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 ρ 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 ρ 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 ρ constant, independently of the orthogonal projection. We discuss some examples of constant mean curvature polytopes.