Comparative grain topology

Space-filling polyhedral networks are commonly studied in biological, physical, and mathematical disciplines. The constraints governing the construction of each network varies considerably under each context, affecting the topological properties of the constituents. A method for mapping the topological symmetry of a space-filling population of polyhedra is presented, relative to all possible polyhedra. This method is applied to the topological comparison of populations generated by seven different processes: (i) natural grain growth in polycrystalline metal, ideal grain growth simulated by (ii) interface-tracking and (iii) phase-field methods, (iv) Poisson–Voronoi and (v) ellipsoid tessellations, and (vi) graph-theoretic and (vii) Monte Carlo enumerations of individual polyhedra. Evidence for topological bias in these populations is discussed.