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0/1-Polytopes in 3D

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The convex hull of a set is the smallest convex set that contains . For instance, the convex hull of three distinct points is a triangle or a line segment.

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A 0/1-polytope is the convex hull of a set of points with coordinates 0 or 1. In other words, a 0/1-polytope is the convex hull of a subset of vertices of a hypercube (the generalization of a cube to any number of dimensions).

A 3D cube has eight vertices, so it has subsets of vertices. This Demonstration shows the corresponding 256 0/1-polytopes.

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Contributed by: George Beck (June 2014)
Open content licensed under CC BY-NC-SA

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In 2D, there are four vertices and convex sets; in 4D, the hypercube has 16 vertices, so there are convex hulls. Neither the 2D nor 4D case is shown here. The 2D case is too easy; the 4D examples would be interesting to project into 3D, say into the 3-space orthogonal to the vector .

Reference

[1] H. Ziegler, Lectures on Polytopes, New York: Springer, 1995 pp. 19–22.

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