Geometric Analysis & Crystalline Anisotropy

The anisoperimetric problem, also known as the crystalline variational problem, may not be as old as the isoperimetric problem, but it has certainly been explored by scientists of different knowledge domains for a while.

Minerals have been formally characterized for the first time by polyhedral shapes by Rene Hauy, in the beginning of the XIX century. Hauy suggested correlations between micro and macroscopic properties of materials, the hypothesis that the smallest units of matter - in his work, molecules - induce the macro behavior. This was an important step for Crystallography.

Under special conditions (equilibrium conditions) such very small scale properties can be used to determine the final shape of a growing crystal - its equilibrium shape. The Wulff construction, published in 1901, presents the steps to build this shape starting from the crystal surface energy, which is a function of the material crystallography.

The problem is not so obvious though for evolution of interfaces happening far from equilibrium, such as the one of a crystal growing from liquid or vapor. The most promising theoretical tool to interpret such conditions is Geometric Measure Theory (GMT).