# 5-orthoplex

In five-dimensional geometry, a **5-orthoplex**, or 5-cross polytope, is a five-dimensional polytope with 10 vertices, 40 edges, 80 triangle faces, 80 tetrahedron cells, 32 5-cell 4-faces.

It is a part of an infinite family of polytopes, called cross-polytopes or *orthoplexes*. The dual polytope is the 5-hypercube or 5-cube.

This configuration matrix represents the 5-orthoplex. The rows and columns correspond to vertices, edges, faces, cells and 4-faces. The diagonal numbers say how many of each element occur in the whole 5-orthoplex. The nondiagonal numbers say how many of the column's element occur in or at the row's element.^{[1]}^{[2]}

Cartesian coordinates for the vertices of a 5-orthoplex, centered at the origin are

There are three Coxeter groups associated with the 5-orthoplex, one regular, dual of the penteract with the C_{5} or [4,3,3,3] Coxeter group, and a lower symmetry with two copies of *5-cell* facets, alternating, with the D_{5} or [3^{2,1,1}] Coxeter group, and the final one as a dual 5-orthotope, called a **5-fusil** which can have a variety of subsymmetries.

This polytope is one of 31 uniform 5-polytopes generated from the B_{5} Coxeter plane, including the regular 5-cube and 5-orthoplex.