6-orthoplex

Alternate names

• Hexacross, derived from combining the family name cross polytope with hex for six (dimensions) in Greek.
• Hexacontatetrapeton as a 64-facetted 6-polytope.
• Acronym: gee (Jonathan Bowers)

As a configuration

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

\begin{bmatrix}\begin{matrix}12 & 10 & 40 & 80 & 80 & 32 \ 2 & 60 & 8 & 24 & 32 & 16 \ 3 & 3 & 160 & 6 & 12 & 8 \ 4 & 6 & 4 & 240 & 4 & 4 \ 5 & 10 & 10 & 5 & 192 & 2 \ 6 & 15 & 20 & 15 & 6 & 64 \end{matrix}\end{bmatrix}

Construction

There are three Coxeter groups associated with the 6-orthoplex, one regular, dual of the hexeract with the C6 or [4,3,3,3,3] Coxeter group, and a half symmetry with two copies of 5-simplex facets, alternating, with the D6 or [33,1,1] Coxeter group. A lowest symmetry construction is based on a dual of a 6-orthotope, called a 6-fusil.

Cartesian coordinates

Cartesian coordinates for the vertices of a 6-orthoplex, centered at the origin are : (±1,0,0,0,0,0), (0,±1,0,0,0,0), (0,0,±1,0,0,0), (0,0,0,±1,0,0), (0,0,0,0,±1,0), (0,0,0,0,0,±1)

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

Images

Related polytopes

The 6-orthoplex can be projected down to 3-dimensions into the vertices of a regular icosahedron.

It is in a dimensional series of uniform polytopes and honeycombs, expressed by Coxeter as 3k1 series. (A degenerate 4-dimensional case exists as 3-sphere tiling, a tetrahedral hosohedron.)

This polytope is one of 63 uniform 6-polytopes generated from the B6 Coxeter plane, including the regular 6-cube or 6-orthoplex.

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. 1966

;Specific

References

1. Coxeter, Regular Polytopes, sec 1.8 Configurations
2. Coxeter, Complex Regular Polytopes, p.117
3. ''[[Quasicrystals and Geometry]]'', Marjorie Senechal, 1996, Cambridge University Press, p. 64. 2.7.1 ''The I₆ crystal''