9-orthoplex
# 9-orthoplex

Regular 9-orthoplex

Ennecross :

| Orthogonal projection inside Petrie polygon ||
| - | - |
| Type | Regular 9-polytope |
| Family | orthoplex |
| Schläfli symbol | ${3^7,4}$ ${3^6,3^{1,1}}$ |
| Coxeter-Dynkin diagrams | |
| 8-faces | 512 ${3^7}$ |
| 7-faces | 2304 ${3^6}$ |
| 6-faces | 4608 ${3^5}$ |
| 5-faces | 5376 ${3^4}$ |
| 4-faces | 4032 ${3^3}$ |
| Cells | 2016 {3,3} |
| Faces | 672 {3} |
| Edges | 144 |
| Vertices | 18 |
| Vertex figure | Octacross |
| Petrie polygon | Octadecagon |
| Coxeter groups | $C_9,$ $[3^7,4]$ $D_9,$ $[3^{6,1,1}]$ |
| Dual | 9-cube |
| Properties | convex, Hanner polytope |

In geometry, a 9-orthoplex or 9-cross polytope, is a regular 9-polytope with 18 vertices, 144 edges, 672 triangle faces, 2016 tetrahedron cells, 4032 5-cells 4-faces, 5376 5-simplex 5-faces, 4608 6-simplex 6-faces, 2304 7-simplex 7-faces, and 512 8-simplex 8-faces.

It has two constructed forms, the first being regular with Schläfli symbol ${3^7,4},$ and the second with alternately labeled (checkerboarded) facets, with Schläfli symbol ${3^6,3^{1,1}}$ or Coxeter symbol $6_{11}.$

It is one of an infinite family of polytopes, called cross-polytopes or orthoplexes. The dual polytope is the 9-hypercube or enneract.

## Alternate names

* Enneacross, derived from combining the family name cross polytope with ennea for nine (dimensions) in Greek
* Pentacosidodecayotton as a 512-facetted 9-polytope (polyyotton)

## Construction

There are two Coxeter groups associated with the 9-orthoplex, one regular, dual of the enneract with the $C_9$ or $[4,3^7]$ symmetry group, and a lower symmetry with two copies of 8-simplex facets, alternating, with the $D_9$ or $[3^{6,1,1}]$ symmetry group.

## Cartesian coordinates

Cartesian coordinates for the vertices of a 9-orthoplex, centered at the origin, are

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

Every vertex pair is connected by an edge, except opposites.

## Images

orthographic projections :

| $B_9$ | $B_9$ | $B_8$ | $B_8$ | $B_7$ | $B_7$ |
| - | - | - | - | - | - |
| [18] | [18] | [16] | [16] | [14] | [14] |
| $B_6$ | $B_6$ | $B_6$ | $B_5$ | $B_5$ | $B_5$ |
| [12] | [12] | [12] | [10] | [10] | [10] |
| $B_4$ | $B_4$ | $B_3$ | $B_3$ | $B_2$ | $B_2$ |
| [8] | [8] | [6] | [6] | [4] | [4] |
| $A_7$ | $A_7$ | $A_5$ | $A_5$ | $A_3$ | $A_3$ |
| [8] | [8] | [6] | [6] | [4] | [4] |
