We formulate and prove a periodic analog of Maxwell’s theorem relating stressed planar frameworks and their liftings to polyhedral surfaces with spherical topology. We use our lifting theorem to prove rigidity-theoretic properties for planar periodic pseudo-triangulations, generalizing their finite counterparts. These properties are then applied to questions originating in mathematical crystallography and materials science, concerning planar periodic auxetic structures and ultrarigid periodic frameworks.
Maxwell’s theorem, periodic framework, periodic stress, liftings, periodic pseudo-triangulation, expansive motion, auxetics, ultrarigidity
Borcea, Ciprian and Streinu, Ileana, "Liftings and Stresses for Planar Periodic Frameworks" (2015). Computer Science: Faculty Publications, Smith College, Northampton, MA.
Author’s submitted manuscript. A second version of the paper, as presented at the Symposium on Computational Geometry (SoCG'14), Kyoto, Japan, 2014, is also included.