Non-commutative Models for Quasicrystals

  • P. Kramer
  • J. Garcia-Escudero
Conference paper
Part of the Centre de Physique des Houches book series (LHWINTER, volume 3)

Abstract

A powerful approach to quasicrystals arises from the use of n-dimensional crystallography: Given a non-crystallographic point group H in R 3, look for the lattice Γ of minimal dimension n which has H as (part of) its point group. The n-dimensional representation of H must be reducible and contain a 3D representation. Split R n into this representation space R 3 and its orthogonal complement, construct discrete quasiperiodic patterns by projection from the lattice to R 3 and use the orthogonal complement for windows and for coding. Periodicity and tranlational symmetry in R 3 are absent by construction. This scheme allows to treat diffraction properties by lifting the Fourier coefficients to the reciprocal lattice Γ R .

Keywords

Propa Lentin 

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References

  1. [1]
    Brown H., Billow R., Neuböser J., Wondratschek H., Zassenhaus H., Crystallographic Groups of Four-Dimensional Space, ( Wiley, New York, 1978 )MATHGoogle Scholar
  2. [2]
    Baake M., Joseph D. and Kramer P., Phys. Lett. A168 (1992) 199–208MathSciNetCrossRefGoogle Scholar
  3. [3]
    Baake M., Kramer P., Schlottmann M. and Zeidler D.Google Scholar
  4. Int. J. Mod. Phys. B4 (1990) 2217–68Google Scholar
  5. [4]
    Garcia-Escudero J. and Kramer P., Anales de Fisica, Monografias 1 vol.1 (1993) 339–42, MadridGoogle Scholar
  6. [5]
    Garcia-Escudero J. and Kramer P., J. Phys. A26 (1993) L1029–35MathSciNetADSGoogle Scholar
  7. [6]
    Garcia-Escudero J. and Kramer P., Proc. Int. Wigner Symposium (1993), OxfordGoogle Scholar
  8. [7]
    Gross M. and Lentin A., Introduction to Formal Grammars, ( Springer, Berlin, 1970 )MATHCrossRefGoogle Scholar
  9. [8]
    Hoperoft J. E. and Ullmann J. D., Einführung in die Automatentheorie, formale Sprachen und Komplexitätstheorie, (Addison-Wesley, Bonn, 1988 )Google Scholar
  10. [9]
    Humphreys J. E., Reflection Groups and Coxeter Groups, ( Cambridge University Press, Cambridge, 1990 )MATHCrossRefGoogle Scholar
  11. [10]
    Janner A., Acta Cryst. A47 (1991) 577–590MathSciNetMATHCrossRefGoogle Scholar
  12. [11]
    Kramer P., Anales de Fisica, Monografias 1, vol.2 (1993) 370–3, MadridGoogle Scholar
  13. [12]
    Kramer P., J. Phys. A26 (1993) 213–228MathSciNetADSMATHGoogle Scholar
  14. [13]
    Kramer P., J. Phys. Lett. A26 (1993) L245 - L250MATHGoogle Scholar
  15. [14]
    Kramer P., J. Phys. A27 (1994) 2011–22MathSciNetADSMATHGoogle Scholar
  16. [15]
    Kramer P. and Wagner H. (1994), preprint TübingenGoogle Scholar
  17. [16]
    Lothaire M., Combinatorics on Words, ( Addison-Wesley, Reading, 1983 )MATHGoogle Scholar
  18. [17]
    Magnus W., Karras A. and Solitar D., Combinatorial Group Theory, ( Dover, New York, 1976 )MATHGoogle Scholar
  19. [18]
    Schwarzenberger R. L. E., N-Dimensional Crystallography, ( Pitman, San Francisco, 1980 )MATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1995

Authors and Affiliations

  • P. Kramer
    • 1
  • J. Garcia-Escudero
    • 2
  1. 1.Institut für Theoretische PhysikUniversität TübingenTübingenGermany
  2. 2.Departamento de FisicaUniversidad de OviedoOviedoSpain

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