# 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$_2$A$_1$ | 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$_1$β$_5$ | t$_2$γ$_5$ | t$_1$γ$_5$ | 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$_5$ |