×

Meyer’s concept of quasicrystal and quasiregular sets. (English) Zbl 0858.52010

Summary: This paper relates two mathematical concepts of long-range order of a set of atoms \(\Lambda\), each of which is based on restrictions on the set of interatomic distances \(\Lambda- \Lambda\). A set \(\Lambda\) in \(\mathbb{R}^n\) is a Meyer set if \(\Lambda\) is a Delone set and there is a finite set \(F\) such that \(\Lambda- \Lambda \subseteq \Lambda+F\). 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 of quasiregular set, which is a set \(\Lambda\) such that both \(\Lambda\) and \(\Lambda- \Lambda\) are Delone sets. This paper shows that these two concepts are identical.

MSC:

52C99 Discrete geometry
11H55 Quadratic forms (reduction theory, extreme forms, etc.)
11E99 Forms and linear algebraic groups
82D25 Statistical mechanics of crystals
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Bombieri, E., Taylor, J.E.: Which distributions diffract? An initial investigation. J. Phys. Colloq.47, C3, 19–28 (1986) · Zbl 0693.52002
[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) · Zbl 0457.05021
[3] de Bruijn, N.G.: Quasicrystals and their Fourier transform. Nederl. Akad. Wetensch. Proc Ser. A89, 123–152 (1986) · Zbl 0617.05022
[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) · Zbl 0655.05025
[5] Chen, L., Moody, R.V., Patera, J.: Non-crystallographic root systems and quasicrystals. Preprint · Zbl 0916.20026
[6] Delaunay, B.N., Delone, B.N.: Neue Darstellung der Geometrischen Kristallographie. Zeit. Kristallographie84, 109–149 (1932) · Zbl 0006.04407
[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) · Zbl 0338.50007
[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)
[10] Elser, V.: The diffraction pattern of projected structures. Acta. Cryst. A42, 36–43 (1986) · Zbl 1176.52006
[11] Engel, P.: Geometric Crystallography – An Axiomatic Introduction to Crystallography. Reidel: Dordrecht, 1986 · Zbl 0659.51001
[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 · Zbl 0804.51032
[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) · Zbl 0598.52012
[14] Galiulin, R.V.: Delone systems. Sov. Phys. Crystallogr.25, No. 5, 517–521 (1980) · Zbl 0466.51013
[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 · Zbl 0839.11009
[18] Hof, A.: Diffraction by aperiodic structures at high temperatures. J. Phys. A: Math Gen.28, 57–62 (1995) · Zbl 0853.60090
[19] Janot, C.: Quasicrystals: A Primer. Oxford: Oxford University Press, 1992 · Zbl 0838.52023
[20] Katz, A.: Theory of matching rules for 3-dimensional Penrose tilings. Commun. Math. Phys.119, 262–268 (1988) · Zbl 0651.52015
[21] Katz, A., Duneau, M.: Quasiperiodic structures determined by the projection method. J. Phys. (Paris) Supp.C47, 103–112 (1987) · Zbl 0693.52004
[22] Kramer, P.: Non-periodic central space filings with icosahedral symmetry using copies of seven elementary cells. Acta Cryst.A 38, 257–264 (1982)
[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)) · Zbl 1176.52010
[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) · Zbl 0842.52011
[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) · Zbl 0807.52017
[27] Levine, D., Steinhardt, P.J.: Quasicrystals: A new class of ordered structures. Phys. Rev. Lett.53, 2477–2480 (1984)
[28] Levitov, L.S.: Local rules for quasicrystals. Commun. Math. Phys.119, 627–666 (1988)
[29] Lunnon, W.F., Pleasants, P.A.B.: Quasicrystallographic tilings. J. Maths. Pures Appl.66, 217–263 (1987) · Zbl 0626.52017
[30] Meyer, Y.: Nombres de Pisot, Nombres de Salem, et analyse harmonique. Lecture Notes in Math. No. 117, Berlin, Heidelberg, New York: Springer, 1970 · Zbl 0189.14301
[31] Meyer, Y.: Algebraic Numbers and Harmonic Analysis. Amsterdam: North-Holland, 1972 · Zbl 0267.43001
[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 · Zbl 0881.11059
[33] Moody, R.V.: Meyer Sets and the Finite Generation of Quasicrystals. In: Symmetries in Science VIII. B. Gruber (ed.) London: Plenum, 1995 · Zbl 0921.52006
[34] Moody, R.V., Patera, J.: Local dynamical generation of quasicrystals. Preprint · Zbl 0847.52023
[35] Radin, C.: Global order from local sources. Bull. Am. Math. Soc.25, 335–364 (1991) · Zbl 0808.52020
[36] Radin, C.: The pinwheel tilings of the plane. Ann. Math.139, 661–702 (1994) · Zbl 0808.52022
[37] Radin, C.: Space tilings and substitutions. Geometriae Dedicata55, 257–264 (1995) · Zbl 0835.52018
[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 · Zbl 0828.52007
[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)
[41] Socolar, J.: Weak matching rules for quasicrystals. Commun. Math. Phys.129, 599–619 (1990) · Zbl 0701.05059
[42] Sohncke, L.: Die regelmässigen ebenen Punktsysteme von unbegrenzter Ausdehnung. J. reine Angew.77, 47–101 (1874) · JFM 05.0511.01
[43] Solomyak, B.: Tiling dynamical systems. Ergod. Th.&Dyn. Sys., to appear · Zbl 1139.37009
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.