  We introduce a general, efficient method to completely describe the topology
of individual grains, bubbles, and cells in three-dimensional polycrystals,
foams, and other multicellular microstructures. This approach is applied to a
pair of three-dimensional microstructures that are often regarded as close
analogues in the literature: one resulting from normal grain growth (mean
curvature flow) and another resulting from a random Poisson-Voronoi
tessellation of space. Grain growth strongly favors particular grain
topologies, compared with the Poisson-Voronoi model. Moreover, the frequencies
of highly symmetric grains are orders of magnitude higher in the the grain
growth microstructure than they are in the Poisson-Voronoi one. Grain topology
statistics provide a strong, robust differentiator of different cellular
microstructures and provide hints to the processes that drive different classes
of microstructure evolution.
