large, single crystals were made. Powder patterns do not reveal the rotational symmetry and can always be fitted to a model if a large enough unit cell is assumed.

Simultaneous with the paper2 announcing this finding there appeared a theoretical paper showing that icosahedral quasicrystals could be understood in terms of cuts of 12-dimensional crystals.29 Because of our residual doubts, the announcement paper received wide circulation before it was submitted for publication and this stimulated an immediate paper by Levine and Steinhardt,30 followed shortly by three other papers3133 that all showed icosahedral symmetry to be consistent with quasiperiodicity and gave diffraction intensities from a quasilattice that were qualitatively in good agreement with the electron diffraction patterns. In the year since publication we have received more than 100 preprints on many aspects of these new materials. Two other point groups, 10/m (decagonal)34 and 12 (dodecagonal)35 have been reported. Icosahedral diffraction patterns have been reported from many different alloys, and the possibility that a stable icosahedral phase (Al5Li3Cu) exists and that large, single crystals can then be grown seems to have been confirmed.36,37 Our research about the structure and properties of these crystals was hampered by the imperfections of rapidly grown crystals, but it can now proceed.



D.Shechtman and I.Blech, Metall. Trans. 16A, 1005 (1985).


D.Shechtman, I.Blech, D.Gratias, and J.W.Cahn, Phys. Rev. Lett. 53, 1951 (1984).


W.Friedrich, P.Knipping, and M.Laue, Ann. Phys. 41, 971 (1913).


C.H.Johansson and J.O.Linde, Ann. Phys. 25, 1 (1936).


H.A.Bohr, Almost Periodic Functions (Chelsea, New York, 1947).


A.S.Besicovitch, Almost Periodic Functions (Cambridge University Press, New York, N.Y., 1932).


P.M.deWolf, in Modulated Structures, NATO ASI Series E 83, edited by Tsakalakos (Martinus Nijhoff, The Hague, 1984).


A.Janner and T.Janssen, Acta Crystallogr. Sect. A 36, 399 (1980).

9. Bruijn, Ned. Akad. Wetensch. A 84, 39 (1981).


F.C.Frank, Proc. R. Soc. London 215, 43 (1952).


D.Nelson and B.I.Halperin, Science 229, 233 (1985).


P.J.Steinhardt, D.R.Nelson, and M.Ronchetti, Phys. Rev. B 28, 784 (1983).


M.Gardner, Sci. Am. 236, 110 (1977).


S.Baer, Zome Primer (Zomework Corp., Albuquerque, 1970).


A.L.Mackay, Physica A 114, 609 (1982).


C.Radin, University of Texas (preprint).


S.Alexander and J.McTague, Phys. Rev. Lett. 41, 702 (1978).


P.Duwez, R.H.Willens, and W.Klement, J. Appl. Phys. 31, 1126 (1960).


A.Wilm, Metallurgie 8, 225 (1911).


P.D.Merica, R.G.Waltenberg, and H.Scott, Scientific papers of the U.S. Bureau of Standards, No. 347 (Washington, D.C., 1919), Vol. 15, p. 271.


L.Bendersky and M.J.Kaufman, Philos. Mag. B 53 (3), L75 (1986).

The National Academies of Sciences, Engineering, and Medicine
500 Fifth St. N.W. | Washington, D.C. 20001

Copyright © National Academy of Sciences. All rights reserved.
Terms of Use and Privacy Statement