Cross-polytope

Last updated October 21, 2024

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,^[1]hyperoctahedron, orthoplex,^[2]staurotope,^[3] 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 ℓ₁-norm on Rⁿ:

\{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]).

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_n, and the simplex family, labeled as α_n. A fourth family, the infinite tessellations of hypercubes, he labeled as δ_n.^[5]

The n-dimensional cross-polytope has 2n vertices, and 2ⁿ 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: 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):

2^{k+1}{n \choose {k+1}}

^[6]

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 k₁₁	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	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} {3^1,1} 3{ }		6	12	8	1
4	β₄ 1₁₁	16-cell 4-orthoplex Tetracross		{3,3,4} {3,3^1,1} 4{ }		8	24	32	16	1
5	β₅ 2₁₁	5-orthoplex Pentacross		{3³,4} {3,3,3^1,1} 5{ }		10	40	80	80	32	1
6	β₆ 3₁₁	6-orthoplex Hexacross		{3⁴,4} {3³,3^1,1} 6{ }		12	60	160	240	192	64	1
7	β₇ 4₁₁	7-orthoplex Heptacross		{3⁵,4} {3⁴,3^1,1} 7{ }		14	84	280	560	672	448	128	1
8	β₈ 5₁₁	8-orthoplex Octacross		{3⁶,4} {3⁵,3^1,1} 8{ }		16	112	448	1120	1792	1792	1024	256	1
9	β₉ 6₁₁	9-orthoplex Enneacross		{3⁷,4} {3⁶,3^1,1} 9{ }		18	144	672	2016	4032	5376	4608	2304	512	1
10	β₁₀ 7₁₁	10-orthoplex Decacross		{3⁸,4} {3⁷,3^1,1} 10{ }		20	180	960	3360	8064	13440	15360	11520	5120	1024	1
...
n	β_n k₁₁	n-orthoplex n-cross		{3^n − 2,4} {3^n − 3,3^1,1} n{}	... ... ...	2n0-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 (L¹ norm). Kusner's conjecture states that this set of 2d points is the largest possible equidistant set for this distance.^[7]

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. β²
_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.^[8] 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.

Citations

↑ Coxeter 1973, pp. 121–122, §7.21. illustration Fig 7-2B.
↑ Conway, J. H.; Sloane, N. J. A. (1991). "The Cell Structures of Certain Lattices". In Hilton, P.; Hirzebruch, F.; Remmert, R. (eds.). Miscellanea Mathematica. Berlin: Springer. pp. 89–90. doi:10.1007/978-3-642-76709-8_5. ISBN 978-3-642-76711-1.
↑ McMullen, Peter (2020). Geometric Regular Polytopes. Cambridge University Press. p. 92. ISBN 978-1-108-48958-4.
↑ Weisstein, Eric W. "Cocktail Party Graph". MathWorld .
↑ 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

Related Research Articles

In elementary geometry, a polytope is a geometric object with flat sides (faces). Polytopes are the generalization of three-dimensional polyhedra to any number of dimensions. Polytopes may exist in any general number of dimensions $n$ as an $n$ -dimensional polytope or $n$ -polytope. For example, a two-dimensional polygon is a 2-polytope and a three-dimensional polyhedron is a 3-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.

<span class="mw-page-title-main">Hypercube</span> Convex polytope, the n-dimensional analogue of a square and a cube

In geometry, a hypercube is an n-dimensional analogue of a square and a cube ; the special case for n = 4 is known as a tesseract. 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. In particular, all its elements or $j$ -faces — cells, faces and so on — are also transitive on the symmetries of the polytope, and are themselves regular polytopes of dimension $j \leq n$ .

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 C₁₆, hexadecachoron, or hexdecahedroid [sic?].

In six-dimensional geometry, a uniform 6-polytope is a six-dimensional uniform polytope. A uniform polypeton is vertex-transitive, and all facets are uniform 5-polytopes.

<span class="mw-page-title-main">Demihypercube</span> Polytope constructed from alternation of an hypercube

In geometry, demihypercubes (also called n-demicubes, n-hemicubes, and half measure polytopes) are a class of n-polytopes constructed from alternation of an n-hypercube, labeled as hγ_n for being half of the hypercube family, γ_n. Half of the vertices are deleted and new facets are formed. The 2n facets become 2n(n−1)-demicubes, and 2ⁿ(n−1)-simplex facets are formed in place of the deleted vertices.

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-cube is a six-dimensional hypercube with 64 vertices, 192 edges, 240 square faces, 160 cubic cells, 60 tesseract 4-faces, and 12 5-cube 5-faces.

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.

<span class="mw-page-title-main">7-orthoplex</span> Regular 7- polytope

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.

<span class="mw-page-title-main">10-orthoplex</span>

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 $for n \geq 4. A lower symmetry form can be created by removing another mirror on an order-4 peak.$

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 2₂₁ polytope is a uniform 6-polytope, constructed within the symmetry of the E₆ group. It was discovered by Thorold Gosset, published in his 1900 paper. He called it an 6-ic semi-regular figure. It is also called the Schläfli polytope.

In 7-dimensional geometry, the 3₂₁ polytope is a uniform 7-polytope, constructed within the symmetry of the E₇ group. It was discovered by Thorold Gosset, published in his 1900 paper. He called it an 7-ic semi-regular figure.

In 8-dimensional geometry, the 4₂₁ is a semiregular uniform 8-polytope, constructed within the symmetry of the E₈ group. It was discovered by Thorold Gosset, published in his 1900 paper. He called it an 8-ic semi-regular figure.

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

Weisstein, Eric W. "Cross polytope". MathWorld .

v t e Fundamental convex regular and uniform polytopes in dimensions 2–10
Family	A_n	B_n	I₂(p) / D_n	E₆ / E₇ / E₈ / F₄ / G₂	H_n
Regular polygon	Triangle	Square	p-gon	Hexagon	Pentagon
Uniform polyhedron	Tetrahedron	Octahedron • Cube	Demicube		Dodecahedron • Icosahedron
Uniform polychoron	Pentachoron	16-cell • Tesseract	Demitesseract	24-cell	120-cell • 600-cell
Uniform 5-polytope	5-simplex	5-orthoplex • 5-cube	5-demicube
Uniform 6-polytope	6-simplex	6-orthoplex • 6-cube	6-demicube	1₂₂ • 2₂₁
Uniform 7-polytope	7-simplex	7-orthoplex • 7-cube	7-demicube	1₃₂ • 2₃₁ • 3₂₁
Uniform 8-polytope	8-simplex	8-orthoplex • 8-cube	8-demicube	1₄₂ • 2₄₁ • 4₂₁
Uniform 9-polytope	9-simplex	9-orthoplex • 9-cube	9-demicube
Uniform 10-polytope	10-simplex	10-orthoplex • 10-cube	10-demicube
Uniform n-polytope	n-simplex	n-orthoplex • n-cube	n-demicube	1_k2 • 2_k1 • k₂₁	n-pentagonal polytope
Topics: Polytope families • Regular polytope • List of regular polytopes and compounds

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[FOOTNOTECoxeter1973121–122§7.21._illustration_Fig_7-2<small>B</small>-1] Coxeter 1973, pp. 121–122, §7.21. illustration Fig 7-2B.

[2] Conway, J. H.; Sloane, N. J. A. (1991). "The Cell Structures of Certain Lattices". In Hilton, P.; Hirzebruch, F.; Remmert, R. (eds.). Miscellanea Mathematica. Berlin: Springer. pp. 89–90. doi:10.1007/978-3-642-76709-8_5. ISBN 978-3-642-76711-1.

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

[4] Weisstein, Eric W. "Cocktail Party Graph". MathWorld .

[FOOTNOTECoxeter1973120–124§7.2-5] Coxeter 1973, pp. 120–124, §7.2.

[FOOTNOTECoxeter1973121§7.2.2.-6] Coxeter 1973, p. 121, §7.2.2..

[7] 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 .

[8] Coxeter, Regular Complex Polytopes, p. 108

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v t e Dimension
Dimensional spaces	Vector space Euclidean space Affine space Projective space Free module Manifold Algebraic variety Spacetime
Other dimensions	Krull Lebesgue covering Inductive Hausdorff Minkowski Fractal Degrees of freedom
Polytopes and shapes	Hyperplane Hypersurface Hypercube Hyperrectangle Demihypercube Hypersphere Cross-polytope Simplex Hyperpyramid
Number systems	Hypercomplex numbers Cayley–Dickson construction
Dimensions by number	Zero One Two Three Four Five Six Seven Eight n-dimensions
See also	Hyperspace Codimension
Category