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 {3<sup>3</sup>,4}, and the second with alternately labeled (checkerboarded) facets, with Schläfli symbol {3,3,3<sup>1,1</sup>} or Coxeter symbol 2<sub>11</sub>.
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.
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<sub>5</sub> or [4,3,3,3] Coxeter group, and a lower symmetry with two copies of 5-cell facets, alternating, with the D<sub>5</sub> or [3<sup>2,1,1</sup>] 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<sub>5</sub> Coxeter plane, including the regular 5-cube and 5-orthoplex.