Cross-polytopes of dimension 2 to 5

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

Ingeometry, across-polytope,^[1]hyperoctahedron,orthoplex,^[2]staurotope,^[3] orcocube is aregular,convex polytope that exists inn-dimensional Euclidean space. A 2-dimensional cross-polytope is a square, a 3-dimensional cross-polytope is a regularoctahedron, and a 4-dimensional cross-polytope is a16-cell. Its facets aresimplexes of the previous dimension, while the cross-polytope'svertex 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 theconvex hull of its vertices. Then-dimensional cross-polytope can also be defined as the closedunit ball (or, according to some authors, its boundary) in theℓ₁-norm onRⁿ, those pointsx = (x₁,x₂...,x_n) satisfying

${\displaystyle |x_1|+|x_2|+\cdots +|x_n|\le 1.}$

Ann-orthoplex can be constructed as abipyramid with an (n−1)-orthoplex base.

The cross-polytope is thedual polytope of thehypercube. Thevertex-edge graph of ann-dimensional cross-polytope is theTurán graphT(2n,n) (also known as acocktail party graph^[4]).

In 1 dimension the cross-polytope is aline segment, which can be chosen as theinterval [−1, +1].

In 2 dimensions the cross-polytope is asquare. If the vertices are chosen as {(±1, 0), (0, ±1)}, the square's sides are at right angles to the axes; in this orientation a square is often called adiamond.

In 3 dimensions the cross-polytope is aregular octahedron—one of the five convex regularpolyhedra known as thePlatonic solids.

The 4-dimensional cross-polytope also goes by the namehexadecachoron or16-cell. It is one of the sixconvex regular 4-polytopes. These4-polytopes were first described by the Swiss mathematicianLudwig Schläfli in the mid-19th century. The vertices of the 4-dimensional hypercube, ortesseract, can be divided into two sets of eight, the convex hull of each set forming a cross-polytope. Moreover, the polytope known as the24-cell can be constructed by symmetrically arranging three cross-polytopes.^[5]

The cross-polytope family is one of threeregular polytope families, labeled byCoxeter asβ_n, the other two being thehypercube family, labeled asγ_n, and thesimplex family, labeled asα_n. A fourth family, theinfinite tessellations of hypercubes, he labeled asδ_n.^[6]

Then-dimensional cross-polytope has 2n vertices, and 2ⁿ facets ((n − 1)-dimensional components) all of which are (n − 1)-simplices. Thevertex figures are all (n − 1)-cross-polytopes. TheSchläfli symbol of the cross-polytope is {3,3,...,3,4}.

Thedihedral angle of then-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 then-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 ofk + 1 orthogonal vertices corresponds to a distinctk-dimensional component which contains them. The number ofk-dimensional components (vertices, edges, faces, ..., facets) in ann-dimensional cross-polytope is thus given by (seebinomial coefficient):

${\displaystyle 2^{k+1}(\genfrac{}{}{0ex}{}{n}{k+1})}$^[7]

The extendedf-vector for ann-orthoplex can be computed by (1,2)ⁿ, like the coefficients ofpolynomial products. For example a 16-cell is (1,2)⁴ = (1,4,4)² = (1,8,24,32,16).

There are many possibleorthographic 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 abipyramid, 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
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
...
n	βn (n−3)11	n-orthoplex n-cross		{3n − 2,4} {3n − 3,31,1} n{}	... ... ...	2n0-faces, ...${\displaystyle 2^{k+1}(\genfrac{}{}{0ex}{}{n}{k+1})}$k-faces ..., 2n(n−1)-faces

The vertices of an axis-aligned cross polytope are all at equal distance from each other in theManhattan distance (L¹ norm).Kusner's conjecture states that this set of 2d points is the largest possibleequidistant set for this distance.^[8]

Regularcomplex polytopes can be defined incomplexHilbert space calledgeneralized orthoplexes (or cross polytopes), β^p
_n =₂{3}₂{3}...₂{4}_p, or... Real solutions exist withp = 2, i.e. β²
_n = β_n =₂{3}₂{3}...₂{4}₂ = {3,3,..,4}. Forp > 2, they exist in${\displaystyle {\mathbb{C}}^n}$. Ap-generalizedn-orthoplex haspn vertices.Generalized orthoplexes have regularsimplexes (real) asfacets.^[9] Generalized orthoplexes makecomplete multipartite graphs, β^p
₂ make K_p,p forcomplete bipartite graph, β^p
₃ make K_p,p,p for complete tripartite graphs β^p
_n creates K_pⁿ orTurán graphs${\displaystyle T(np,n)}$. Anorthogonal projection can be defined that maps all the vertices equally-spaced on a circle, with all pairs of vertices connected, except multiples ofn. Theregular polygon perimeter in these orthogonal projections is called apetrie 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	${\displaystyle {\mathbb{C}}^2}$	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	${\displaystyle {\mathbb{C}}^3}$	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	${\displaystyle {\mathbb{C}}^4}$	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	${\displaystyle {\mathbb{C}}^5}$	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	${\displaystyle {\mathbb{C}}^6}$	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

Cross-polytopes can be combined with their dual cubes to form compound polytopes:

• In two dimensions, we obtain theoctagrammic star figure {⁠8/2⁠},
• In three dimensions we obtain thecompound of cube and octahedron,
• In four dimensions we obtain the compound of tesseract and 16-cell.

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

• Coxeter, H.S.M. (1973).Regular Polytopes (3rd ed.). New York: Dover.
  • pp. 121–122, §7.21, illustration Fig 7.2B
  • p. 296, Table I (iii): Regular Polytopes, three regular polytopes in n-dimensions (n≥5)

Wikimedia Commons has media related toCross-polytope graphs.

• Weisstein, Eric W."Cross polytope".MathWorld.

• Regular polytopes
• Multi-dimensional geometry

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