Symmetry groups for beta-lattices

International audienceWe present a construction of symmetry plane-groups for quasiperiodic point-sets named beta-lattices. The framework is issued from beta-integers counting systems. Beta-lattices are vector superpositions of beta-integers. When $\beta>1$ is a quadratic Pisot-Vijayaraghavan (PV) algebraic unit, the set of beta-integers can be equipped with an abelian group structure and an internal multiplicative law. When $\beta=(1+\sqrt{5})/2$, $1+\sqrt{2}$ and $2+\sqrt{3}$, we show that these arithmetic and algebraic structures lead to freely generated symmetry plane-groups for beta-lattices. These plane-groups are based on repetitions of discrete adapted rotations and translations we shall refer to as ``beta-rotations" and ``beta-translations". Hence beta-lattices, endowed with beta-rotations and beta-translations, can be viewed like lattices. The quasiperiodic function $\rho_S(n)$, defined on the set of beta-integers as counting the number of small tiles between the origin and the $n^{\rm th}$ beta-integer, plays a central part in these new group structures. In particular, this function behaves asymptotically like a linear function. As an interesting consequence, beta-lattices and their symmetries behave asymptotically like lattices and lattice symmetries respectively