# 6-orthoplex

In geometry, a **6-orthoplex**, or 6-cross polytope, is a regular 6-polytope with 12 vertices, 60 edges, 160 triangle faces, 240 tetrahedron cells, 192 5-cell *4-faces*, and 64 *5-faces*.

It is a part of an infinite family of polytopes, called cross-polytopes or *orthoplexes*. The dual polytope is the 6-hypercube, or hexeract.

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.^{[1]}^{[2]}

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

Cartesian coordinates for the vertices of a 6-orthoplex, centered at the origin are

The 6-orthoplex can be projected down to 3-dimensions into the vertices of a regular icosahedron.^{[3]}

It is in a dimensional series of uniform polytopes and honeycombs, expressed by Coxeter as 3_{k1} 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 B_{6} Coxeter plane, including the regular 6-cube or 6-orthoplex.