| bgcolor=#e7dcc3 colspan=2 | 6-orthoplex Hexacross | |
|---|---|---|
| bgcolor=#ffffff align=center colspan=2 | Orthogonal projection inside Petrie polygon | |
| Type | Regular 6-polytope | |
| Family | orthoplex | |
| Schläfli symbols | ||
| Coxeter-Dynkin diagrams | = | |
| 5-faces | ||
| 4-faces | ||
| Cells | ||
| Faces | ||
| Edges | 60 | |
| Vertices | 12 | |
| Vertex figure | 5-orthoplex | |
| Petrie polygon | dodecagon | |
| Coxeter groups | B6, [4,3<sup>4</sup>] D6, [3<sup>3,1,1</sup>] | |
| Dual | 6-cube | |
| Properties | convex, Hanner polytope |
It has two constructed forms, the first being regular with Schläfli symbol, and the second with alternately labeled (checkerboarded) 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.
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.[1] [2]
\begin{bmatrix}\begin{matrix}12&10&40&80&80&32\ 2&60&8&24&32&16\ 3&3&160&6&12&8\ 4&6&4&240&4&4\ 5&10&10&5&192&2\ 6&15&20&15&6&64\end{matrix}\end{bmatrix}
There are three Coxeter groups associated with the 6-orthoplex, one regular, dual of the hexeract with the C6 or [4,3,3,3,3] Coxeter group, and a half symmetry with two copies of 5-simplex facets, alternating, with the D6 or [3<sup>3,1,1</sup>] 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 | [4,3,3,3,3] | 46080 | ||
| Quasiregular 6-orthoplex | [3,3,3,3<sup>1,1</sup>] | 23040 | ||
| 6-fusil | + | [4,3,3,3,3] | 7680 | |
| + | [4,3,3,2,4] | 3072 | ||
| 2 | [4,3,2,4,3] | 2304 | ||
| +2 | [4,3,3,2,2] | 1536 | ||
| ++ | [4,3,2,4,2] | 768 | ||
| 3 | [4,2,4,2,4] | 512 | ||
| +3 | [4,3,2,2,2] | 384 | ||
| 2+2 | [4,2,4,2,2] | 256 | ||
| +4 | [4,2,2,2,2] | 128 | ||
| 6 | [2,2,2,2,2] | 64 |
Cartesian coordinates for the vertices of a 6-orthoplex, centered at the origin are
(±1,0,0,0,0,0), (0,±1,0,0,0,0), (0,0,±1,0,0,0), (0,0,0,±1,0,0), (0,0,0,0,±1,0), (0,0,0,0,0,±1)
Every vertex pair is connected by an edge, except opposites.
The 6-orthoplex can be projected down to 3-dimensions into the vertices of a regular icosahedron.[3]
It is in a dimensional series of uniform polytopes and honeycombs, expressed by Coxeter as 3k1 series. (A degenerate 4-dimensional case exists as 3-sphere tiling, a tetrahedral hosohedron.)
This polytope is one of 63 uniform 6-polytopes generated from the B6 Coxeter plane, including the regular 6-cube or 6-orthoplex.
