7-orthoplex


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, and the second with alternately labeled facets, with Schläfli symbol 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.

Alternate names

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.

Images

Construction

There are two Coxeter groups associated with the 7-orthoplex, one regular, dual of the hepteract with the C7 or symmetry group, and a half symmetry with two copies of 6-simplex facets, alternating, with the D7 or symmetry group. A lowest symmetry construction is based on a dual of a 7-orthotope, called a 7-fusil.
NameCoxeter diagramSchläfli symbolSymmetryOrderVertex figure
regular 7-orthoplex645120
Quasiregular 7-orthoplex322560
7-fusil7128

Cartesian coordinates

for the vertices of a 7-orthoplex, centered at the origin are
Every vertex pair is connected by an edge, except opposites.