Isomorphism classes of cut loci for a cube

Abstract

We prove that a face of a cube can be optimally partitioned into connected 193 sets on which the cut locus, or ridge tree, is constant up to isomorphism as a labeled graph. These are 60 connected open sets, curves bounding them, and intersection points of curves. Polynomial equations for the curves are provided. Sixteen pairs of sets support the same cut locus class. We present the 177 distinct cut locus classes.