# Cross-polytope

In geometry, a **cross-polytope**,[1] **hyperoctahedron**, **orthoplex**,[2] or **cocube** is a regular, convex polytope that exists in *n*-dimensions. 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.

2 dimensions square |
3 dimensions octahedron |

4 dimensions 16-cell |
5 dimensions 5-orthoplex |

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}:

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 a *n*-dimensional cross-polytope is a Turán graph *T*(2*n*,*n*).

## 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.

## 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*, and the simplices, labeled as

_{n}*α*. A fourth family, the infinite tessellations of hypercubes, he labeled as

_{n}*δ*.[3]

_{n}The *n*-dimensional cross-polytope has 2*n* 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 . 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

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):

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.

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-orthoplexPentacross |
{3^{3},4}{3,3,3 ^{1,1}}5{ } |
10 | 40 | 80 | 80 | 32 | 1 | |||||||

6 | β_{6}3 _{11} |
6-orthoplexHexacross |
{3^{4},4}{3 ^{3},3^{1,1}}6{ } |
12 | 60 | 160 | 240 | 192 | 64 | 1 | ||||||

7 | β_{7}4 _{11} |
7-orthoplexHeptacross |
{3^{5},4}{3 ^{4},3^{1,1}}7{ } |
14 | 84 | 280 | 560 | 672 | 448 | 128 | 1 | |||||

8 | β_{8}5 _{11} |
8-orthoplexOctacross |
{3^{6},4}{3 ^{5},3^{1,1}}8{ } |
16 | 112 | 448 | 1120 | 1792 | 1792 | 1024 | 256 | 1 | ||||

9 | β_{9}6 _{11} |
9-orthoplexEnneacross |
{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-orthoplexDecacross |
{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-orthoplexn-cross |
{3^{n − 2},4}{3 ^{n − 3},3^{1,1}}n{} |
2n 0-faces, ... ..., 2k-faces^{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 2*d* points is the largest possible equidistant set for this distance.[5]

## 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 *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 . A *p*-generalized *n*-orthoplex has *pn* vertices. *Generalized orthoplexes* have regular simplexes (real) as facets.[6] 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.

p=2 | p=3 | p=4 | p=5 | p=6 | p=7 | p=8 | ||
---|---|---|---|---|---|---|---|---|

_{2}{4}_{2} = {4} = K _{2,2} |
_{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} | ||

_{2}{3}_{2}{4}_{2} = {3,4} = K _{2,2,2} |
_{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} | ||

_{2}{3}_{2}{3}_{2}{3,3,4} = K _{2,2,2,2} |
_{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} | ||

_{2}{3}_{2}{3}_{2}{3}_{2}{4}_{2}{3,3,3,4} = K _{2,2,2,2,2} |
_{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} | ||

_{2}{3}_{2}{3}_{2}{3}_{2}{3}_{2}{4}_{2}{3,3,3,3,4} = K _{2,2,2,2,2,2} |
_{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.

## See also

- List of regular polytopes
- Hyperoctahedral group, the symmetry group of the cross-polytope

## Citations

- Coxeter 1973, pp. 121-122, §7.21. illustration Fig 7-2B.
- Conway calls it an n-
**orthoplex**for*orthant complex*. - Coxeter 1973, pp. 120-124, §7.2.
- Coxeter 1973, p. 121, §7.2.2..
- Guy, Richard K. (1983), "An olla-podrida of open problems, often oddly posed",
*American Mathematical Monthly*,**90**(3): 196–200, doi:10.2307/2975549, JSTOR 2975549. - Coxeter, Regular Complex Polytopes, p. 108

## References

- Coxeter, H.S.M. (1973).
*Regular Polytopes*(3rd ed.). New York: Dover.- pp. 121-122, §7.21. see illustration Fig 7.2B
- p. 296, Table I (iii): Regular Polytopes, three regular polytopes in n-dimensions (n≥5)

## External links

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