Cellular microstructures are ubiquitous in nature. They can be found in polycrystalline microstructures, foams, plant epidermis, ferroelectrics, complex fluids, and even in ice cream.

In many situations, the cell/grain/bubble walls move to reduce their surface area (a surface tension effect), with a velocity proportional to the wall’s mean curvature. As a result, the cells evolve and coarsen.

Using this relation, and little else, von Neumann gave an exact formula for the growth rate of a cell in a 2-d cellular structure, which is the basis of modern grain growth theory. Borrowing ideas from geometric probability, we present an exact solution for the same problem in 3-d using the “mean width.” We then describe why the mean width is the natural linear measure of grain size and topology and is useful across broad swaths of the sciences. Next, we extend this 50 year old theory into all d≥2. Finally, we discuss using these ideas to more efficiently simulate grain growth.

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