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

Materials Science