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 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 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.

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):

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

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	βn 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} {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
7	β7 411	7-orthoplex Heptacross		{35,4} {34,31,1} 7{ }		14	84	280	560	672	448	128	1
8	β8 511	8-orthoplex Octacross		{36,4} {35,31,1} 8{ }		16	112	448	1120	1792	1792	1024	256	1
9	β9 611	9-orthoplex Enneacross		{37,4} {36,31,1} 9{ }		18	144	672	2016	4032	5376	4608	2304	512	1
10	β10 711	10-orthoplex Decacross		{38,4} {37,31,1} 10{ }		20	180	960	3360	8064	13440	15360	11520	5120	1024	1
...
n	βn k11	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 (L¹ 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. 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. 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} = K2,2	$\mathbb {\mathbb {C} } ^{2}$	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	$\mathbb {\mathbb {C} } ^{3}$	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	$\mathbb {\mathbb {C} } ^{4}$	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	$\mathbb {\mathbb {C} } ^{5}$	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	$\mathbb {\mathbb {C} } ^{6}$	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 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

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.