Medial surfaces of hyperbolic structures

We describe an algorithm for numerical computation of a medial surface and an associated medial graph for three-dimensional shapes bounded by oriented triangulated surface manifolds in three-dimensional Euclidean space (domains). We apply the construction to bicontinuous domain shapes found in molecular self-assemblies, the cubic infinite periodic minimal surfaces of genus three: Gyroid (G), Diamond (D) and Primitive (P) surfaces. The medial surface is the locus of centers of maximal spheres, i.e. spheres wholly contained within the domains which graze the surface tangentially and are not contained in any other such sphere. The construction of a medial surface is a natural generalization of Voronoi diagrams to continuous surfaces. The medial surface provides an explicit construction of the volume element associated with a patch of the bounding surface, leading to a robust measure of the surface to volume ratio for complex forms. It also allows for sensible definition of a line graph (the medial graph), particularly useful for domains consisting of connected channels, and not reliant on symmetries of the domains. In addition, the medial surface construction produces a length associated with any point on the surface. Variations of this length give a useful measure of global homogeneity of topologically complex morphologies. Comparison of medial surfaces for the P, D and G surfaces reveal the Gyroid to be the most globally homogeneous of these cubic bicontinuous forms (of genus three). This result is compared with the ubiquity of the G surface morphology in soft mesophases, including lyotropic liquid crystals and block copolymers.