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Smectic liquid crystals are remarkable, beautiful examples of materialsmicrostructure, with ordered patterns of geometrically perfect ellipses andhyperbolas. The solution of the complex problem of filling three-dimensionalspace with domains of focal conics under constraining boundary conditionsyields a set of strict rules, which are similar to the compatibility conditionsin a martensitic crystal. Here we present the rules giving compatibleconditions for the concentric circle domains found at two-dimensional smecticinterfaces with planar boundary conditions. Using configurations generated bynumerical simulations, we develop a clustering algorithm to decompose theplanar boundaries into domains. The interfaces between different domains agreewell with the smectic compatibility conditions. We also discuss generalizationsof our approach to describe the full three-dimensional smectic domains, wherethe variant symmetry group is the Weyl-Poincar\'e group of Lorentz boosts,translations, rotations, and dilatations.

The weirdest martensite: Smectic liquid crystal microstructure and Weyl-Poincar\'e invariance