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Quasiperiodic tilings and quasicrystals. (English) Zbl 0639.52014

Translation from Zap. Nauchn. Semin. Leningr. Otd. Mat. Inst. Steklova 155, 116-135 (Russian) (1986; Zbl 0618.52012).

MSC:

52C17 Packing and covering in \(n\) dimensions (aspects of discrete geometry)
05B45 Combinatorial aspects of tessellation and tiling problems
82D25 Statistical mechanics of crystals

Citations:

Zbl 0618.52012
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References:

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[3] R. Penrose, ?The role of aesthetics in pure and applied mathematical research,? Bull. Inst. Math. Appl.,10, No. 7?8, 266?271 (1974).
[4] M. Gardner, ?Extraordinary nonperiodic tiling that enriches the theory of tiles,? Sci. Am.,236, No. 1, 110?121 (1977). · doi:10.1038/scientificamerican0177-110
[5] N. G. de Bruijn, ?Algebraic theory of Penrose’s nonperiodic tiling of the plane,? Nederl. Akad. Wetensch. Proc. Ser. A,43, 39?66 (1981).
[6] A. L. Mackay, Kristallografia,26, 910 (1981).
[7] P. Kramer and R. Neri, ?On periodic and nonperiodic space fillings of Em obtained by projection,? Acta Crystallogr. Sec. A,40, 580?587 (1984). · Zbl 1176.52010 · doi:10.1107/S0108767384001203
[8] P. A. Kalugin, A. Yu. Kitaev, and L. S. Levitov, ?Six-dimensional properties of AlMn alloy,? J. Physique Lett.,46, 601?607 (1985). · doi:10.1051/jphyslet:019850046013060100
[9] D. Levine and P. J. Steinhardt, ?Quasicrystals: A new class of ordered structures,? Phys. Rev.,53, 2477?2480 (1984).
[10] R. M. Robinson, ?Seven polygons which permit only nonperiodic tilings of the plane,? Not. Am. Math. Soc.,14, 835 (1967).
[11] D. Schechtman, I. Blech, D. Gratia, and J. W. Cahn, ?Metallic phase with long-range orientational order and no translational symmetry,? Phys. Rev. Lett.,53, 1951?1953 (1984). · doi:10.1103/PhysRevLett.53.1951
[12] V. Elser, ?The diffraction pattern of projected structures,? Phys. Rev. B,32, 4892 (1985). · Zbl 1176.52006 · doi:10.1103/PhysRevB.32.4892
[13] A. Katz and M. Duneau, ?Quasiperiodic patterns and icosahedral symmetry,? J. Physique,47, 181?196 (1986). · Zbl 0693.52003 · doi:10.1051/jphys:01986004702018100
[14] M. M. Skriganov, Geometric and Arithmetic Methods in the Spectral Theory of Multidimensional Periodic Operators [in Russian], Leningrad (1985). · Zbl 0567.47004
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