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Quasicrystals and their symmetries


This paper is a survey of results on symmetries of crystals and of some results on a mathematical approach to the theory of quasicrystals and their symmetries.

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  1. 1.

    H. Abels, “Properly discontinuous groups of affine transformations: A survey,” Geom. Dedicata, 87, 309–333 (2001).

    MATH  MathSciNet  Article  Google Scholar 

  2. 2.

    H. Abels, G. A. Margulis, and G. A. Soifer, “Properly discontinuous groups of affine transformations with orthogonal linear part,” C. R. Acad. Sci. Paris (I), 324, 253–258 (1997).

    MATH  MathSciNet  Google Scholar 

  3. 3.

    G. Aragón, J. L. Aragón, F. Davila, A. Gomez, and M. A. Rodriguez, “Modern geometric calculations in crystallography,” in: E. Bayro Corrochano and G. Sobczyk, eds., Geometric Algebra with Applications in Science and Engineering, Birkhäuser, Boston (2001), Chapter 18, pp. 371–386.

    Google Scholar 

  4. 4.

    L. Auslander, “The structures of compact locally affine manifolds,” Topology, 3, 131–139 (1964).

    MATH  MathSciNet  Article  Google Scholar 

  5. 5.

    B. Delone, N. Padurov, and A. Alexandrov, Mathematical Foundations of a Structure Analysis of Crystals [in Russian], ONTI, GTTI, Moscow (1934).

    Google Scholar 

  6. 6.

    D. Fried and W. D. Goldman, “Three-dimensional affine crystallographic groups,” Adv. Math., 47, 1–49 (1983).

    MATH  MathSciNet  Article  Google Scholar 

  7. 7.

    A. Janner, “Crystallographic symmetries of quasicrystals,” Phase Transitions, 43, 35–47 (1993).

    Google Scholar 

  8. 8.

    C. Janot, Quasicrystals: A Primer, Clarendon Press, Oxford (1994).

    Google Scholar 

  9. 9.

    A. G. Kurosh, General Algebra. Lectures of 1969–1970 Academic Year [in Russian], Nauka, Moscow (1974).

    Google Scholar 

  10. 10.

    Le Ty Kyok Thang, S. A. Piunikhin, and V. A. Sadov, “Geometry of quasicrystals,” Russian Math. Surveys, 48, No. 1, 41–102 (1993).

    MathSciNet  Article  Google Scholar 

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Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 10, No. 3, pp. 3–10, 2004.

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Artamonov, V.A. Quasicrystals and their symmetries. J Math Sci 139, 6657–6662 (2006).

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  • Symmetry Group
  • Inverse Semigroup
  • Linear Span
  • Rotation Group
  • Dihedral Group