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Most metallic and ceramic materials are comprised of a space-fillingcollection of crystalline grains separated by grain boundaries. While thisgrain structure has been studied for more than a century, there few rigorousresults regarding its global properties available in the literature. We presenta new, rigorous result for three-dimensional grain structures that relates theintegral of the Gaussian curvature over the grain boundaries to the numbers ofgrains and quadruple junctions. The result is numerically verified for a grainstructure consisting of periodic truncated octahedra.

A constant of motion for ideal grain growth in three dimensions