Cross-polytope

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In geometry, a cross-polytope, or orthoplex, is a regular, convex polytope that exists in any number of dimensions. The vertices of a cross-polytope consist of all permutations of (±1, 0, 0, …, 0). The cross-polytope is the convex hull of its vertices. (Note: some authors define a convex-polytope only as the boundary of this region).

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 regular polyhedra known as the Platonic solids. Higher dimensional cross-polytopes are generalizations of these.

Image:Square diamond (shape).png Image:Octahedron.jpg Image:Cell16-4dpolytope.png
2 dimensions 3 dimensions 4 dimensions


The cross-polytope can also be characterized as the dual polytope of the hypercube.

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4 dimensions

The 4-dimensional cross-polytope also goes by the name hexadecachoron or 16-cell. It is one of six regular convex polychora. These polychora were first described by the Swiss mathematician Ludwig Schläfli in the mid-19th century.

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Higher dimensions

In n > 4 dimensions there are only three regular polytopes: the simplex, hypercube, and the cross-polytope, of which the last two are duals. The simplex is self-dual.

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

<math>2^{k+1}{n \choose {k+1}}</math>

For the first few n and k we have:

n\k 0 1 2 3 4
1 2
2 4 4
3 6 12 8
4 8 24 32 16
5 10 40 80 80 32
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See also

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External links

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