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

Hexacross, derived from combining the family name cross polytope with hex for six in Greek. Hexacontatetrapeton as a 64-facetted 6-polytope.

Acronym: gee

As a configuration

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.

Name	Coxeter	Schläfli	Symmetry	Order
Regular 6-orthoplex				46080
Quasiregular 6-orthoplex				23040
6-fusil		+		7680
6-fusil		+		3072
6-fusil		2		2304
6-fusil		+2		1536
6-fusil		++		768
6-fusil		3		512
6-fusil		+3		384
6-fusil		2+2		256
6-fusil		+4		128
6-fusil		6		64

Cartesian coordinates

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

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 3_k1 series.
This polytope is one of 63 uniform 6-polytopes generated from the B₆ Coxeter plane, including the regular 6-cube or 6-orthoplex.