# 6-orthoplex

6-orthoplex Hexacross :
| Orthogonal projection inside Petrie polygon ||
| Type | Regular 6-polytope |
| Family | orthoplex |
| Schläfli symbols | {3,3,3,3,4} {3,3,3,3$^{1,1}$} |
| Coxeter-Dynkin diagrams | = |
| 5-faces | 64 {3$^4$} |
| 4-faces | 192 {3$^3$} |
| Cells | 240 {3,3} |
| Faces | 160 {3} |
| Edges | 60 |
| Vertices | 12 |
| Vertex figure | 5-orthoplex |
| Petrie polygon | dodecagon |
| Coxeter groups | B$_6$, [4,3$^4$] D$_6$, [3$^{3,1,1}$] |
| Dual | 6-cube |
| Properties | convex, Hanner polytope |
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 {3$^4$,4}, and the second with alternately labeled (checkerboarded) facets, with Schläfli symbol {3,3,3,3$^{1,1}$} or Coxeter symbol 3$_{11}$.
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 (dimensions) in Greek.
* Hexacontitetrapeton as a 64-facetted 6-polytope.

## 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.
${\begin{bmatrix}{\begin{matrix}12&10&40&80&80&32\\2&60&8&24&32&16\\3&3&160&6&12&8\\4&6&4&240&4&4\\5&10&10&5&192&2\\6&15&20&15&6&64\end{matrix}}\end{bmatrix}}$

## Construction

There are three Coxeter groups associated with the 6-orthoplex, one regular, dual of the hexeract with the C$_6$ or [4,3,3,3,3] Coxeter group, and a half symmetry with two copies of 5-simplex facets, alternating, with the D$_6$ or [3$^{3,1,1}$] Coxeter group. A lowest symmetry construction is based on a dual of a 6-orthotope, called a 6-fusil.
| Name | Schläfli | Symmetry | Order |
| Regular 6-orthoplex | {3,3,3,3,4} | [4,3,3,3,3] | 46080 |
| Quasiregular 6-orthoplex | {3,3,3,3$^{1,1}$} | [3,3,3,3$^{1,1}$] | 23040 |
| 6-fusil | {3,3,3,4}+ | [4,3,3,3,3] | 7680 |
| 6-fusil | {3,3,4}+{4} | [4,3,3,2,4] | 3072 |
| 6-fusil | 2{3,4} | [4,3,2,4,3] | 2304 |
| 6-fusil | {3,3,4}+2 | [4,3,3,2,2] | 1536 |
| 6-fusil | {3,4}+{4}+ | [4,3,2,4,2] | 768 |
| 6-fusil | 3{4} | [4,2,4,2,4] | 512 |
| 6-fusil | {3,4}+3 | [4,3,2,2,2] | 384 |
| 6-fusil | 2{4}+2 | [4,2,4,2,2] | 256 |
| 6-fusil | {4}+4 | [4,2,2,2,2] | 128 |
| 6-fusil | 6 | [2,2,2,2,2] | 64 |

## Cartesian coordinates

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.

## Images

orthographic projections :
| Coxeter plane | B$_6$ | B$_5$ | B$_4$ |
| Graph | | | |
| Dihedral symmetry | [12] | [10] | [8] |
| Coxeter plane | B$_3$ | B$_2$ |
| Graph | | |
| Dihedral symmetry | [6] | [4] |
| Coxeter plane | A$_5$ | A$_3$ |
| Graph | | |
| Dihedral symmetry | [6] | [4] |

## Related polytopes

The 6-orthoplex can be projected down to 3-dimensions into the vertices of a regular icosahedron.
| 2D | 2D | 3D | 3D |
| Icosahedron {3,5} = H$_3$ Coxeter plane | 6-orthoplex {3,3,3,3$^{1,1}$} = D$_6$ Coxeter plane | Icosahedron | 6-orthoplex |
| This construction can be geometrically seen as the 12 vertices of the 6-orthoplex projected to 3 dimensions as the vertices of a regular icosahedron. This represents a geometric folding of the D$_6$ to H$_3$ Coxeter groups: : to . On the left, seen by these 2D Coxeter plane orthogonal projections, the two overlapping central vertices define the third axis in this mapping. Every pair of vertices of the 6-orthoplex are connected, except opposite ones: 30 edges are shared with the icosahedron, while 30 more edges from the 6-orthoplex project to the interior of the icosahedron. ||||
It is in a dimensional series of uniform polytopes and honeycombs, expressed by Coxeter as 3$_{k1}$ series. (A degenerate 4-dimensional case exists as 3-sphere tiling, a tetrahedral hosohedron.)
3$_{k1}$ dimensional figures :
| Space | Finite | Finite | Finite | Finite | Euclidean | Hyperbolic |
| n | 4 | 5 | 6 | 7 | 8 | 9 |
| Coxeter group | A$_3$A$_1$ | A$_5$ | D$_6$ | E$_7$ | ${\tilde {E}}_{7}$=E$_7$$^+$ | ${\bar {T}}_{8}$=E$_7$$^{++}$ |
| Coxeter diagram | | | | | | |
| Symmetry | [3$^{−1,3,1}$] | [3$^{0,3,1}$] | [[3$^{1,3,1}$]] = [4,3,3,3,3] | [3$^{2,3,1}$] | [3$^{3,3,1}$] | [3$^{4,3,1}$] |
| Order | 48 | 720 | 46,080 | 2,903,040 | ∞ | ∞ |
| Graph | | | | | - | - |
| Name | 3$_{1,-1}$ | 3$_{10}$ | 3$_{11}$ | 3$_{21}$ | 3$_{31}$ | 3$_{41}$ |
This polytope is one of 63 uniform 6-polytopes generated from the B$_6$ Coxeter plane, including the regular 6-cube or 6-orthoplex.
B6 polytopes :
| β$_6$ | t$_1$β$_6$ | t$_2$β$_6$ | t$_2$γ$_6$ | t$_1$γ$_6$ | γ$_6$ | t$_{0,1}$β$_6$ | t$_{0,2}$β$_6$ |
| t$_{1,2}$β$_6$ | t$_{0,3}$β$_6$ | t$_{1,3}$β$_6$ | t$_{2,3}$γ$_6$ | t$_{0,4}$β$_6$ | t$_{1,4}$γ$_6$ | t$_{1,3}$γ$_6$ | t$_{1,2}$γ$_6$ |
| t$_{0,5}$γ$_6$ | t$_{0,4}$γ$_6$ | t$_{0,3}$γ$_6$ | t$_{0,2}$γ$_6$ | t$_{0,1}$γ$_6$ | t$_{0,1,2}$β$_6$ | t$_{0,1,3}$β$_6$ | t$_{0,2,3}$β$_6$ |
| t$_{1,2,3}$β$_6$ | t$_{0,1,4}$β$_6$ | t$_{0,2,4}$β$_6$ | t$_{1,2,4}$β$_6$ | t$_{0,3,4}$β$_6$ | t$_{1,2,4}$γ$_6$ | t$_{1,2,3}$γ$_6$ | t$_{0,1,5}$β$_6$ |
| t$_{0,2,5}$β$_6$ | t$_{0,3,4}$γ$_6$ | t$_{0,2,5}$γ$_6$ | t$_{0,2,4}$γ$_6$ | t$_{0,2,3}$γ$_6$ | t$_{0,1,5}$γ$_6$ | t$_{0,1,4}$γ$_6$ | t$_{0,1,3}$γ$_6$ |
| t$_{0,1,2}$γ$_6$ | t$_{0,1,2,3}$β$_6$ | t$_{0,1,2,4}$β$_6$ | t$_{0,1,3,4}$β$_6$ | t$_{0,2,3,4}$β$_6$ | t$_{1,2,3,4}$γ$_6$ | t$_{0,1,2,5}$β$_6$ | t$_{0,1,3,5}$β$_6$ |
| t$_{0,2,3,5}$γ$_6$ | t$_{0,2,3,4}$γ$_6$ | t$_{0,1,4,5}$γ$_6$ | t$_{0,1,3,5}$γ$_6$ | t$_{0,1,3,4}$γ$_6$ | t$_{0,1,2,5}$γ$_6$ | t$_{0,1,2,4}$γ$_6$ | t$_{0,1,2,3}$γ$_6$ |
| t$_{0,1,2,3,4}$β$_6$ | t$_{0,1,2,3,5}$β$_6$ | t$_{0,1,2,4,5}$β$_6$ | t$_{0,1,2,4,5}$γ$_6$ | t$_{0,1,2,3,5}$γ$_6$ | t$_{0,1,2,3,4}$γ$_6$ | t$_{0,1,2,3,4,5}$γ$_6$ |