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,^[1] orthoplex,^[2] hyperoctahedron, or cocube is a regular, convex polytope that exists in n-dimensions. A 2-orthoplex is a square, a 3-orthoplex is a regular octahedron, and a 4-orthoplex 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 are 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. Higher-dimensional cross-polytopes are generalizations of these.

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 ${\displaystyle \delta _{n}=\arccos \left({\frac {2-n}{n}}\right)}$. 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 volume of the n-dimensional cross-polytope is

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

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

$2^{k+1}{n \choose {k+1}}$^[3]

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.^[4]

Generalized orthoplex

Regular complex polytopes can be defined in complex Hilbert space called generalized orthoplexes (or cross polytopes), βp
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 $\mathbb {C} ^{n}$. A p-generalized n-orthoplex has pn vertices. Generalized orthoplexes have regular simplexes (real) as facets.^[5] Generalized orthoplexes make complete multipartite graphs, βp
2 make K_p,p for complete bipartite graph, βp
3 make K_p,p,p for complete tripartite graphs. βp
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	$\mathbb{C}^2$	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	$\mathbb {C} ^{3}$	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	${\displaystyle \mathbb {C} ^{4}}$	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	$\mathbb{C}^5$	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	${\displaystyle \mathbb {C} ^{6}}$	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

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. Coxeter, Regular polytopes, p.120
4. 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.
5. Coxeter, Regular Complex Polytopes, p. 108

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". Glossary for Hyperspace. Archived from the original on 4 February 2007.

Fundamental convex regular and uniform polytopes in dimensions 2–10
Family	An	Bn	I2(p) / Dn	E6 / E7 / E8 / E9 / E10 / F4 / G2	Hn
Regular polygon	Triangle	Square	p-gon	Hexagon	Pentagon
Uniform polyhedron	Tetrahedron	Octahedron • Cube	Demicube		Dodecahedron • Icosahedron
Uniform 4-polytope	5-cell	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	1k2 • 2k1 • k21	n-pentagonal polytope
Topics: Polytope families • Regular polytope • List of regular polytopes and compounds