Cross-polytope

In geometry, a cross-polytope,^[1] orthoplex,^[2] hyperoctahedron, or cocube is a regular, convex polytope that exists in any number of dimensions. The vertices of a cross-polytope are all the permutations of (±1, 0, 0, …, 0). The cross-polytope is the convex hull of its vertices. Its facets are simplexes of the previous dimension, while the cross-polytope's vertex figure is another cross-polytope from the previous dimension.

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:

$\{x\in\mathbb R^n : \|x\|_1 \le 1\}.$

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. Higher-dimensional cross-polytopes are generalizations of these.

2 dimensions square		3 dimensions octahedron		4 dimensions 16-cell

The cross-polytope is the dual polytope of the hypercube. The 1-skeleton of a n-dimensional cross-polytope is a Turán graph T(2n,n).

4 dimensions

The 4-dimensional cross-polytope also goes by the name hexadecachoron or 16-cell. It is one of 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 simplices, 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

$\arccos\left(\frac{2-n}{n}\right)$.

The number of k-dimensional components (vertices, edges, faces, …, facets) in an n-dimensional cross-polytope is given by (see binomial coefficient):

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

The volume of the n-dimensional cross-polytope is

$\frac{2^n}{n!}.$

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
1	β1	Line segment 1-orthoplex		{ }		2
2	β2 −111	square 2-orthoplex Bicross		{4} 2{ } = { }+{ }		4	4
3	β3 011	octahedron 3-orthoplex Tricross		{3,4} {3^0,1,1} 3{ }		6	12	8
4	β4 111	16-cell 4-orthoplex Tetracross		{3,3,4} {3^1,1,1} 4{ }		8	24	32	16
5	β5 211	5-orthoplex Pentacross		{3^3,4} {3^2,1,1} 5{ }		10	40	80	80	32
6	β6 311	6-orthoplex Hexacross		{3^4,4} {3^3,1,1} 6{ }		12	60	160	240	192	64
7	β7 411	7-orthoplex Heptacross		{3^5,4} {3^4,1,1} 7{ }		14	84	280	560	672	448	128
8	β8 511	8-orthoplex Octacross		{3^6,4} {3^5,1,1} 8{ }		16	112	448	1120	1792	1792	1024	256
9	β9 611	9-orthoplex Enneacross		{3^7,4} {3^6,1,1} 9{ }		18	144	672	2016	4032	5376	4608	2304	512
10	β10 711	10-orthoplex Decacross		{3^8,4} {3^7,1,1} 10{ }		20	180	960	3360	8064	13440	15360	11520	5120	1024
...
n	βn k11	n-orthoplex n-cross		{3^n − 2,4} {3^n − 3,1,1} n{}	... ... ...	2n 0-faces, ... $2^{k+1}{n\choose k+1}$ k-faces ..., 2^n (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.^[3]

See also

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

Notes

1. Elte, E. L. (1912), The Semiregular Polytopes of the Hyperspaces, Groningen: University of Groningen Chapter IV, five dimensional semiregular polytope
2. Conway calls it an n-orthoplex for orthant complex.
3. 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.

References

• Coxeter, H. S. M. (1973). Regular Polytopes (3rd ed.). New York: Dover Publications. pp. 121–122. ISBN 0-486-61480-8. p. 296, Table I (iii): Regular Polytopes, three regular polytopes in n-dimensions (n>=5)

External links

• Weisstein, Eric W., "Cross polytope", MathWorld.
• Polytope Viewer (Click <polytopes...> to select cross polytope.)
• Olshevsky, George, Cross polytope at Glossary for Hyperspace.