Cross Polytope

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The cross polytope beta_n is the regular polytope in n dimensions corresponding to the convex hull of the points formed by permuting the coordinates (+/-1, 0, 0, ..., 0). A cross-polytope (also called an orthoplex) is denoted beta_n and has 2n vertices and Schläfli symbol {3,...,3_()_(n-2),4}. The cross polytope is named because its 2n vertices are located equidistant from the origin along the Cartesian axes in Euclidean space, which each such axis perpendicular to all others. A cross polytope is bounded by 2^n (n-1)-simplexes, and is a dipyramid erected (in both directions) into the nth dimension, with an (n-1)-dimensional cross polytope as its base.

LineSegmentSquareOctahedron16Cell

In one dimension, the cross polytope is the line segment [-1,1]. In two dimensions, the cross polytope {4} is the filled square with vertices (-1,0), (0,-1), (1,0), (0,1). In three dimensions, the cross polytope {3,4} is the convex hull of the octahedron with vertices (-1,0,0), (0,-1,0), (0,0,-1), (1,0,0), (0,1,0), (0,0,1). In four dimensions, the cross polytope {3,3,4} is the 16-cell, depicted in the above figure by projecting onto one of the four mutually perpendicular three-spaces within the four-space obtained by dropping one of the four vertex components (R. Towle).

CrossPolytopeGraphs

The skeleton of beta_n is isomorphic with the circulant graph Ci_(2n)(1,2,...,n-1), also known as the cocktail party graph K_(n×2).

For all dimensions, the dual of the cross polytope is the hypercube (and vice versa). Consequently, the number of k-simplices contained in an n-cross polytope is (n; k+1)2^(k+1).

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