Symmetry groups for beta-lattices

AbstractWe 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 β>1 is a quadratic Pisot–Vijayaraghavan algebraic unit, the set of beta-integers can be equipped with an abelian group structure and an internal multiplicative law. When β=(1+5)2, 1+2 and 2+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 ρS(n), defined on the set of beta-integers as counting the number of small tiles between the origin and the nth 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