# Cross-polytope

Cross-polytopes of dimension 2 to 5 :
| 2 dimensions square | 3 dimensions octahedron |
| 4 dimensions 16-cell | 5 dimensions 5-orthoplex |
In geometry, a cross-polytope, hyperoctahedron, orthoplex, staurotope, or cocube is a regular, convex polytope that exists in n-dimensional Euclidean space. A 2-dimensional cross-polytope is a square, a 3-dimensional cross-polytope is a regular octahedron, and a 4-dimensional cross-polytope is a 16-cell. Its facets are simplexes of the previous dimension, while the cross-polytope's vertex figure is another cross-polytope from the previous dimension.
The vertices of a cross-polytope can be chosen as the unit vectors pointing along each co-ordinate axis – i.e. all the permutations of (±1, 0, 0, ..., 0). The cross-polytope is the convex hull of its vertices. The n-dimensional cross-polytope can also be defined as the closed unit ball (or, according to some authors, its boundary) in the ℓ$_1$-norm on R$^n$:
    $\{x\in \mathbb {R} ^{n}:\|x\|_{1}\leq 1\}.$
In 1 dimension the cross-polytope is simply the line segment [−1, +1], in 2 dimensions it is a square (or diamond) with vertices {(±1, 0), (0, ±1)}. In 3 dimensions it is an octahedron—one of the five convex regular polyhedra known as the Platonic solids. This can be generalised to higher dimensions with an n-orthoplex being constructed as a bipyramid with an (n−1)-orthoplex base.
The cross-polytope is the dual polytope of the hypercube. The 1-skeleton of an n-dimensional cross-polytope is the Turán graph T(2n, n) (also known as a cocktail party graph).

## 4 dimensions

The 4-dimensional cross-polytope also goes by the name hexadecachoron or 16-cell. It is one of the six convex regular 4-polytopes. These 4-polytopes were first described by the Swiss mathematician Ludwig Schläfli in the mid-19th century.
The vertices of the 4-dimensional hypercube, or tesseract, can be divided into two sets of eight, the convex hull of each set forming a cross-polytope. Moreover, the polytope known as the 24-cell can be constructed by symmetrically arranging three cross-polytopes.

## Higher dimensions

The cross-polytope family is one of three regular polytope families, labeled by Coxeter as β$_n$, the other two being the hypercube family, labeled as γ$_n$, and the simplex family, labeled as α$_n$. A fourth family, the infinite tessellations of hypercubes, he labeled as δ$_n$.
The n-dimensional cross-polytope has 2n vertices, and 2$^n$ facets ((n − 1)-dimensional components) all of which are (n − 1)-simplices. The vertex figures are all (n − 1)-cross-polytopes. The Schläfli symbol of the cross-polytope is {3,3,...,3,4}.
The dihedral angle of the n-dimensional cross-polytope is $\delta _{n}=\arccos \left({\frac {2-n}{n}}\right)$. This gives: δ$_2$ = arccos(0/2) = 90°, δ$_3$ = arccos(−1/3) = 109.47°, δ$_4$ = arccos(−2/4) = 120°, δ$_5$ = arccos(−3/5) = 126.87°, ... δ$_∞$ = arccos(−1) = 180°.
The hypervolume of the n-dimensional cross-polytope is
    ${\frac {2^{n}}{n!}}.$
For each pair of non-opposite vertices, there is an edge joining them. More generally, each set of k + 1 orthogonal vertices corresponds to a distinct k-dimensional component which contains them. The number of k-dimensional components (vertices, edges, faces, ..., facets) in an n-dimensional cross-polytope is thus given by (see binomial coefficient):
    $2^{k+1}{n \choose {k+1}}$
The extended f-vector for an n-orthoplex can be computed by (1,2)$^n$, like the coefficients of polynomial products. For example a 16-cell is (1,2)$^4$ = (1,4,4)$^2$ = (1,8,24,32,16).
There are many possible orthographic projections that can show the cross-polytopes as 2-dimensional graphs. Petrie polygon projections map the points into a regular 2n-gon or lower order regular polygons. A second projection takes the 2(n−1)-gon petrie polygon of the lower dimension, seen as a bipyramid, projected down the axis, with 2 vertices mapped into the center.
Cross-polytope elements :
| n | β$_n$ k$_{11}$ | Name(s) Graph | Graph 2n-gon | Schläfli | Coxeter-Dynkin diagrams | Vertices | Edges | Faces | Cells | 4-faces | 5-faces | 6-faces | 7-faces | 8-faces | 9-faces | 10-faces |
| 0 | β$_0$ | Point 0-orthoplex | . | ( ) | | 1 | | | | | | | | | | |
| 1 | β$_1$ | Line segment 1-orthoplex | | { } | | 2 | 1 | | | | | | | | | |
| 2 | β$_2$ −1$_{11}$ | Square 2-orthoplex Bicross | | {4} 2{ } = { }+{ } | | 4 | 4 | 1 | | | | | | | | |
| 3 | β$_3$ 0$_{11}$ | Octahedron 3-orthoplex Tricross | | {3,4} {3$^{1,1}$} 3{ } | | 6 | 12 | 8 | 1 | | | | | | | |
| 4 | β$_4$ 1$_{11}$ | 16-cell 4-orthoplex Tetracross | | {3,3,4} {3,3$^{1,1}$} 4{ } | | 8 | 24 | 32 | 16 | 1 | | | | | | |
| 5 | β$_5$ 2$_{11}$ | 5-orthoplex Pentacross | | {3$^3$,4} {3,3,3$^{1,1}$} 5{ } | | 10 | 40 | 80 | 80 | 32 | 1 | | | | | |
| 6 | β$_6$ 3$_{11}$ | 6-orthoplex Hexacross | | {3$^4$,4} {3$^3$,3$^{1,1}$} 6{ } | | 12 | 60 | 160 | 240 | 192 | 64 | 1 | | | | |
| 7 | β$_7$ 4$_{11}$ | 7-orthoplex Heptacross | | {3$^5$,4} {3$^4$,3$^{1,1}$} 7{ } | | 14 | 84 | 280 | 560 | 672 | 448 | 128 | 1 | | | |
| 8 | β$_8$ 5$_{11}$ | 8-orthoplex Octacross | | {3$^6$,4} {3$^5$,3$^{1,1}$} 8{ } | | 16 | 112 | 448 | 1120 | 1792 | 1792 | 1024 | 256 | 1 | | |
| 9 | β$_9$ 6$_{11}$ | 9-orthoplex Enneacross | | {3$^7$,4} {3$^6$,3$^{1,1}$} 9{ } | | 18 | 144 | 672 | 2016 | 4032 | 5376 | 4608 | 2304 | 512 | 1 | |
| 10 | β$_{10}$ 7$_{11}$ | 10-orthoplex Decacross | | {3$^8$,4} {3$^7$,3$^{1,1}$} 10{ } | | 20 | 180 | 960 | 3360 | 8064 | 13440 | 15360 | 11520 | 5120 | 1024 | 1 |
| n | β$_n$ k$_{11}$ | n-orthoplex n-cross | | {3$^{n−2}$,4} {3$^{n−3}$,3$^{1,1}$} n | ... ... ... | 2n 0-faces, ... $2^{k+1}{n \choose k+1}$ k-faces ..., 2$^n$ (n−1)-faces | 2n 0-faces, ... $2^{k+1}{n \choose k+1}$ k-faces ..., 2$^n$ (n−1)-faces | 2n 0-faces, ... $2^{k+1}{n \choose k+1}$ k-faces ..., 2$^n$ (n−1)-faces | 2n 0-faces, ... $2^{k+1}{n \choose k+1}$ k-faces ..., 2$^n$ (n−1)-faces | 2n 0-faces, ... $2^{k+1}{n \choose k+1}$ k-faces ..., 2$^n$ (n−1)-faces | 2n 0-faces, ... $2^{k+1}{n \choose k+1}$ k-faces ..., 2$^n$ (n−1)-faces | 2n 0-faces, ... $2^{k+1}{n \choose k+1}$ k-faces ..., 2$^n$ (n−1)-faces | 2n 0-faces, ... $2^{k+1}{n \choose k+1}$ k-faces ..., 2$^n$ (n−1)-faces | 2n 0-faces, ... $2^{k+1}{n \choose k+1}$ k-faces ..., 2$^n$ (n−1)-faces | 2n 0-faces, ... $2^{k+1}{n \choose k+1}$ k-faces ..., 2$^n$ (n−1)-faces | 2n 0-faces, ... $2^{k+1}{n \choose k+1}$ k-faces ..., 2$^n$ (n−1)-faces |
The vertices of an axis-aligned cross polytope are all at equal distance from each other in the Manhattan distance (L$^1$ norm). Kusner's conjecture states that this set of 2d points is the largest possible equidistant set for this distance.

## Generalized orthoplex

Regular complex polytopes can be defined in complex Hilbert space called generalized orthoplexes (or cross polytopes), β$^p$ $_n$ = $_2${3}$_2${3}...$_2${4}$_p$, or ... Real solutions exist with p = 2, i.e. β$^2$ $_n$ = β$_n$ = $_2${3}$_2${3}...$_2${4}$_2$ = {3,3,..,4}. For p > 2, they exist in $\mathbb {\mathbb {C} } ^{n}$. A p-generalized n-orthoplex has pn vertices. Generalized orthoplexes have regular simplexes (real) as facets. Generalized orthoplexes make complete multipartite graphs, β$^p$ $_2$ make K$_{p,p}$ for complete bipartite graph, β$^p$ $_3$ make K$_{p,p,p}$ for complete tripartite graphs. β$^p$ $_n$ creates K$_{pn}$. An orthogonal projection can be defined that maps all the vertices equally-spaced on a circle, with all pairs of vertices connected, except multiples of n. The regular polygon perimeter in these orthogonal projections is called a petrie polygon.
Generalized orthoplexes :
| | p = 2 | | p = 3 | p = 4 | p = 5 | p = 6 | p = 7 | p = 8 |
| $\mathbb {R} ^{2}$ | $_2${4}$_2$ = {4} = K$_{2,2}$ | $\mathbb {\mathbb {C} } ^{2}$ | $_2${4}$_3$ = K$_{3,3}$ | $_2${4}$_4$ = K$_{4,4}$ | $_2${4}$_5$ = K$_{5,5}$ | $_2${4}$_6$ = K$_{6,6}$ | $_2${4}$_7$ = K$_{7,7}$ | $_2${4}$_8$ = K$_{8,8}$ |
| $\mathbb {R} ^{3}$ | $_2${3}$_2${4}$_2$ = {3,4} = K$_{2,2,2}$ | $\mathbb {\mathbb {C} } ^{3}$ | $_2${3}$_2${4}$_3$ = K$_{3,3,3}$ | $_2${3}$_2${4}$_4$ = K$_{4,4,4}$ | $_2${3}$_2${4}$_5$ = K$_{5,5,5}$ | $_2${3}$_2${4}$_6$ = K$_{6,6,6}$ | $_2${3}$_2${4}$_7$ = K$_{7,7,7}$ | $_2${3}$_2${4}$_8$ = K$_{8,8,8}$ |
| $\mathbb {R} ^{4}$ | $_2${3}$_2${3}$_2$ {3,3,4} = K$_{2,2,2,2}$ | $\mathbb {\mathbb {C} } ^{4}$ | $_2${3}$_2${3}$_2${4}$_3$ K$_{3,3,3,3}$ | $_2${3}$_2${3}$_2${4}$_4$ K$_{4,4,4,4}$ | $_2${3}$_2${3}$_2${4}$_5$ K$_{5,5,5,5}$ | $_2${3}$_2${3}$_2${4}$_6$ K$_{6,6,6,6}$ | $_2${3}$_2${3}$_2${4}$_7$ K$_{7,7,7,7}$ | $_2${3}$_2${3}$_2${4}$_8$ K$_{8,8,8,8}$ |
| $\mathbb {R} ^{5}$ | $_2${3}$_2${3}$_2${3}$_2${4}$_2$ {3,3,3,4} = K$_{2,2,2,2,2}$ | $\mathbb {\mathbb {C} } ^{5}$ | $_2${3}$_2${3}$_2${3}$_2${4}$_3$ K$_{3,3,3,3,3}$ | $_2${3}$_2${3}$_2${3}$_2${4}$_4$ K$_{4,4,4,4,4}$ | $_2${3}$_2${3}$_2${3}$_2${4}$_5$ K$_{5,5,5,5,5}$ | $_2${3}$_2${3}$_2${3}$_2${4}$_6$ K$_{6,6,6,6,6}$ | $_2${3}$_2${3}$_2${3}$_2${4}$_7$ K$_{7,7,7,7,7}$ | $_2${3}$_2${3}$_2${3}$_2${4}$_8$ K$_{8,8,8,8,8}$ |
| $\mathbb {R} ^{6}$ | $_2${3}$_2${3}$_2${3}$_2${3}$_2${4}$_2$ {3,3,3,3,4} = K$_{2,2,2,2,2,2}$ | $\mathbb {\mathbb {C} } ^{6}$ | $_2${3}$_2${3}$_2${3}$_2${3}$_2${4}$_3$ K$_{3,3,3,3,3,3}$ | $_2${3}$_2${3}$_2${3}$_2${3}$_2${4}$_4$ K$_{4,4,4,4,4,4}$ | $_2${3}$_2${3}$_2${3}$_2${3}$_2${4}$_5$ K$_{5,5,5,5,5,5}$ | $_2${3}$_2${3}$_2${3}$_2${3}$_2${4}$_6$ K$_{6,6,6,6,6,6}$ | $_2${3}$_2${3}$_2${3}$_2${3}$_2${4}$_7$ K$_{7,7,7,7,7,7}$ | $_2${3}$_2${3}$_2${3}$_2${3}$_2${4}$_8$ K$_{8,8,8,8,8,8}$ |

## Related polytope families

Cross-polytopes can be combined with their dual cubes to form compound polytopes:
* In two dimensions, we obtain the octagrammic star figure {⁠8/2⁠},
* In three dimensions we obtain the compound of cube and octahedron,
* In four dimensions we obtain the compound of tesseract and 16-cell.