In geometry, a 7-orthoplex, or 7-cross polytope, is a regular 7-polytope with 14 vertices, 84 edges, 280 triangle faces, 560 tetrahedron cells, 672 5-cell 4-faces, 448 5-faces, and 128 6-faces.
It has two constructed forms, the first being regular with Schläfli symbol {3<sup>5</sup>,4}, and the second with alternately labeled (checkerboarded) facets, with Schläfli symbol {3,3,3,3,3<sup>1,1</sup>} or Coxeter symbol 4<sub>11</sub>.
It is a part of an infinite family of polytopes, called cross-polytopes or orthoplexes. The dual polytope is the 7-hypercube, or hepteract.
This configuration matrix represents the 7-orthoplex. The rows and columns correspond to vertices, edges, faces, cells, 4-faces, 5-faces and 6-faces. The diagonal numbers say how many of each element occur in the whole 7-orthoplex. The nondiagonal numbers say how many of the column's element occur in or at the row's element.
There are two Coxeter groups associated with the 7-orthoplex, one regular, dual of the hepteract with the C<sub>7</sub> or [4,3,3,3,3,3] symmetry group, and a half symmetry with two copies of 6-simplex facets, alternating, with the D<sub>7</sub> or [3<sup>4,1,1</sup>] symmetry group. A lowest symmetry construction is based on a dual of a 7-orthotope, called a 7-fusil.
Cartesian coordinates for the vertices of a 7-orthoplex, centered at the origin are
Every vertex pair is connected by an edge, except opposites.