2 dimensions square | 3 dimensions octahedron |

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

In geometry, a **cross-polytope**,^{ [1] }**hyperoctahedron**, **orthoplex**,^{ [2] } 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.

- 4 dimensions
- Higher dimensions
- Generalized orthoplex
- Related polytope families
- See also
- Citations
- References
- External links

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*).

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 **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

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):

^{ [4] }

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 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, projected down the axis, with 2 vertices mapped into the center.

n | β_{n}k_{11} | Name(s) Graph | Graph 2 n-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-orthoplexn-cross | {3^{n − 2},4}{3 ^{n − 3},3^{1,1}}n{} | ... ... ... | 2n0-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] }

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 . 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} |

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.

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

- ↑ 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

In geometry, a **cube** is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex.

In elementary geometry, a **polytope** is a geometric object with flat sides (*faces*). It is a generalization in any number of dimensions of the three-dimensional polyhedron. Polytopes may exist in any general number of dimensions n as an n-dimensional polytope or **n-polytope**. In this context, "flat sides" means that the sides of a (*k* + 1)-polytope consist of k-polytopes that may have (*k* – 1)-polytopes in common. For example, a two-dimensional polygon is a 2-polytope and a three-dimensional polyhedron is a 3-polytope.

In geometry, a **hypercube** is an *n*-dimensional analogue of a square and a cube. It is a closed, compact, convex figure whose 1-skeleton consists of groups of opposite parallel line segments aligned in each of the space's dimensions, perpendicular to each other and of the same length. A unit hypercube's longest diagonal in *n* dimensions is equal to .

In mathematics, a **regular polytope** is a polytope whose symmetry group acts transitively on its flags, thus giving it the highest degree of symmetry. All its elements or j-faces — cells, faces and so on — are also transitive on the symmetries of the polytope, and are regular polytopes of dimension ≤ *n*.

In geometry, the **16-cell** is the regular convex 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol {3,3,4}. It is one of the six regular convex 4-polytopes first described by the Swiss mathematician Ludwig Schläfli in the mid-19th century. It is also called **C _{16}**,

In geometry, a **five-dimensional polytope** is a polytope in five-dimensional space, bounded by (4-polytope) facets, pairs of which share a polyhedral cell.

In six-dimensional geometry, a **uniform polypeton** is a six-dimensional uniform polytope. A uniform polypeton is vertex-transitive, and all facets are uniform 5-polytopes.

In five-dimensional geometry, a **5-cube** is a name for a five-dimensional hypercube with 32 vertices, 80 edges, 80 square faces, 40 cubic cells, and 10 tesseract 4-faces.

In five-dimensional geometry, a **demipenteract** or **5-demicube** is a semiregular 5-polytope, constructed from a *5-hypercube* (penteract) with alternated vertices removed.

In five-dimensional geometry, a **5-orthoplex**, or 5-cross polytope, is a five-dimensional polytope with 10 vertices, 40 edges, 80 triangle faces, 80 tetrahedron cells, 32 5-cell 4-faces.

In geometry, a **complex polytope** is a generalization of a polytope in real space to an analogous structure in a complex Hilbert space, where each real dimension is accompanied by an imaginary one.

In geometry, a **6-orthoplex**, or 6-cross polytope, is a regular 6-polytope with 12 vertices, 60 edges, 160 triangle faces, 240 tetrahedron cells, 192 5-cell *4-faces*, and 64 *5-faces*.

In geometry, a **7-orthoplex**, or 7-cross polytope, is a regular 7-polytope with 14 vertices, 84 edges, 280 triangle faces, 560 tetrahedron cells, 672 5-cells *4-faces*, 448 *5-faces*, and 128 *6-faces*.

In geometry, a **10-orthoplex** or 10-cross polytope, is a regular 10-polytope with 20 vertices, 180 edges, 960 triangle faces, 3360 octahedron cells, 8064 5-cells *4-faces*, 13440 *5-faces*, 15360 *6-faces*, 11520 *7-faces*, 5120 *8-faces*, and 1024 *9-faces*.

In geometry, the **alternated hypercube honeycomb** is a dimensional infinite series of honeycombs, based on the hypercube honeycomb with an alternation operation. It is given a Schläfli symbol h{4,3...3,4} representing the regular form with half the vertices removed and containing the symmetry of Coxeter group for n ≥ 4. A lower symmetry form can be created by removing another mirror on an order-4 peak.

In geometry, a **Petrie polygon** for a regular polytope of n dimensions is a skew polygon in which every *n* – 1 consecutive sides belongs to one of the facets. The **Petrie polygon** of a regular polygon is the regular polygon itself; that of a regular polyhedron is a skew polygon such that every two consecutive sides belongs to one of the faces. Petrie polygons are named for mathematician John Flinders Petrie.

In 6-dimensional geometry, the **2 _{21}** polytope is a uniform 6-polytope, constructed within the symmetry of the E

In 7-dimensional geometry, the **3 _{21}** polytope is a uniform 7-polytope, constructed within the symmetry of the E

In 8-dimensional geometry, the **4 _{21}** is a semiregular uniform 8-polytope, constructed within the symmetry of the E

- 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)

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