Symmetry groups for beta-lattices

We present a construction of symmetry plane-groups for quasiperiodic point-sets in the plane, named beta-lattices. The algebraic framework is issued from counting systems called beta-integers, determined by Pisot-Vijayaraghavan (PV) algebraic integers β > 1. The beta-integer sets can be equipped with abelian group structures and internal multiplicative laws. These arithmetic structures lead to freely generated symmetry plane-groups for beta-lattices, based on repetitions of discrete “adapted rotations and translations” in the plane. Hence beta-lattices, endowed with these adapted rotations and translations, can be viewed like lattices. Moreover, beta-lattices tend to behave asymptotically like lattices.