Cross-polytope
2 dimensions square |
3 dimensions octahedron |
4 dimensions 16-cell |
5 dimensions 5-orthoplex |
In geometry, a cross-polytope,^{[1]} hyperoctahedron, orthoplex,^{[2]} 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.
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 ℓ_{1}-norm on R^{n}:
- [math]\displaystyle{ \{x\in\mathbb R^n : \|x\|_1 \le 1\}. }[/math]
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 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 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}.^{[3]}
The n-dimensional cross-polytope has 2n vertices, and 2^{n} 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 [math]\displaystyle{ \delta_n = \arccos\left(\frac{2-n}{n}\right) }[/math]. This gives: δ_{2} = arccos(0/2) = 90°, δ_{3} = arccos(−1/3) = 109.47°, δ_{4} = arccos(−2/4) = 120°, δ_{5} = arccos(−3/5) = 126.87°, ... δ_{∞} = arccos(−1) = 180°.
The hypervolume of the n-dimensional cross-polytope is
- [math]\displaystyle{ \frac{2^n}{n!}. }[/math]
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):
- [math]\displaystyle{ 2^{k+1}{n \choose {k+1}} }[/math]^{[4]}
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.
n | β_{n} k_{11} |
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 | β_{0} | Point 0-orthoplex |
. | ( ) | 1 | |||||||||||
1 | β_{1} | Line segment 1-orthoplex |
{ } | 2 | 1 | |||||||||||
2 | β_{2} −1_{11} |
square 2-orthoplex Bicross |
{4} 2{ } = { }+{ } |
4 | 4 | 1 | ||||||||||
3 | β_{3} 0_{11} |
octahedron 3-orthoplex Tricross |
{3,4} {3^{1,1}} 3{ } |
6 | 12 | 8 | 1 | |||||||||
4 | β_{4} 1_{11} |
16-cell 4-orthoplex Tetracross |
{3,3,4} {3,3^{1,1}} 4{ } |
8 | 24 | 32 | 16 | 1 | ||||||||
5 | β_{5} 2_{11} |
5-orthoplex Pentacross |
{3^{3},4} {3,3,3^{1,1}} 5{ } |
10 | 40 | 80 | 80 | 32 | 1 | |||||||
6 | β_{6} 3_{11} |
6-orthoplex Hexacross |
{3^{4},4} {3^{3},3^{1,1}} 6{ } |
12 | 60 | 160 | 240 | 192 | 64 | 1 | ||||||
7 | β_{7} 4_{11} |
7-orthoplex Heptacross |
{3^{5},4} {3^{4},3^{1,1}} 7{ } |
14 | 84 | 280 | 560 | 672 | 448 | 128 | 1 | |||||
8 | β_{8} 5_{11} |
8-orthoplex Octacross |
{3^{6},4} {3^{5},3^{1,1}} 8{ } |
16 | 112 | 448 | 1120 | 1792 | 1792 | 1024 | 256 | 1 | ||||
9 | β_{9} 6_{11} |
9-orthoplex Enneacross |
{3^{7},4} {3^{6},3^{1,1}} 9{ } |
18 | 144 | 672 | 2016 | 4032 | 5376 | 4608 | 2304 | 512 | 1 | |||
10 | β_{10} 7_{11} |
10-orthoplex Decacross |
{3^{8},4} {3^{7},3^{1,1}} 10{ } |
20 | 180 | 960 | 3360 | 8064 | 13440 | 15360 | 11520 | 5120 | 1024 | 1 | ||
... | ||||||||||||||||
n | β_{n} k_{11} |
n-orthoplex n-cross |
{3^{n − 2},4} {3^{n − 3},3^{1,1}} n{} |
... ... ... |
2n 0-faces, ... [math]\displaystyle{ 2^{k+1}{n\choose k+1} }[/math] 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 (L^{1} norm). Kusner's conjecture states that this set of 2d points is the largest possible equidistant set for this distance.^{[5]}
Generalized orthoplex
Regular complex polytopes can be defined in complex Hilbert space called generalized orthoplexes (or cross polytopes), βpn = _{2}{3}_{2}{3}..._{2}{4}_{p}, or ... Real solutions exist with p = 2, i.e. β2n = β_{n} = _{2}{3}_{2}{3}..._{2}{4}_{2} = {3,3,..,4}. For p > 2, they exist in [math]\displaystyle{ \mathbb{\Complex}^n }[/math]. A p-generalized n-orthoplex has pn vertices. Generalized orthoplexes have regular simplexes (real) as facets.^{[6]} Generalized orthoplexes make complete multipartite graphs, βp2 make K_{p,p} for complete bipartite graph, βp3 make K_{p,p,p} for complete tripartite graphs. βpn 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.
p = 2 | p = 3 | p = 4 | p = 5 | p = 6 | p = 7 | p = 8 | ||
---|---|---|---|---|---|---|---|---|
[math]\displaystyle{ \mathbb{R}^2 }[/math] | _{2}{4}_{2} = {4} = K_{2,2} |
[math]\displaystyle{ \mathbb{\Complex}^2 }[/math] | _{2}{4}_{3} = K_{3,3} |
_{2}{4}_{4} = K_{4,4} |
_{2}{4}_{5} = K_{5,5} |
_{2}{4}_{6} = K_{6,6} |
_{2}{4}_{7} = K_{7,7} |
_{2}{4}_{8} = K_{8,8} |
[math]\displaystyle{ \mathbb{R}^3 }[/math] | _{2}{3}_{2}{4}_{2} = {3,4} = K_{2,2,2} |
[math]\displaystyle{ \mathbb{\Complex}^3 }[/math] | _{2}{3}_{2}{4}_{3} = K_{3,3,3} |
_{2}{3}_{2}{4}_{4} = K_{4,4,4} |
_{2}{3}_{2}{4}_{5} = K_{5,5,5} |
_{2}{3}_{2}{4}_{6} = K_{6,6,6} |
_{2}{3}_{2}{4}_{7} = K_{7,7,7} |
_{2}{3}_{2}{4}_{8} = K_{8,8,8} |
[math]\displaystyle{ \mathbb{R}^4 }[/math] | _{2}{3}_{2}{3}_{2} {3,3,4} = K_{2,2,2,2} |
[math]\displaystyle{ \mathbb{\Complex}^4 }[/math] | _{2}{3}_{2}{3}_{2}{4}_{3} K_{3,3,3,3} |
_{2}{3}_{2}{3}_{2}{4}_{4} K_{4,4,4,4} |
_{2}{3}_{2}{3}_{2}{4}_{5} K_{5,5,5,5} |
_{2}{3}_{2}{3}_{2}{4}_{6} K_{6,6,6,6} |
_{2}{3}_{2}{3}_{2}{4}_{7} K_{7,7,7,7} |
_{2}{3}_{2}{3}_{2}{4}_{8} K_{8,8,8,8} |
[math]\displaystyle{ \mathbb{R}^5 }[/math] | _{2}{3}_{2}{3}_{2}{3}_{2}{4}_{2} {3,3,3,4} = K_{2,2,2,2,2} |
[math]\displaystyle{ \mathbb{\Complex}^5 }[/math] | _{2}{3}_{2}{3}_{2}{3}_{2}{4}_{3} K_{3,3,3,3,3} |
_{2}{3}_{2}{3}_{2}{3}_{2}{4}_{4} K_{4,4,4,4,4} |
_{2}{3}_{2}{3}_{2}{3}_{2}{4}_{5} K_{5,5,5,5,5} |
_{2}{3}_{2}{3}_{2}{3}_{2}{4}_{6} K_{6,6,6,6,6} |
_{2}{3}_{2}{3}_{2}{3}_{2}{4}_{7} K_{7,7,7,7,7} |
_{2}{3}_{2}{3}_{2}{3}_{2}{4}_{8} K_{8,8,8,8,8} |
[math]\displaystyle{ \mathbb{R}^6 }[/math] | _{2}{3}_{2}{3}_{2}{3}_{2}{3}_{2}{4}_{2} {3,3,3,3,4} = K_{2,2,2,2,2,2} |
[math]\displaystyle{ \mathbb{\Complex}^6 }[/math] | _{2}{3}_{2}{3}_{2}{3}_{2}{3}_{2}{4}_{3} K_{3,3,3,3,3,3} |
_{2}{3}_{2}{3}_{2}{3}_{2}{3}_{2}{4}_{4} K_{4,4,4,4,4,4} |
_{2}{3}_{2}{3}_{2}{3}_{2}{3}_{2}{4}_{5} K_{5,5,5,5,5,5} |
_{2}{3}_{2}{3}_{2}{3}_{2}{3}_{2}{4}_{6} K_{6,6,6,6,6,6} |
_{2}{3}_{2}{3}_{2}{3}_{2}{3}_{2}{4}_{7} K_{7,7,7,7,7,7} |
_{2}{3}_{2}{3}_{2}{3}_{2}{3}_{2}{4}_{8} K_{8,8,8,8,8,8} |
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
- ↑ Coxeter 1973, pp. 121-122, §7.21. illustration Fig 7-2B.
- ↑ Conway, J. H.; Sloane, N. J. A. (1991). "The Cell Structures of Certain Lattices". in Hilton, P.; Hirzebruch, F.; Remmert, R.. Miscellanea Mathematica. Berlin: Springer. pp. 89–90. doi:10.1007/978-3-642-76709-8_5. ISBN 978-3-642-76711-1.
- ↑ Coxeter 1973, pp. 120-124, §7.2.
- ↑ Coxeter 1973, p. 121, §7.2.2..
- ↑ Guy, Richard K. (1983), "An olla-podrida of open problems, often oddly posed", American Mathematical Monthly 90 (3): 196–200, doi:10.2307/2975549.
- ↑ 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
Original source: https://en.wikipedia.org/wiki/Cross-polytope.
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