6-orthoplex


In geometry, a 6-orthoplex, or 6-cross polytope, is a regular 6-polytope with 12 vertices, 60 edges, 160 triangle faces, 240 tetrahedron cells, 192 5-cell 4-faces, and 64 5-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 311.
It is a part of an infinite family of polytopes, called cross-polytopes or orthoplexes. The dual polytope is the 6-hypercube, or hexeract.

Alternate names

This configuration matrix represents the 6-orthoplex. The rows and columns correspond to vertices, edges, faces, cells, 4-faces and 5-faces. The diagonal numbers say how many of each element occur in the whole 6-orthoplex. The nondiagonal numbers say how many of the column's element occur in or at the row's element.

Construction

There are three Coxeter groups associated with the 6-orthoplex, one regular, dual of the hexeract with the C6 or Coxeter group, and a half symmetry with two copies of 5-simplex facets, alternating, with the D6 or Coxeter group. A lowest symmetry construction is based on a dual of a 6-orthotope, called a 6-fusil.
NameCoxeterSchläfliSymmetryOrder
Regular 6-orthoplex46080
Quasiregular 6-orthoplex23040
6-fusil+7680
6-fusil+3072
6-fusil22304
6-fusil+21536
6-fusil++768
6-fusil3512
6-fusil+3384
6-fusil2+2256
6-fusil+4128
6-fusil664

Cartesian coordinates

for the vertices of a 6-orthoplex, centered at the origin are
Every vertex pair is connected by an edge, except opposites.

Images

Related polytopes

The 6-orthoplex can be projected down to 3-dimensions into the vertices of a regular icosahedron.
It is in a dimensional series of uniform polytopes and honeycombs, expressed by Coxeter as 3k1 series.
This polytope is one of 63 uniform 6-polytopes generated from the B6 Coxeter plane, including the regular 6-cube or 6-orthoplex.