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A New Method to Generate Quasicrystalline Structures : Examples in 2D Tilings Jean-François Sadoc, R

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A new method to generate quasicrystalline structures : examples in 2D tilings Jean-François Sadoc, R. Mosseri To cite this version: Jean-François Sadoc, R. Mosseri. A new method to generate quasicrystalline structures : examples in 2D tilings. Journal de Physique, 1990, 51 (3), pp.205-221. 10.1051/jphys:01990005103020500. jpa-00212361 HAL Id: jpa-00212361 https://hal.archives-ouvertes.fr/jpa-00212361 Submitted on 1 Jan 1990 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. J. Phys. France 51 (1990) 205-221 1er FÉVRIER 1990, 205 Classification Physics Abstracts 61.40 A new method to generate quasicrystalline structures : examples in 2D tilings Jean-François Sadoc (1) and R. Mosseri (2) (1) Laboratoire de Physique des Solides, Université de Paris-Sud et CNRS, 91405 Orsay, France (2) Laboratoire de Physique des Solides de Bellevue-CNRS, 92195 Meudon Cedex, France (Reçu le 18 juillet 1989, révisé et accepté le 20 octobre 1989) Résumé. 2014 Nous présentons un nouvel algorithme pour la génération des structures quasi- cristallines. Il est relié à la méthode de coupe et projection, mais il permet une génération directement dans l’espace « physique » E de la structure. La sélection des sites dans l’espace orthogonal est remplacée par un test directement dans une grille de domaines d’acceptance dans l’espace E. Cette méthode montre qu’il y a une sorte de réseau cristallin sous-jacent au quasi- cristal. Nous illustrons la construction dans le cas 4D-2D avec les symétries d’ordre 5, 8, 10 et 12 qui sont obtenues par projection de 4D à 2D. Par la même méthode d’autres types de quasi- cristaux avec une symétrie plus basse, ayant un réseau moyen, sont construits. Nous présentons un exemple de symétrie 4. Les points de ce quasi-cristal sont un sous-ensemble des points du quasi-cristal ayant la symétrie complète d’ordre 8. Abstract. 2014 We present a new algorithm for the generation of quasicrystalline structures. It is related to the cut and projection method, but allows a direct generation of the structure in the « physical » space E. The orthogonal space site selection is replaced by a direct check in a periodic array of « acceptance » regions in E. This method shows that there is a sort of underlying crystalline lattice in quasicrystals. We illustrate the construction in the 4D-2D cases with the 5-, 8-, 10- and 12-fold symmetries which can be obtained by projection from 4D to 2D. Using this new method we also generate quasicrystals with a lower symmetry which have simple mean lattices. We present for instance a quasicrystal with a 4-fold symmetry. The points of this quasicrystal are a subset of the quasicrystal which has the whole 8-fold symmetry. 1. Introduction. In this paper we present a new algorithm for the generation of quasicrystalline structures, closely related to the cut and projection (C.P.) method [1], but which allows a direct generation of the structure in the « physical » space E. The orthogonal space site selection is replaced by a direct check in a periodic array of « acceptance » regions in E as will be explain below. This method, which is valid for projection from any dimension, is presented in the next section. We then illustrate the construction in the 4D-2D cases. There is a theorem [2] which allows one to determine the smallest dimension required in order to obtain a 2D quasicrystal with a Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01990005103020500 206 given symmetry. If the symmetry is of order s the number of its prime integers smaller than s give the dimension of the space. We are interested in quasicrystals obtained from 4D. Consider the different possible symmetries which are not crystalline : 5, 7, 8, 9, 10, 11, 12, ... Other symmetries give more than 4 numbers. So the 5, 8, 10 and 12 symmetries exhaust the quasicrystalline symmetries which can be obtained by projection from 4D to 2D. In the present paper we shall discuss mainly the octagonal and the dodecagonal quasicrystals with a few comments on the two other cases. We also present in an appendix a method which allows one to determine rapidly the space on which it is interesting to project the 4D structure, and which explains the determination of the projection matrix. In the usual C.P. framework the determination of the physical space is done by considering representations of the symmetry group operations in the n-dimensional space. Here we adopt a more geometrical approach by considering the symmetry of the Petrie polygon attached to the lattice. 2. Description of the algorithm. Let us first recall a few facts and definitions from the C.P. method. Let L be an n-dimensional lattice embedded in Rn. The « physical » space E, in which the quasiperiodic tiling is to be generated, is a p-dimensional space. For simplicity we shall suppose that n = 2 p. Let f2 be a bounded volume in Rn- n is shifted along E, defining a « strip ». All vertices of L which fall in the strip are selected and orthogonally mapped onto E, giving the vertices of the quasiperiodic tiling. For the « standard » tilings, 5li is the L unit cell or the Voronoi region. In the language of the 2D case, such quasiperiodic tilings are obtained by mapping on E a monolayer surface selected in the strip (this surface is tiled by square faces when the lattice L is a cubic lattice). The selection algorithm usually proceeds as follows : the lattice vertices to be selected are precisely those which fall, by orthogonal projection onto the space E’ orthogonal to E, inside a « window » W which is the hull of the projection onto E’ of 5li ., They are therefore obtained by a systematic inspection of all the L vertices. The modified version will proceed differently for the site selection and in most cases will be more efficient. Specifically, the standard C.P. algorithm speed is proportional to the n-th power of the size which is inspected. The modified version runs at only the p-th power of this size. The basic idea is to split L into reticular sub-lattices, i.e. to express L as the product of two complementary p-dimensional lattices, the base B and the fibre F : B and F are and The respectively generated by {bl, ..., b.p} {f1, ..., fp} . simplest example is, if L is the hypercubic lattice Z", to take as B and F the lattices respectively generated by and where is the canonical basis in Rn. Now the site selection (ei , ... , ep} {ep + 1, ..., en} {ei} is done fibre after fibre. One copy of F, called F lo l is attached.to each point Q of the base B 207 (see Fig.1 ) . Let us call W {Q} the intersection of F { Q } the strip S generated by translating llÎ along E : Fig. 1. - Scheme of the modified projection method. B is the base, F the fibre, E the physical space and A is the centre of an acceptance domain. ’ The points of F {Q} which are to be selected are precisely those which are inside W {Q} . The entire procedure can be done directly on the physical space E. If II is the orthogonal projection onto E, we define : {Q} The Z form a periodic array of « acceptance domains » (A.D.), which is generally not a tiling. Indeed neighbouring A.D. could .overlap in some cases. To each point of U one attaches a copy of the lattice V {Q} and we select those points of V {Q} which fall inside the A.D. Z {Q} . The quasiperiodic structure X reads formally : The advantage is that it is easy to calculate a priori which point of V lo l is closest to the center of the A.D. Z {O}. It is then sufficient to test only a small subset T of V lo l surrounding this point. This set T is the maximal subset of V lo l which can fit into the A.D. All these steps will be illustrated below in 2D examples. 3. Lattices, honeycombs and polytopes in R4. It is always possible to characterise the local order of a simple crystalline structure by one, or a few, polyhedron (a polytope in 4D) [3] : this could be the coordination shell, the Voronoi cell or interstices in the structure. If we are interested in 2D structures mapped from R", such local polytopes are mapped in the plane E inside polygons using the cut and projections method. Therefore it is interesting to consider in R" a polygonal line contained in the local polytope and to study how it could give a polygon with the required symmetry for the quasicrystal when it is projected on a plane. If we want to obtain all the points of the 208 polygonal line in R4 selected by the strip, the plane E must remain close to the polygonal line, so this line gives information on the position of the projection plane in R4. There is a polygonal line which strongly characterizes a polytope : the Petrie polygon [3]. The Petrie polygon of a polyhedron is a skew polygon such that every two consecutive sides, but not three, belong to a face of the polyhedron.
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