5-orthoplex
# 5-orthoplex

Regular 5-orthoplex (pentacross) :

| Orthogonal projection inside Petrie polygon ||
| - | - |
| Type | Regular 5-polytope |
| Family | orthoplex |
| Schläfli symbol | {3,3,3,4} ${3,3,3^{1,1}}$ |
| Coxeter-Dynkin diagrams | |
| 4-faces | 32 ${3^3}$ |
| Cells | 80 {3,3} |
| Faces | 80 {3} |
| Edges | 40 |
| Vertices | 10 |
| Vertex figure | 16-cell |
| Petrie polygon | decagon |
| Coxeter groups | $BC_5,$ [3,3,3,4] $D_5,$ $[3^{2,1,1}]$ |
| Dual | 5-cube |
| Properties | convex, Hanner polytope |

In five-dimensional geometry, a 5-orthoplex, or 5-cross polytope, is a five-dimensional polytope with 10 vertices, 40 edges, 80 triangle faces, 80 tetrahedron cells, 32 5-cell 4-faces.

It has two constructed forms, the first being regular with Schläfli symbol ${3^3,4},$ and the second with alternately labeled (checkerboarded) facets, with Schläfli symbol ${3,3,3^{1,1}}$ or Coxeter symbol $2_{11}.$

It is a part of an infinite family of polytopes, called cross-polytopes or orthoplexes. The dual polytope is the 5-hypercube or 5-cube.

## Alternate names

* pentacross, derived from combining the family name cross polytope with pente for five (dimensions) in Greek.
* Triacontaditeron (or triacontakaiditeron) - as a 32-facetted 5-polytope (polyteron).

## As a configuration

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

${\begin{bmatrix}{\begin{matrix}10&8&24&32&16\\2&40&6&12&8\\3&3&80&4&4\\4&6&4&80&2\\5&10&10&5&32\end{matrix}}\end{bmatrix}}$

## Cartesian coordinates

Cartesian coordinates for the vertices of a 5-orthoplex, centered at the origin are

    (±1,0,0,0,0), (0,±1,0,0,0), (0,0,±1,0,0), (0,0,0,±1,0), (0,0,0,0,±1)

## Construction

There are three Coxeter groups associated with the 5-orthoplex, one regular, dual of the penteract with the $C_5$ or [4,3,3,3] Coxeter group, and a lower symmetry with two copies of 5-cell facets, alternating, with the $D_5$ or $[3^{2,1,1}]$ Coxeter group, and the final one as a dual 5-orthotope, called a 5-fusil which can have a variety of subsymmetries.

| Name | Schläfli symbol | Symmetry | Order |
| - | - | - | - |
| regular 5-orthoplex | {3,3,3,4} | [3,3,3,4] | 3840 |
| Quasiregular 5-orthoplex | ${3,3,3^{1,1}}$ | $[3,3,3^{1,1}]$ | 1920 |
| 5-fusil |
| 5-fusil | {3,3,3,4} | [4,3,3,3] | 3840 |
| 5-fusil | {3,3,4}+ | [4,3,3,2] | 768 |
| 5-fusil | {3,4}+{4} | [4,3,2,4] | 384 |
| 5-fusil | {3,4}+2 | [4,3,2,2] | 192 |
| 5-fusil | 2{4}+ | [4,2,4,2] | 128 |
| 5-fusil | {4}+3 | [4,2,2,2] | 64 |
| 5-fusil | 5 | [2,2,2,2] | 32 |

## Other images

orthographic projections :

| Coxeter plane | $B_5$ | $B_4$ / $D_5$ | $B_3$ / $D_4$ / $A_2$ |
| - | - | - | - |
| Graph | | | |
| Dihedral symmetry | [10] | [8] | [6] |
| Coxeter plane | $B_2$ | $A_3$ |
| Graph | | |
| Dihedral symmetry | [4] | [4] |

* The perspective projection (3D to 2D) of a stereographic projection (4D to 3D) of the Schlegel diagram (5D to 4D) of the 5-orthoplex. 10 sets of 4 edges form 10 circles in the 4D Schlegel diagram: two of these circles are straight lines in the stereographic projection because they contain the center of projection.

## Related polytopes and honeycombs

$2_{k1}$ figures in n dimensions :

| Space | Finite | Finite | Finite | Finite | Finite | Finite | Euclidean | Hyperbolic |
| - | - | - | - | - | - | - | - | - |
| n | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
| Coxeter group | $E_3=A_2A_1$ | $E_4=A_4$ | $E_5=D_5$ | $E_6$ | $E_7$ | $E_8$ | $E_9$ = ${\tilde {E}}_{8}$ = $E_8^+$ | $E_{10}$ = ${\bar {T}}_{8}$ = $E_8^{++}$ |
| Coxeter diagram | | | | | | | | |
| Symmetry | $[3^{−1,2,1}]$ | $[3^{0,2,1}]$ | $[[3^{1,2,1}]]$ | $[3^{2,2,1}]$ | $[3^{3,2,1}]$ | $[3^{4,2,1}]$ | $[3^{5,2,1}]$ | $[3^{6,2,1}]$ |
| Order | 12 | 120 | 384 | 51,840 | 2,903,040 | 696,729,600 | ∞ | ∞ |
| Graph | | | | | | | - | - |
| Name | $2_{−1,1}$ | $2_{01}$ | $2_{11}$ | $2_{21}$ | $2_{31}$ | $2_{41}$ | $2_{51}$ | $2_{61}$ |

This polytope is one of 31 uniform 5-polytopes generated from the $B_5$ Coxeter plane, including the regular 5-cube and 5-orthoplex.

B5 polytopes :

| $β_5$ | $t_1β_5$ | $t_2γ_5$ | $t_1γ_5$ | $γ_5$ | $t_{0,1}β_5$ | $t_{0,2}β_5$ | $t_{1,2}β_5$ |
| - | - | - | - | - | - | - | - |
| $t_{0,3}β_5$ | $t_{1,3}γ_5$ | $t_{1,2}γ_5$ | $t_{0,4}γ_5$ | $t_{0,3}γ_5$ | $t_{0,2}γ_5$ | $t_{0,1}γ_5$ | $t_{0,1,2}β_5$ |
| $t_{0,1,3}β_5$ | $t_{0,2,3}β_5$ | $t_{1,2,3}γ_5$ | $t_{0,1,4}β_5$ | $t_{0,2,4}γ_5$ | $t_{0,2,3}γ_5$ | $t_{0,1,4}γ_5$ | $t_{0,1,3}γ_5$ |
| $t_{0,1,2}γ_5$ | $t_{0,1,2,3}β_5$ | $t_{0,1,2,4}β_5$ | $t_{0,1,3,4}γ_5$ | $t_{0,1,2,4}γ_5$ | $t_{0,1,2,3}γ_5$ | $t_{0,1,2,3,4}γ_5$ |
