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Communications in Mathematical Physics

, Volume 129, Issue 3, pp 599–619 | Cite as

Weak matching rules for quasicrystals

  • Joshua E. S. Socolar
Article

Abstract

Weak matching rules for a quasicrystalline tiling are local rules that ensure that fluctuations in “perp-space” are uniformly bounded. It is shown here that weak matching rules exist forN-fold symmetric tilings, whereN is any integer not divisible by four. The result suggests that, contrary to previous indications, quasicrystalline ground states are not confined to those symmetries for which the incommensurate ratios of wavevectors are quadratic irrationals. An explicit method of constructing weak matching rules forN-fold symmetric tilings in two dimensions is presented. It is shown that the generalization of the construction yields weak matching rules in the case of icosahedral symmetry as well.

Keywords

Neural Network Statistical Physic Complex System Nonlinear Dynamics Quantum Computing 
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.
    Steinhardt, P.J., Ostlund, S.: The physics of quasicrystals. Singapore: World Scientific 1987Google Scholar
  2. 2.
    Penrose, R.: Bull. Inst. Math. Appl.10, 266 (1974)Google Scholar
  3. 3.
    de Bruijn, N.: Proc. Kon. Nederl. Akad. Wetensch. A84, 39 (1981)Google Scholar
  4. 4.
    Socolar, J.E.S.: Phys. Rev. B39, 10519 (1989)CrossRefGoogle Scholar
  5. 5.
    Katz, A.: Commun. Math. Phys.118, 263 (1988)CrossRefGoogle Scholar
  6. 6.
    Socolar, J.E.S., Steinhardt, P.J.: Phys. Rev. B34, 617 (1986)CrossRefGoogle Scholar
  7. 7.
    Levitov, L.S.: Commun. Math. Phys.119, 627 (1988)CrossRefGoogle Scholar
  8. 8.
    Socolar, J.E.S.: In Quasicrystals — Adriatico Anniversary Research Conference. Jaric, M.V., Lundqvist, S. (eds.). Spingapore: World Scientific (to appear) (1989)Google Scholar
  9. 9.
    Kleman, M., Pavlovitch, A.: J. Phys. C3, Workshop on Aperiodic Crystals: 229 (1986)Google Scholar
  10. 10.
    Elser, V.: J. Phys. A. Math. Gen.17, 1509 (1984)CrossRefGoogle Scholar
  11. 11.
    Henley, C.L.: J. Phys. A. Math. Gen.21, 1649 (1988)CrossRefGoogle Scholar
  12. 12.
    Gahler, F., Rhyner, J.: J. Math. Phys. A19, 267 (1986)CrossRefGoogle Scholar
  13. 13.
    Lubensky, T.C.: In Aperiodicity and Order, vol.1, Jaric, M.V. (ed.). Boston: Academic Press 1988Google Scholar
  14. 14.
    One could equally well consider aci-row border, wherec is any number relatively prime toQ, but only one case is necessary for the proofGoogle Scholar
  15. 15.
    Katz, A.: UnpublishedGoogle Scholar
  16. 16.
    Levine, D., Steinhardt, P.J.: Phys. Rev. B34, 596 (1986)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag 1990

Authors and Affiliations

  • Joshua E. S. Socolar
    • 1
  1. 1.Department of PhysicsHarvard UniversityCambridgeUSA

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