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On finite symmetry groups of some models of three-dimensional quasicrystals

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We give a full description of the finite symmetry groups of the cut-and-project model for three-dimensional quasicrystals under the assumption that the phase space dimension is at most 3.

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References

  1. Artamonov V. A., “On symmetries of quasicrystals,” in: Algebraic Structures and Their Representations, Amer. Math. Soc., Providence, RI, 2005, pp. 175–188 (Contemp. Math.; V. 376.).

    Google Scholar 

  2. Artamonov V. A. and Slovokhotov Yu. L., Groups and Their Applications in Physics, Chemistry, and Crystallography [in Russian], Akademiya, Moscow (2005).

    Google Scholar 

  3. Hermisson J., Richard Ch., and Baake M., “A guide to the symmetry structure of quasiperiodic tiling classes,” J. Phys. I France, 7, 1003–1018 (1997).

    Article  MathSciNet  Google Scholar 

  4. Baake M., Hermisson J., and Pleasants A. B., “The torus parametrization of quasiperiodic LI-classes,” J. Phys. A, 30, 3029–3056 (1997).

    Article  MATH  MathSciNet  Google Scholar 

  5. Moody R. V., “Model sets: a survey,” in: Quasicrystals to More Complex Systems, Les Houches, 1998 (F. Alex and J.-P. Gazeau, eds.), Centre de Physique des Houches, Springer-Verlag, Berlin, 2000, 13, pp. 145–166 (arXiv:math.MG/0002020 v1).

    Google Scholar 

  6. Quasicrystals and Discrete Geometry, (Patera J., ed.), Amer. Math. Soc., Providence, RI (1998) (Fields Inst. Monogr.).

    MATH  Google Scholar 

  7. Rabson D. R., Mermin N. D., Rokhaar D. S., and Wright D. C., “The space groups of axial crystals and quasicrystals,” Rev. Modern Phys., 63, No. 3, 699–733 (1991).

    Article  MathSciNet  Google Scholar 

  8. Mermin N. D., “The space groups of icosahedral quasicrystals and cubic, orthorhombic, monoclinic and triclinic crystals,,” Rev. Modern Phys., 64, No. 1, 3–51 (1992).

    Article  MathSciNet  Google Scholar 

  9. Artamonov V. A. and Sànchez S., “On symmetry groups of quasicrystals,” Math. Notes, 87, No. 3, 303–308 (2010).

    Article  MATH  MathSciNet  Google Scholar 

  10. Lě Thang Tu Quoc, Piunikhin S. A., and Sadov V. A., “The geometry of quasicrystals,” Russian Math. Surveys, 48, No. 1, 37–100 (1993).

    Article  MathSciNet  Google Scholar 

  11. International Tables for Crystallography (Theo Hahn, ed.), 2nd Rev. Edition, Int. Union Crystallography, V. A Reidel. Norwell, MA (1987).

    Google Scholar 

  12. Artamonov V. A. and Sànchez S., “Remarks on symmetries of 2D-quasicrystals (SI-CMMSE-2006),” Int. J. Comput. Math., 85, No. 3–4, 319–328 (2008).

    Article  MATH  MathSciNet  Google Scholar 

  13. Petrashen’ M. I. and Trifonov E. D., Application of Group Theory in Quantum Mechanics [in Russian], URSS, Moscow (1999).

    Google Scholar 

  14. Shafarevich I. R., Basic Concepts of Algebra [in Russian], VINITI, Moscow (1986) (Itogi Nauki i Tekhniki, Sovremennye Problemy Matematiki, Fundamental’nye Napravleniya, 11).

    Google Scholar 

  15. Niizeki K., “Self-similarity of quasilattices in two dimensions. I. The n-gonal quasilattice,” J. Phys. A: Math. General, 22, 193–204 (1989).

    Article  MATH  MathSciNet  Google Scholar 

  16. Horne Clare E., Geometrical Symmetry in Patterns and Tilings, Woodgead Publ., Cambridge, England (2000).

    Book  Google Scholar 

  17. Fujiwara T. and Ishii Y., Quasicrystals. A Handbook of Metal Physics, Elsevier, Amsterdam (2008).

    Google Scholar 

  18. Deza M. M. and Deza E., Encyclopedia of Distance, Springer-Verlag, Dordrecht, Heidelberg, London, and New York (2009).

    Book  Google Scholar 

  19. Senechal M., Quasicrystals and Geometry, Cambridge Univ. Press, Cambridge (1995).

    MATH  Google Scholar 

  20. Quasicrystals, Structure and Physical Properties (Trebin H.-R., ed.), Wiley-VCH GmbH & Co. KGaA, Stuttgart (2003).

    Google Scholar 

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Correspondence to V. A. Artamonov.

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Original Russian Text Copyright © 2011 Artamonov V. A. and Sánchez S.

The authors were supported by the Russian Foundation for Basic Research (Grant 09-01-00058).

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Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 52, No. 6, pp. 1221–1233, November–December, 2011.

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Artamonov, V.A., Sánchez, S. On finite symmetry groups of some models of three-dimensional quasicrystals. Sib Math J 52, 969–979 (2011). https://doi.org/10.1134/S0037446611060024

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  • DOI: https://doi.org/10.1134/S0037446611060024

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