# 8-orthoplex

8-orthoplex Octacross :
| Orthogonal projection inside Petrie polygon ||
| Type | Regular 8-polytope |
| Family | orthoplex |
| Schläfli symbol | {3$^6$,4} {3,3,3,3,3,3$^{1,1}$} |
| Coxeter-Dynkin diagrams | |
| 7-faces | 256 {3$^6$} |
| 6-faces | 1024 {3$^5$} |
| 5-faces | 1792 {3$^4$} |
| 4-faces | 1792 {3$^3$} |
| Cells | 1120 {3,3} |
| Faces | 448 {3} |
| Edges | 112 |
| Vertices | 16 |
| Vertex figure | 7-orthoplex |
| Petrie polygon | hexadecagon |
| Coxeter groups | C$_8$, [3$^6$,4] D$_8$, [3$^{5,1,1}$] |
| Dual | 8-cube |
| Properties | convex, Hanner polytope |
In geometry, an 8-orthoplex or 8-cross polytope is a regular 8-polytope with 16 vertices, 112 edges, 448 triangle faces, 1120 tetrahedron cells, 1792 5-cells 4-faces, 1792 5-faces, 1024 6-faces, and 256 7-faces.
It has two constructive forms, the first being regular with Schläfli symbol {3$^6$,4}, and the second with alternately labeled (checkerboarded) facets, with Schläfli symbol {3,3,3,3,3,3$^{1,1}$} or Coxeter symbol 5$_{11}$.
It is a part of an infinite family of polytopes, called cross-polytopes or orthoplexes. The dual polytope is an 8-hypercube, or octeract.

## Alternate names

* Octacross, derived from combining the family name cross polytope with oct for eight (dimensions) in Greek
* Diacosipentacontahexazetton as a 256-facetted 8-polytope (polyzetton)

## As a configuration

This configuration matrix represents the 8-orthoplex. The rows and columns correspond to vertices, edges, faces, cells, 4-faces, 5-faces, 6-faces and 7-faces. The diagonal numbers say how many of each element occur in the whole 8-orthoplex. The nondiagonal numbers say how many of the column's element occur in or at the row's element.
${\begin{bmatrix}{\begin{matrix}16&14&84&280&560&672&448&128\\2&112&12&60&160&240&192&64\\3&3&448&10&40&80&80&32\\4&6&4&1120&8&24&32&16\\5&10&10&5&1792&6&12&8\\6&15&20&15&6&1792&4&4\\7&21&35&35&21&7&1024&2\\8&28&56&70&56&28&8&256\end{matrix}}\end{bmatrix}}$
The diagonal f-vector numbers are derived through the Wythoff construction, dividing the full group order of a subgroup order by removing individual mirrors.
| B$_8$ | k-face | f$_k$ | f$_0$ | f$_1$ | f$_2$ | f$_3$ | f$_4$ | f$_5$ | f$_6$ | f$_7$ | k-figure | notes |
| B$_7$ | ( ) | f$_0$ | 16 | 14 | 84 | 280 | 560 | 672 | 448 | 128 | {3,3,3,3,3,4} | B$_8$/B$_7$ = 2^8*8!/2^7/7! = 16 |
| A$_1$B$_6$ | { } | f$_1$ | 2 | 112 | 12 | 60 | 160 | 240 | 192 | 64 | {3,3,3,3,4} | B$_8$/A$_1$B$_6$ = 2^8*8!/2/2^6/6! = 112 |
| A$_2$B$_5$ | {3} | f$_2$ | 3 | 3 | 448 | 10 | 40 | 80 | 80 | 32 | {3,3,3,4} | B$_8$/A$_2$B$_5$ = 2^8*8!/3!/2^5/5! = 448 |
| A$_3$B$_4$ | {3,3} | f$_3$ | 4 | 6 | 4 | 1120 | 8 | 24 | 32 | 16 | {3,3,4} | B$_8$/A$_3$B$_4$ = 2^8*8!/4!/2^4/4! = 1120 |
| A$_4$B$_3$ | {3,3,3} | f$_4$ | 5 | 10 | 10 | 5 | 1792 | 6 | 12 | 8 | {3,4} | B$_8$/A$_4$B$_3$ = 2^8*8!/5!/8/3! = 1792 |
| A$_5$B$_2$ | {3,3,3,3} | f$_5$ | 6 | 15 | 20 | 15 | 6 | 1792 | 4 | 4 | {4} | B$_8$/A$_5$B$_2$ = 2^8*8!/6!/4/2 = 1792 |
| A$_6$A$_1$ | {3,3,3,3,3} | f$_6$ | 7 | 21 | 35 | 35 | 21 | 7 | 1024 | 2 | { } | B$_8$/A$_6$A$_1$ = 2^8*8!/7!/2 = 1024 |
| A$_7$ | {3,3,3,3,3,3} | f$_7$ | 8 | 28 | 56 | 70 | 56 | 28 | 8 | 256 | ( ) | B$_8$/A$_7$ = 2^8*8!/8! = 256 |

## Construction

There are two Coxeter groups associated with the 8-cube, one regular, dual of the octeract with the C$_8$ or [4,3,3,3,3,3,3] symmetry group, and a half symmetry with two copies of 7-simplex facets, alternating, with the D$_8$ or [3$^{5,1,1}$] symmetry group. A lowest symmetry construction is based on a dual of an 8-orthotope, called an 8-fusil.
| Name | Schläfli symbol | Symmetry | Order |
| regular 8-orthoplex | {3,3,3,3,3,3,4} | [3,3,3,3,3,3,4] | 10321920 |
| Quasiregular 8-orthoplex | {3,3,3,3,3,3$^{1,1}$} | [3,3,3,3,3,3$^{1,1}$] | 5160960 |
| 8-fusil | 8 | [2$^7$] | 256 |

## Cartesian coordinates

Cartesian coordinates for the vertices of an 8-cube, centered at the origin are
    (±1,0,0,0,0,0,0,0), (0,±1,0,0,0,0,0,0), (0,0,±1,0,0,0,0,0), (0,0,0,±1,0,0,0,0),
    (0,0,0,0,±1,0,0,0), (0,0,0,0,0,±1,0,0), (0,0,0,0,0,0,0,±1), (0,0,0,0,0,0,0,±1)
Every vertex pair is connected by an edge, except opposites.

## Images

orthographic projections :
| B$_8$ | B$_8$ | B$_8$ | B$_7$ | B$_7$ | B$_7$ |
| [16] | [16] | [16] | [14] | [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] |
It is used in its alternated form 5$_{11}$ with the 8-simplex to form the 5$_{21}$ honeycomb.