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Article Dans Une Revue Theoretical Computer Science Année : 2004

Symmetry groups for beta-lattices

Avi Elkharrat
  • Fonction : Auteur
Jean-Pierre Gazeau
  • Fonction : Auteur
Jean-Louis Verger-Gaugry

Résumé

We 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.
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Dates et versions

hal-00159658 , version 1 (03-07-2007)

Identifiants

  • HAL Id : hal-00159658 , version 1

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Avi Elkharrat, Christiane Frougny, Jean-Pierre Gazeau, Jean-Louis Verger-Gaugry. Symmetry groups for beta-lattices. Theoretical Computer Science, 2004, 319, pp.281-305. ⟨hal-00159658⟩
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