Cross-polytope

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.

Cross-polytopes of dimension 2 to 5

2 dimensions square	3 dimensions octahedron
4 dimensions 16-cell	5 dimensions 5-orthoplex

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:

${\displaystyle \{x\in \mathbb {R} ^{n}:\|x\|_{1}\leq 1\}.}$

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

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

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):

${\displaystyle 2^{k+1}{n \choose {k+1}}}$^[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	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, ... ${\displaystyle 2^{k+1}{n \choose k+1}}$ 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 ${\displaystyle \mathbb {\mathbb {C} } ^{n}}$. 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
${\displaystyle \mathbb {R} ^{2}}$	2{4}2 = {4} = K2,2	${\displaystyle \mathbb {\mathbb {C} } ^{2}}$	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
${\displaystyle \mathbb {R} ^{3}}$	2{3}2{4}2 = {3,4} = K2,2,2	${\displaystyle \mathbb {\mathbb {C} } ^{3}}$	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
${\displaystyle \mathbb {R} ^{4}}$	2{3}2{3}2 {3,3,4} = K2,2,2,2	${\displaystyle \mathbb {\mathbb {C} } ^{4}}$	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
${\displaystyle \mathbb {R} ^{5}}$	2{3}2{3}2{3}2{4}2 {3,3,3,4} = K2,2,2,2,2	${\displaystyle \mathbb {\mathbb {C} } ^{5}}$	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
${\displaystyle \mathbb {R} ^{6}}$	2{3}2{3}2{3}2{3}2{4}2 {3,3,3,3,4} = K2,2,2,2,2,2	${\displaystyle \mathbb {\mathbb {C} } ^{6}}$	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

Citations

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.
5. Coxeter 1973, pp. 120–124, §7.2.
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

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	An	Bn	I2(p) / Dn	E6 / E7 / E8 / F4 / G2	Hn
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	1k2 • 2k1 • k21	n-pentagonal polytope
Topics: Polytope families • Regular polytope • List of regular polytopes and compounds