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-cells 4-faces, 448 5-faces, and 128 6-faces.
It has two constructed forms, the first being regular with Schläfli symbol {35,4}, and the second with alternately labeled (checkerboarded) facets, with Schläfli symbol {3,3,3,3,31,1} or Coxeter symbol 411.
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.
{\displaystyle {\begin{bmatrix}{\begin{matrix}14&12&60&160&240&192&64\\2&84&10&40&80&80&32\\3&3&280&8&24&32&16\\4&6&4&560&6&12&8\\5&10&10&5&672&4&4\\6&15&20&15&6&448&2\\7&21&35&35&21&7&128\end{matrix}}\end{bmatrix}}}
There are two Coxeter groups associated with the 7-orthoplex, one regular, dual of the hepteract with the C7 or [4,3,3,3,3,3] symmetry group, and a half symmetry with two copies of 6-simplex facets, alternating, with the D7 or [34,1,1] 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.