Ultrarigid periodic frameworks

We give an algebraic characterization of when a \(d\)-dimensional periodic framework has no non-trivial, symmetry preserving, motion for any choice of periodicity lattice. Our condition is decidable, and we provide a simple algorithm that does not require complicated algebraic computations. In dimension \(d=2\), we give a combinatorial characterization in the special case when the the number of edge orbits is the minimum possible for ultrarigidity. All our results apply to a fully flexible, fixed area, or fixed periodicity lattice.