Quasiperiodicity and randomness in tilings of the plane |
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| Authors: | C. Godrèche J. M. Luck |
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| Affiliation: | (1) Service de Physique du Solide et de Résonance Magnétique, CEN Saclay, 91191 Gif-sur-Yvette Cedex, France;(2) Service de Physique Théorique (Laboratoire de l'Institut de Recherche Fondamentale du Commissariat à l'Energie Atomique), CEN Saclay, 91191 Gif-sur-Yvette Cedex, France;(3) Department of Theoretical Physics, University of Oxford, 1 Keble Road, OX1 3NP Oxford, Great Britain;(4) Department of Physics, University of Edinburgh, The King's Buildings, Mayfield Road, EH9 3JZ Edinburgh, Great Britain |
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| Abstract: | We define new tilings of the plane with Robinson triangles, by means of generalized inflation rules, and study their Fourier spectrum. Penrose's matching rules are not obeyed; hence the tilings exhibit new local environments, such as three different bond lengths, as well as new patterns at all length scales. Several kinds of such generalized tilings are considered. A large class of deterministic tilings, including chiral tilings, is strictly quasiperiodic, with a tenfold rotationally symmetric Fourier spectrum. Random tilings, either locally (with extensive entropy) or globally random (without extensive entropy), exhibit a mixed (discrete+continuous) diffraction spectrum, implying a partial perfect long-range order. |
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| Keywords: | Aperiodic tilings inflation rules quasiperiodicity |
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