6-orthoplex
# 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 | $A_3A_1$ | $A_5$ | $D_6$ | $E_7$ | ${\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 :

| $β_6$ | $t_1β_6$ | $t_2β_6$ | $t_2γ_6$ | $t_1γ_6$ | $γ_6$ | $t_{0,1}β_6$ | $t_{0,2}β_6$ |
| - | - | - | - | - | - | - | - |
| $t_{1,2}β_6$ | $t_{0,3}β_6$ | $t_{1,3}β_6$ | $t_{2,3}γ_6$ | $t_{0,4}β_6$ | $t_{1,4}γ_6$ | $t_{1,3}γ_6$ | $t_{1,2}γ_6$ |
| $t_{0,5}γ_6$ | $t_{0,4}γ_6$ | $t_{0,3}γ_6$ | $t_{0,2}γ_6$ | $t_{0,1}γ_6$ | $t_{0,1,2}β_6$ | $t_{0,1,3}β_6$ | $t_{0,2,3}β_6$ |
| $t_{1,2,3}β_6$ | $t_{0,1,4}β_6$ | $t_{0,2,4}β_6$ | $t_{1,2,4}β_6$ | $t_{0,3,4}β_6$ | $t_{1,2,4}γ_6$ | $t_{1,2,3}γ_6$ | $t_{0,1,5}β_6$ |
| $t_{0,2,5}β_6$ | $t_{0,3,4}γ_6$ | $t_{0,2,5}γ_6$ | $t_{0,2,4}γ_6$ | $t_{0,2,3}γ_6$ | $t_{0,1,5}γ_6$ | $t_{0,1,4}γ_6$ | $t_{0,1,3}γ_6$ |
| $t_{0,1,2}γ_6$ | $t_{0,1,2,3}β_6$ | $t_{0,1,2,4}β_6$ | $t_{0,1,3,4}β_6$ | $t_{0,2,3,4}β_6$ | $t_{1,2,3,4}γ_6$ | $t_{0,1,2,5}β_6$ | $t_{0,1,3,5}β_6$ |
| $t_{0,2,3,5}γ_6$ | $t_{0,2,3,4}γ_6$ | $t_{0,1,4,5}γ_6$ | $t_{0,1,3,5}γ_6$ | $t_{0,1,3,4}γ_6$ | $t_{0,1,2,5}γ_6$ | $t_{0,1,2,4}γ_6$ | $t_{0,1,2,3}γ_6$ |
| $t_{0,1,2,3,4}β_6$ | $t_{0,1,2,3,5}β_6$ | $t_{0,1,2,4,5}β_6$ | $t_{0,1,2,4,5}γ_6$ | $t_{0,1,2,3,5}γ_6$ | $t_{0,1,2,3,4}γ_6$ | $t_{0,1,2,3,4,5}γ_6$ |
