Constant mean curvature polytopes and hypersurfaces via projections

Given a regular polytope in Euclidean space and an orthogonal projection to the complex plane, the function which assigns to each vertex its projected value satisfies a quadratic difference equation. The form of the equation is the same, whatever the polytope, except for a real parameter rho which varies from polytope to polytope. It is independent of the projection used and the size of the polytope. When we consider an orthogonal projection of a smooth hypersurface in Euclidean space, remarkably we find the same phenomena, namely that a smooth version of the equation is satisfied independently of the projection, where the parameter rho depends only on the mean curvature. We therefore make an unconventional definition of a constant mean-curvature polytope as one which satisfies this same equation with rho constant, independently of the orthogonal projection. We discuss some examples of constant mean curvature polytopes. (C) 2013 Elsevier B.V. All rights reserved.