# Cross-polytope

> Mediated Wiki article. Canonical URL: https://mediated.wiki/source/Cross-polytope
> Markdown URL: https://mediated.wiki/source/Cross-polytope.md
> Source: https://en.wikipedia.org/wiki/Cross-polytope
> Source revision: 1350932366
> License: Creative Commons Attribution-ShareAlike 4.0 International (https://creativecommons.org/licenses/by-sa/4.0/)

Cross-polytopes of dimension 2 to 5 2 dimensions square 3 dimensions octahedron 4 dimensions 16-cell 5 dimensions 5-orthoplex

In [geometry](/source/Geometry), a **cross-polytope**,[1] **hyperoctahedron**, **orthoplex**,[2] **staurotope**,[3] or **cocube** is a [regular](/source/Regular_polytope), [convex polytope](/source/Convex_polytope) that exists in *n*-[dimensional Euclidean space](/source/Dimensions). A 2-dimensional cross-polytope is a square, a 3-dimensional cross-polytope is a regular [octahedron](/source/Octahedron), and a 4-dimensional cross-polytope is a [16-cell](/source/16-cell). Its facets are [simplexes](/source/Simplex) of the previous dimension, while the cross-polytope's [vertex figure](/source/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](/source/Convex_hull) of its vertices. The *n*-dimensional cross-polytope can also be defined as the closed [unit ball](/source/Unit_ball) (or, according to some authors, its boundary) in the [ℓ1-norm](/source/L1-norm) on **R***n*, those points *x* = (*x*1, *x*2..., *x**n*) satisfying

- |x_1| + |x_2| + \cdots + |x_n| \le 1.

An *n*-orthoplex can be constructed as a [bipyramid](/source/Bipyramid#Other_dimensions) with an (*n*−1)-orthoplex base.

The cross-polytope is the [dual polytope](/source/Dual_polytope) of the [hypercube](/source/Hypercube). The [vertex-edge graph](/source/Graph_of_a_polytope) of an *n*-dimensional cross-polytope is the [Turán graph](/source/Tur%C3%A1n_graph) *T*(2*n*, *n*) (also known as a *cocktail party graph* [4]).

## Low-dimensional examples

In 1 dimension the cross-polytope is a [line segment](/source/Line_segment), which can be chosen as the [interval](/source/Real_interval) [−1, +1].

In 2 dimensions the cross-polytope is a [square](/source/Square_(geometry)). If the vertices are chosen as {(±1, 0), (0, ±1)}, the square's sides are at right angles to the axes; in this orientation a square is often called a *diamond*.

In 3 dimensions the cross-polytope is a [regular octahedron](/source/Regular_octahedron)—one of the five convex regular [polyhedra](/source/Polyhedron) known as the [Platonic solids](/source/Platonic_solid).

The 4-dimensional cross-polytope also goes by the name **hexadecachoron** or **[16-cell](/source/16-cell)**. It is one of the six [convex regular 4-polytopes](/source/Convex_regular_4-polytope). These [4-polytopes](/source/4-polytope) were first described by the Swiss mathematician [Ludwig Schläfli](/source/Ludwig_Schl%C3%A4fli) in the mid-19th century. The vertices of the 4-dimensional hypercube, or [tesseract](/source/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](/source/24-cell) can be constructed by symmetrically arranging three cross-polytopes.[5]

## *n* dimensions

The cross-polytope family is one of three [regular polytope](/source/Regular_polytope) families, labeled by [Coxeter](/source/Coxeter) as *βn*, the other two being the [hypercube](/source/Hypercube) family, labeled as *γn*, and the [simplex](/source/Simplex) family, labeled as *αn*. A fourth family, the [infinite tessellations of hypercubes](/source/Hypercubic_honeycomb), he labeled as *δn*.[6]

The *n*-dimensional cross-polytope has 2*n* vertices, and 2*n* facets ((*n* − 1)-dimensional components) all of which are (*n* − 1)-[simplices](/source/Simplex). The [vertex figures](/source/Vertex_figure) are all (*n* − 1)-cross-polytopes. The [Schläfli symbol](/source/Schl%C3%A4fli_symbol) of the cross-polytope is {3,3,...,3,4}.

The [dihedral angle](/source/Dihedral_angle#Geometry) of the *n*-dimensional cross-polytope is \delta_n = \arccos\left(\frac{2-n}{n}\right). This gives: δ2 = arccos(0/2) = 90°, δ3 = arccos(−1/3) = 109.47°, δ4 = arccos(−2/4) = 120°, δ5 = 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](/source/Binomial_coefficient)):

- 2^{k+1}{n \choose {k+1}}[7]

The extended [f-vector](/source/F-vector) for an *n*-orthoplex can be computed by (**1**,2)*n*, like the coefficients of [polynomial products](/source/Polynomial#Multiplication). For example a 16-cell is (**1**,2)4 = (**1**,4,4)2 = (**1**,8,24,32,16).

There are many possible [orthographic projections](/source/Orthographic_projection) that can show the cross-polytopes as 2-dimensional graphs. [Petrie polygon](/source/Petrie_polygon) projections map the points into a regular 2*n*-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](/source/Bipyramid), projected down the axis, with 2 vertices mapped into the center.

Cross-polytope elements n βn k11 Name(s) Graph Graph 2n-gon Schläfli Coxeter-Dynkin diagrams Vertices Edges Faces Cells 4-faces 5-faces 6-faces 0 β0 Point 0-orthoplex ( ) 1 1 β1 Line segment 1-orthoplex { } 2 1 2 β2 −111 Square 2-orthoplex Bicross {4} 2{ } = { }+{} 4 4 1 3 β3 011 Octahedron 3-orthoplex Tricross {3,4} {31,1} 3{ } 6 12 8 1 4 β4 111 16-cell 4-orthoplex Tetracross {3,3,4} {3,31,1} 4{ } 8 24 32 16 1 5 β5 211 5-orthoplex Pentacross {33,4} {3,3,31,1} 5{ } 10 40 80 80 32 1 6 β6 311 6-orthoplex Hexacross {34,4} {33,31,1} 6{ } 12 60 160 240 192 64 1 ... n βn (n−3)11 n-orthoplex n-cross {3n − 2,4} {3n − 3,31,1} n{} ... ... ... 2n 0-faces, ... 2^{k+1}{n\choose k+1} k-faces ..., 2n (n−1)-faces

The vertices of an axis-aligned cross polytope are all at equal distance from each other in the [Manhattan distance](/source/Taxicab_geometry) ([L1 norm](/source/Lp_space)). [Kusner's conjecture](/source/Kusner's_conjecture) states that this set of 2*d* points is the largest possible [equidistant set](/source/Equidistant_set) for this distance.[8]

## Generalized orthoplex

Regular [complex polytopes](/source/Complex_polytope#Regular_complex_polytopes) can be defined in [complex](/source/Complex_number) [Hilbert space](/source/Hilbert_space) called *generalized orthoplexes* (or cross polytopes), β = 2{3}2{3}...2{4}*p*, or ... Real solutions exist with *p* = 2, i.e. β = β*n* = 2{3}2{3}...2{4}2 = {3,3,..,4}. For *p* > 2, they exist in \mathbb{\Complex}^n. A *p*-generalized *n*-orthoplex has *pn* vertices. *Generalized orthoplexes* have regular [simplexes](/source/Simplex) (real) as [facets](/source/Facet_(geometry)).[9] Generalized orthoplexes make [complete multipartite graphs](/source/Complete_multipartite_graph), β make K*p*,*p* for [complete bipartite graph](/source/Complete_bipartite_graph), β make K*p*,*p*,*p* for complete tripartite graphs β creates K*p**n* or [Turán graphs](/source/Tur%C3%A1n_graph) T(np,n). An [orthogonal projection](/source/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](/source/Regular_polygon) perimeter in these orthogonal projections is called a [petrie polygon](/source/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} = K2,2 \mathbb{\Complex}^2 2{4}3 = K3,3 2{4}4 = K4,4 2{4}5 = K5,5 2{4}6 = K6,6 2{4}7 = K7,7 2{4}8 = K8,8 \mathbb{R}^3 2{3}2{4}2 = {3,4} = K2,2,2 \mathbb{\Complex}^3 2{3}2{4}3 = K3,3,3 2{3}2{4}4 = K4,4,4 2{3}2{4}5 = K5,5,5 2{3}2{4}6 = K6,6,6 2{3}2{4}7 = K7,7,7 2{3}2{4}8 = K8,8,8 \mathbb{R}^4 2{3}2{3}2 {3,3,4} = K2,2,2,2 \mathbb{\Complex}^4 2{3}2{3}2{4}3 K3,3,3,3 2{3}2{3}2{4}4 K4,4,4,4 2{3}2{3}2{4}5 K5,5,5,5 2{3}2{3}2{4}6 K6,6,6,6 2{3}2{3}2{4}7 K7,7,7,7 2{3}2{3}2{4}8 K8,8,8,8 \mathbb{R}^5 2{3}2{3}2{3}2{4}2 {3,3,3,4} = K2,2,2,2,2 \mathbb{\Complex}^5 2{3}2{3}2{3}2{4}3 K3,3,3,3,3 2{3}2{3}2{3}2{4}4 K4,4,4,4,4 2{3}2{3}2{3}2{4}5 K5,5,5,5,5 2{3}2{3}2{3}2{4}6 K6,6,6,6,6 2{3}2{3}2{3}2{4}7 K7,7,7,7,7 2{3}2{3}2{3}2{4}8 K8,8,8,8,8 \mathbb{R}^6 2{3}2{3}2{3}2{3}2{4}2 {3,3,3,3,4} = K2,2,2,2,2,2 \mathbb{\Complex}^6 2{3}2{3}2{3}2{3}2{4}3 K3,3,3,3,3,3 2{3}2{3}2{3}2{3}2{4}4 K4,4,4,4,4,4 2{3}2{3}2{3}2{3}2{4}5 K5,5,5,5,5,5 2{3}2{3}2{3}2{3}2{4}6 K6,6,6,6,6,6 2{3}2{3}2{3}2{3}2{4}7 K7,7,7,7,7,7 2{3}2{3}2{3}2{3}2{4}8 K8,8,8,8,8,8

## Related polytope families

Cross-polytopes can be combined with their dual cubes to form compound polytopes:

- In two dimensions, we obtain the [octagrammic](/source/Octagram) star figure ,
- In three dimensions we obtain the [compound of cube and octahedron](/source/Compound_of_cube_and_octahedron),
- In four dimensions we obtain the compound of tesseract and 16-cell.

## See also

- [List of regular polytopes](/source/List_of_regular_polytopes)
- [Hyperoctahedral group](/source/Hyperoctahedral_group), the symmetry group of the cross-polytope

## Citations

1. Coxeter 1973, pp. 121–122.

1. Conway, J. H. & Sloane, N. J. A. (1991). "The Cell Structures of Certain Lattices". *Miscellanea Mathematica*. Hilton, P. (ed.). Berlin: Springer. pp. 89–90. [doi:10.1007/978-3-642-76709-8_5](https://doi.org/10.1007/978-3-642-76709-8_5). ISBN 978-3-642-76711-1.

1. McMullen, Peter (2020). *Geometric Regular Polytopes*. Cambridge University Press. p. 92. ISBN 978-1-108-48958-4.

1. Bengtsson, Ingemar & Życzkowski, Karol (2017). *Geometry of Quantum States: An Introduction to Quantum Entanglement*. 2nd ed. Cambridge University Press. p. 162. ISBN 978-1-107-02625-4.

1. Coxeter 1973, pp. 120–124.

1. Coxeter 1973, 121.

1. Guy, Richard K. (1983), "An olla-podrida of open problems, often oddly posed", *American Mathematical Monthly*. **90** (3): 196–200, [doi:10.2307/2975549](https://doi.org/10.2307/2975549). [JSTOR 2975549](https://www.jstor.org/stable/2975549).

1. Coxeter, Regular Complex Polytopes, p. 108

## References

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

---
Adapted from the Wikipedia article [Cross-polytope](https://en.wikipedia.org/wiki/Cross-polytope) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Cross-polytope?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.