Cross-polytope

In geometry, a cross-polytope, orthoplex, hyperoctahedron, or cocube is a regular, convex polytope that exists in any number of dimensions. The vertices of a cross-polytope consist of all permutations of (±1, 0, 0, …, 0). The cross-polytope is the convex hull of its vertices. (Note: some authors define a cross-polytope only as the boundary of this region.)

The n-dimensional cross-polytope can also be defined as the closed unit ball in the ℓ₁-norm on Rⁿ:

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 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-polytope. These polychora were first described by the Swiss mathematician Ludwig Schläfli in the mid-19th century.

Higher dimensions

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

The volume of the n-dimensional cross-polytope is

For n ≠ 1, a two dimensional graph of the edges of the n-dimensional cross-polytope can be constructed by drawing 2n vertices on a circle and connecting all pairs of vertices except for vertices exactly on opposite sides of the circle. (These unattached pairs represent the vertex pairs on opposite directions of one coordinate axis of the polytope.) To put this more abstractly, the graph is the complement of a matching of n edges.

Cross-polytope elements

n	βn k11	Name(s) Graph	Graph	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	Bicross square 2-orthoplex		{4} {} x {}		4	4
3	β3 011	Tricross octahedron 3-orthoplex		{3,4} {30,1,1}		6	12	8
4	β4 111	Tetracross 16-cell 4-orthoplex		{3,3,4} {31,1,1}		8	24	32	16
5	β5 211	Pentacross 5-orthoplex		{33,4} {32,1,1}		10	40	80	80	32
6	β6 311	Hexacross 6-orthoplex		{34,4} {33,1,1}		12	60	160	240	192	64
7	β7 411	Heptacross 7-orthoplex		{35,4} {34,1,1}		14	84	280	560	672	448	128
8	β8 511	Octacross 8-orthoplex		{36,4} {35,1,1}		16	112	448	1120	1792	1792	1024	256
9	β9 611	Enneacross 9-orthoplex		{37,4} {36,1,1}		18	144	672	2016	4032	5376	4608	2304	512
10	β10 711	Decacross 10-orthoplex		{38,4} {37,1,1}		20	180	960	3360	8064	13440	15360	11520	5120	1024
...
n	βn (n − 3)11	n-cross n-orthoplex		{3n − 2,4} {3n − 3,1,1}	... ...	2n

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