# Introduction to the Mathematics of Quasicrystals by Marko V. Jaric

By Marko V. Jaric

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I. and Galiulin, R. V. (1976). A local criterion for regularity of a system of points. R. (in Russian) 227. (English translation: Soviet Math. Dokl. 17, No. 2, 319-322. Elser, V. (1986). The diffraction pattern of projected structures. Acta Crystallographica A42, 36-43. Engel, P. (1981). Uber Wirkungsbereichasteilungen mit kubischer Symmetrie. Zeitschrift fur Kristallographie 157, 259-275. Engel, P. (1986). Geometrical Crystallography. Riedel, Dordrecht. Fedorov, E. A. (1885). Introduction to the Theory of Figures.

Journal fur die reine und angewandte Mathematik 352, 161-183 . Senechal, M . (1986) . Geometr y an d crysta l symmetry . Computers and Mathematics with Applications 12B , Nos . 3/4 , 565-578 ; reprinte d i n Symmetry: Unifying Human Understanding (I . ) . Pergamo n Press , Ne w York , 1986 . Senechal, M . A brie f histor y o f geometrica l crystallograhy . An Historical Atlas of Crystallography (edite d b y J . Lim a d e Faria) , t o b e published . Stogrin, M . I . (1973) . Regula r Dirichlet-Vorono i partition s fo r th e secon d triclini c group.

Consider, for example, the monohedral tiling shown in Fig. 34. Its p r o totile is composed of six equilateral triangles. The tiling has the composition property: any tile can be grouped together with three neighboring tiles to form a larger tile similar to itself. Notice also that, within the tiling, this grouping into larger tiles can be done in only one way. Thus the original tiling contains within it a unique larger copy of itself. Clearly we can repeat this process, grouping together four large tiles t o form even larger tiles, again in a unique way.