6-orthoplex

• 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 ==

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 also the regular 6-cube. == Notes ==