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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

Point Symmetry Coxeter Group Monoid Structure Formal Grammar Input Tape 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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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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