Cross-polytope
In geometry, a cross-polytope,^{[1]} orthoplex,^{[2]} hyperoctahedron, or cocube is a regular, convex polytope that exists in any number of dimensions. 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. Its facets are simplexes of the previous dimension, while the cross-polytope's vertex figure is another cross-polytope from the previous dimension.
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}:
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.
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-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 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
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
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 |
Graph 2(n-1)-gon |
Schläfli | Coxeter-Dynkin diagrams |
Vertices | Edges | Faces | Cells | 4-faces | 5-faces | 6-faces | 7-faces | 8-faces | 9-faces | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
1 | β_{1} | Line segment 1-orthoplex |
{} | Template:CDD Template:CDD |
2 | ||||||||||||
2 | β_{2} −1_{11} |
square 2-orthoplex Bicross |
{4} {}+{} |
Template:CDD Template:CDD |
4 | 4 | |||||||||||
3 | β_{3} 0_{11} |
octahedron 3-orthoplex Tricross |
{3,4} {3^{0,1,1}} {}+{}+{} |
Template:CDD Template:CDD Template:CDD |
6 | 12 | 8 | ||||||||||
4 | β_{4} 1_{11} |
16-cell 4-orthoplex Tetracross |
{3,3,4} {3^{1,1,1}} 4{} |
Template:CDD Template:CDD Template:CDD |
8 | 24 | 32 | 16 | |||||||||
5 | β_{5} 2_{11} |
5-orthoplex Pentacross |
{3^{3},4} {3^{2,1,1}} 5{} |
Template:CDD Template:CDD Template:CDD |
10 | 40 | 80 | 80 | 32 | ||||||||
6 | β_{6} 3_{11} |
6-orthoplex Hexacross |
{3^{4},4} {3^{3,1,1}} 6{} |
Template:CDD Template:CDD Template:CDD |
12 | 60 | 160 | 240 | 192 | 64 | |||||||
7 | β_{7} 4_{11} |
7-orthoplex Heptacross |
{3^{5},4} {3^{4,1,1}} 7{} |
Template:CDD Template:CDD Template:CDD |
14 | 84 | 280 | 560 | 672 | 448 | 128 | ||||||
8 | β_{8} 5_{11} |
8-orthoplex Octacross |
{3^{6},4} {3^{5,1,1}} 8{} |
Template:CDD Template:CDD Template:CDD |
16 | 112 | 448 | 1120 | 1792 | 1792 | 1024 | 256 | |||||
9 | β_{9} 6_{11} |
9-orthoplex Enneacross |
{3^{7},4} {3^{6,1,1}} 9{} |
Template:CDD Template:CDD Template:CDD |
18 | 144 | 672 | 2016 | 4032 | 5376 | 4608 | 2304 | 512 | ||||
10 | β_{10} 7_{11} |
10-orthoplex Decacross |
{3^{8},4} {3^{7,1,1}} 10{} |
Template:CDD Template:CDD Template:CDD |
20 | 180 | 960 | 3360 | 8064 | 13440 | 15360 | 11520 | 5120 | 1024 | |||
... | |||||||||||||||||
n | β_{n} k_{11} |
n-orthoplex n-cross |
{3^{n − 2},4} {3^{n − 3,1,1}} n{} |
Template:CDD...Template:CDD Template:CDD...Template:CDD Template:CDD...Template:CDD |
2n 0-faces, ... 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.^{[3]}
See also
- List of regular polytopes
- Hyperoctahedral group, the symmetry group of the cross-polytope
Notes
References
- {{#invoke:citation/CS1|citation
|CitationClass=book }} p. 296, Table I (iii): Regular Polytopes, three regular polytopes in n-dimensions (n>=5)
External links
- Weisstein, Eric W., "Cross polytope", MathWorld.
- Polytope ViewerTemplate:Dead link (Click <polytopes...> to select cross polytope.)
- Template:GlossaryForHyperspace