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 has two constructed forms, the first being regular with Schläfli symbol {33,4}, and the second with alternately labeled (checkerboarded) facets, with Schläfli symbol {3,3,31,1} or Coxeter symbol 211.

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.

Alternate names

 * pentacross, derived from combining the family name cross polytope with pente for five (dimensions) in Greek.
 * Triacontaditeron (or triacontakaiditeron) - as a 32-facetted 5-polytope (polyteron).

As a configuration
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.

$$\begin{bmatrix}\begin{matrix} 10 & 8 & 24 & 32 & 16 \\ 2 & 40 & 6 & 12 & 8 \\ 3 & 3 & 80 & 4 & 4 \\ 4 & 6 & 4 & 80 & 2 \\ 5 & 10 & 10 & 5 & 32 \end{matrix}\end{bmatrix}$$

Cartesian coordinates
Cartesian coordinates for the vertices of a 5-orthoplex, centered at the origin are
 * (±1,0,0,0,0), (0,±1,0,0,0), (0,0,±1,0,0), (0,0,0,±1,0), (0,0,0,0,±1)

Construction
There are three Coxeter groups associated with the 5-orthoplex, one regular, dual of the penteract with the C5 or [4,3,3,3] Coxeter group, and a lower symmetry with two copies of 5-cell facets, alternating, with the D5 or [32,1,1] Coxeter group, and the final one as a dual 5-orthotope, called a 5-fusil which can have a variety of subsymmetries.

Related polytopes and honeycombs
This polytope is one of 31 uniform 5-polytopes generated from the B5 Coxeter plane, including the regular 5-cube and 5-orthoplex.