Skip to main content
SpringerLink
Log in

Meyer's concept of quasicrystal and quasiregular sets

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

Abstract

This paper relates two mathematical concepts of long-range order of a set of atoms Λ, each of which is based on restrictions on the set of interatomic distances Λ−Λ. A set Λ in ℝn is aMeyer set if Λ is a Delone set and there is a finite setF such that\(\Lambda - \Lambda \subseteq \Lambda + F.{\text{ Y}}\). Meyer proposed that such sets include the possible structures of quasicrystals. He obtained a structure theory for such sets, which reformulates results that he obtained in harmonic analysis around 1970, and which relates these sets to cut-and-project sets. In geometric crystallography V.I. Galiulin introduced the concept ofquasiregular set, which is a set Λ such that both Λ and Λ−Λ are Delone sets. This paper shows that these two concepts are identical.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Bombieri, E., Taylor, J.E.: Which distributions diffract? An initial investigation. J. Phys. Colloq.47, C3, 19–28 (1986)

    Google Scholar 

  2. de Bruijn, N.G.: Algebraic theory of Penrose's nonperiodic tiling of the plane I, II. Nederl. Akad. Wetensch. Proc. Series A84, 39–52 and 53–66 (1981)

    Google Scholar 

  3. de Bruijn, N.G.: Quasicrystals and their Fourier transform. Nederl. Akad. Wetensch. Proc Ser. A89, 123–152 (1986)

    Google Scholar 

  4. Burkov, S.E.: Absence of weak local rules for the planar quasicrystallographic tiling with 8-fold symmetry. Commun. Math. Phys.117, 667–675 (1988)

    Article  Google Scholar 

  5. Chen, L., Moody, R.V., Patera, J.: Non-crystallographic root systems and quasicrystals. Preprint

  6. Delaunay, B.N., Delone, B.N.: Neue Darstellung der Geometrischen Kristallographie. Zeit. Kristallographie84, 109–149 (1932)

    Google Scholar 

  7. Delone, B.N., Dolbilin, N.P., Štogrin, M.I., Galiulin, R.V.: A local criterion for regularity in a system of points. Sov. Math. Dokl.17, 319–322 (1976)

    Google Scholar 

  8. Dolbilin, N.P., Lagarias, J.C., Senechal, M.: Multiregular Point Systems. Preprint

  9. Duneau, M., Katz, A.: Quasiperiodic structures. Phys. Rev. Lett.54, 2688–2691 (1985)

    Article  Google Scholar 

  10. Elser, V.: The diffraction pattern of projected structures. Acta. Cryst. A42, 36–43 (1986)

    Article  Google Scholar 

  11. Engel, P.: Geometric Crystallography — An Axiomatic Introduction to Crystallography. Reidel: Dordrecht, 1986

    Google Scholar 

  12. Engel, P.: Geometric Crystallography. In: Handbook of Convex Geometry, Volume B.P. Gruber and J.M. Wills (eds.) Amsterdam: North-Holland, 1993, pp. 991–1041

    Google Scholar 

  13. Gähler, F., Rhyner, J.: Equivalence of the generalized grid and projection methods for the construction of quasiperiodic tilings. J. Phys. A19, 267–277 (1986)

    Google Scholar 

  14. Galiulin, R.V.: Delone systems. Sov. Phys. Crystallogr.25, No. 5, 517–521 (1980)

    Google Scholar 

  15. Galiulin, R.V.: Zonohedral Delone systems. In: Collected Abstracts. XII European Crystallog. Meeting, Moscow, Vol. I. 1989, p. 21

  16. Hof, A.: Quasicrystals, Aperiodicity and Lattice Systems. Thesis, U. of Groningen, 1992

  17. Hof, A.: On diffraction by aperiodic structures. Commun. Math. Phys.,174, 149–159, 1995

    Article  Google Scholar 

  18. Hof, A.: Diffraction by aperiodic structures at high temperatures. J. Phys. A: Math Gen.28, 57–62 (1995)

    Article  Google Scholar 

  19. Janot, C.: Quasicrystals: A Primer. Oxford: Oxford University Press, 1992

    Google Scholar 

  20. Katz, A.: Theory of matching rules for 3-dimensional Penrose tilings. Commun. Math. Phys.119, 262–268 (1988)

    Google Scholar 

  21. Katz, A., Duneau, M.: Quasiperiodic structures determined by the projection method. J. Phys. (Paris) Supp.C47, 103–112 (1987)

    Google Scholar 

  22. Kramer, P.: Non-periodic central space filings with icosahedral symmetry using copies of seven elementary cells. Acta Cryst.A 38, 257–264 (1982)

    Article  Google Scholar 

  23. Kramer, P., Neri, R.: On periodic and nonperiodic space fillings of\(\mathbb{E}^n \) obtained by projection. Acta Cryst.A 40, 580–587 (1984), (Erratum: Acta Cryst.A 41, 619 (1985))

    Article  Google Scholar 

  24. Kramer, P., Papadopolis, Z., Moody, R.V.: A growth mechanism for theT 2F) tiling. Preprint

  25. Le, T.Q.T.: Local rules for pentagonal quasicrystals. Disc. & Comp. Geom.14, 31–70 (1995)

    Google Scholar 

  26. Le, T.Q.T., Plunikhin, S., Sadov, V.: Geometry of quasicrystals. Uspeki. Math. Nauk.48, 41–102 (1993) (in Russian). English translation: Russian Math. Surveys48, 37–100 (1993)

    Google Scholar 

  27. Levine, D., Steinhardt, P.J.: Quasicrystals: A new class of ordered structures. Phys. Rev. Lett.53, 2477–2480 (1984)

    Article  Google Scholar 

  28. Levitov, L.S.: Local rules for quasicrystals. Commun. Math. Phys.119, 627–666 (1988)

    Article  Google Scholar 

  29. Lunnon, W.F., Pleasants, P.A.B.: Quasicrystallographic tilings. J. Maths. Pures Appl.66, 217–263 (1987)

    Google Scholar 

  30. Meyer, Y.: Nombres de Pisot, Nombres de Salem, et analyse harmonique. Lecture Notes in Math. No. 117, Berlin, Heidelberg, New York: Springer, 1970

    Google Scholar 

  31. Meyer, Y.: Algebraic Numbers and Harmonic Analysis. Amsterdam: North-Holland, 1972

    Google Scholar 

  32. Meyer, Y.: Quasicrystals, Diophantine Approximation and Algebraic Numbers. In: Beyond Quasicrystals. F. Axel, D. Gratias (eds.) Les Editions de Physique. Berlin, Heidelberg, New York: Springer, 1995, pp. 3–16

    Google Scholar 

  33. Moody, R.V.: Meyer Sets and the Finite Generation of Quasicrystals. In: Symmetries in Science VIII. B. Gruber (ed.) London: Plenum, 1995

    Google Scholar 

  34. Moody, R.V., Patera, J.: Local dynamical generation of quasicrystals. Preprint

  35. Radin, C.: Global order from local sources. Bull. Am. Math. Soc.25, 335–364 (1991)

    Google Scholar 

  36. Radin, C.: The pinwheel tilings of the plane. Ann. Math.139, 661–702 (1994)

    Google Scholar 

  37. Radin, C.: Space tilings and substitutions. Geometriae Dedicata55, 257–264 (1995)

    Article  Google Scholar 

  38. Robinson, E.A., Jr.: The Dynamical Theory of Tilings and Quasicrystallography. In: Multidimensional Symbolic Dynamics: Proceedings of the Special Year Warwick 1994

  39. Senechal, M.: Quasicrystals and Geometry. Cambridge: Cambridge University Press, 1995

    Google Scholar 

  40. Shechtman, D., Blech, I., Gratias, D., Cahn, J.W.: Metallic phase with long-range orientational order and no translational symmetry. Phys. Rev. Lett.53, 1951–1953 (1984)

    Article  Google Scholar 

  41. Socolar, J.: Weak matching rules for quasicrystals. Commun. Math. Phys.129, 599–619 (1990)

    Article  Google Scholar 

  42. Sohncke, L.: Die regelmässigen ebenen Punktsysteme von unbegrenzter Ausdehnung. J. reine Angew.77, 47–101 (1874)

    Google Scholar 

  43. Solomyak, B.: Tiling dynamical systems. Ergod. Th.&Dyn. Sys., to appear

Download references

Author information

Authors and Affiliations

  1. AT&T Bell Laboratories, 07974, Murray Hill, New Jersey, USA

    J. C. Lagarias

Authors
  1. J. C. Lagarias
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Communicated by J.L. Lebowitz

Rights and permissions

Reprints and permissions

About this article

Cite this article

Lagarias, J.C. Meyer's concept of quasicrystal and quasiregular sets. Commun.Math. Phys. 179, 365–376 (1996). https://doi.org/10.1007/BF02102593

Download citation

  • Received:

  • Accepted:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02102593

Keywords

Search

Navigation