2 D quasi-periodic Rauzy tiling as a section of 3 D periodic tiling
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RY OF CRYSTAL STRUCTURES
2D QuasiPeriodic Rauzy Tiling As a Section of 3D Periodic Tiling A. V. Maleev, A. V. Shutov, and V. G. Zhuravlev Vladimir State Humanitarian University, Vladimir, 600024 Russia email: [email protected] Received October 15, 2009
Abstract—A method for constructing a 2D quasiperiodic Rauzy tiling Til as a section of some 3D periodic 3D 3D tiling Til is considered. The translation lattice of the tiling Til and its connectivity graph are constructed using the discrete modeling of packings. The calculation of the layerbylayer growth polyhedron for the tiling 3D Til made it possible to estimate from upper the shape of a growth polygon for the tiling Til. As a result, the growth shape in six out of eight growth sectors has been rigorously proven. A set of quasiperiodic tilings (locally indistinguishable from the Rauzy tiling T il ), including seven centrosymmetric tilings, has been obtained. DOI: 10.1134/S1063774510050019
INTRODUCTION De Bruijn [1], followed by Janssen [2], was the first to note that quasiperiodic tilings can be obtained as irrational sections of periodic tilings. Oguey, Duneau, and Katz [3] developed a general design for construct ing quasiperiodic tilings of arbitrary dimensions on the basis of sections of multidimensional skew tori. The essence of this design is as follows: when the reduction to a smaller dimension is performed, a finite (accurate to shift) set of tiles arises in the section. This leads to tilings of the Penrose mosaic type [4], i.e., til ings with flat boundaries and local noncrystallo graphic finiteorder symmetry transformations. Another type of quasiperiodic tilings is the Rauzy tiling T il [5]. It belongs to the set of 2D quasiperiodic tilings constructed on the basis of cubic irrationalities [6–8]. One characteristic feature of such tilings, when compared with other periodic and quasiperiodic ones, is that they have fractal boundaries and generally possess rich semigroup similarity [9, 10]. In this study we explicitly constructed a 3D peri 3D odic tiling Til with the following specific features: it has fractal boundaries and one of its sections coincides with the Rauzy tiling T il . Siegel [11] was the first to note the possibility of obtaining a Rauzy tiling using sections, like in the case of Penrose tilings. However, the problem of implementing this concept turned out to be far from trivial. A discrete modeling method of packings [12, 13], which includes an approximation of bodies by discrete models (polycubes) and the genera tion of all possible packings of these polycubes with a specified packing factor, was proposed to solve it [14]. In this study we considered some other sections of 3D the periodic tiling Til which generate a set of quasi periodic tilings that are locally isomorphic to the
Rauzy tiling but cannot be brought into coincidence 3D with it by any steps. The tiling Til turned out to be centrosymmetric; i.e., from the point of view of struc tural crystallography, it is described by the sp. gr. P 1 . Therefore, among this
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