \ 8 & 28 & 56 & 70 & 56 & 28 & 8 & 256 \end{matrix}\end{bmatrix}

The diagonal f-vector numbers are derived through the Wythoff construction, dividing the full group order of a subgroup order by removing individual mirrors.

Construction

There are two Coxeter groups associated with the 8-cube, one regular, dual of the octeract with the C8 or [4,3,3,3,3,3,3] symmetry group, and a half symmetry with two copies of 7-simplex facets, alternating, with the D8 or [35,1,1] symmetry group. A lowest symmetry construction is based on a dual of an 8-orthotope, called an 8-fusil.

Cartesian coordinates

Cartesian coordinates for the vertices of an 8-cube, centered at the origin are : (±1,0,0,0,0,0,0,0), (0,±1,0,0,0,0,0,0), (0,0,±1,0,0,0,0,0), (0,0,0,±1,0,0,0,0), : (0,0,0,0,±1,0,0,0), (0,0,0,0,0,±1,0,0), (0,0,0,0,0,0,0,±1), (0,0,0,0,0,0,0,±1)

Every vertex pair is connected by an edge, except opposites.

Images

It is used in its alternated form 511 with the 8-simplex to form the 521 honeycomb.

References

• 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, wiley.com,
    • (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.

References

1. {{KlitzingPolytopes. ../incmats/ek.htm. x3o3o3o3o3o3o4o - ek
2. Coxeter, Regular Polytopes, sec 1.8 Configurations
3. Coxeter, Complex Regular Polytopes, p.117