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geometry.college, Mon, 15 Nov 1993

Starting in 1912, X-ray diffraction made it possible to study crystal structure. Although it does not show the actual crystal, diffraction indirectly indicates some of the structure, by giving information about the Fourier transform of the density distribution. If the density in a substance has a lot of symmetry, the X-rays will reinforce or cancel, as they stay in phase or go out of phase with each other; as a result, the diffraction pattern will exhibit symmetry as well.

If all of the density in a substance were to occur at the vertices of a lattice, the diffraction pattern would exhibit hexagonal, square, or rectangular symmetry. Until 1984, these were the only three observed symmetries. Using this and further evidence, physicists thought that crystals had a lattice structure at the atomic level. In fact, they considered this to be one of the defining characteristics of a crystal.

In 1984, a paper of Shechtman, Blech, Gratias, and Cahn disproved the assumption that substances arranged themselves in lattices; they found a metal alloy with with five-fold symmetry in its diffraction pattern. Since this is not a possible symmetry for the diffraction pattern of a lattice structure, the alloy must have a different structure. This discovery gave rise to the new research area of quasicrystals. In the last ten years, there has been much progress in this area, but there are still many open questions; even the structure of this first metal alloy is still an open question.

The study of quasicrystals uses many different branches of mathematics and physics, including the study of Penrose tilings. This is a nonperiodic way to tile the plane or three-dimensional space. It uses only two rhombus-shaped objects in two dimensions or two rhombohedrons in three dimensions. The tilings must obey certain strict matching rules, only allowing for a finite number of different ways to put the two kinds of tiles together. Although a Penrose tiling is not periodic, it has something called quasiperiodicity. This means that patterns repeat themselves quite often; in fact, given any pattern in the tiling, there is an R>0 such that it is possible to find a repeat of the pattern within any ball of radius R. Thus Penrose tilings still retain a lot of order.

It is quite amazing that this abstract geometric construction of Penrose tiles relates to the study of quasicrystals; using the vertices of a certain three-dimensional Penrose tiling as the points of density of a theoretical substance, the resulting diffraction pattern (in other words, a planar slice of the square modulus of its Fourier transform), exhibits five-fold symmetry and looks quite similar to that of the alloy discovered in 1985.

Here is an even further amazing fact related to the diffraction of Penrose tiles; in 1982 N.G. de Bruijn showed that all Penrose tilings of the plane result from the projection of part of a five-dimensional lattice onto the plane. Likewise, all Penrose tilings in three dimensions result from the projection of part of a six-dimensional lattice onto three-dimensional space. Thus the new alloy, although no longer a lattice, has a diffraction pattern similar to that of the projection of part of a six-dimensional lattice.

This discovery of nonperiodic tilings with symmetry in the diffraction pattern resulted in the following mathematical question: how ordered does a pattern of densities have to be so that the diffraction pattern still has bright spots? Physically, we know that there must be some order; otherwise the waves will have no strong reinforcements and cancellations needed for dark and bright spots. What are the conditions for these reinforcements and cancellations to occur in the Fourier transforms of projections of portions of arbitrary dimensional lattices? In what ways can one predict a diffraction pattern based on knowledge of the densities?

Mathematics professor Marjorie Senechal tries to answer these questions. In order to simplify the question, she only looks at diffraction patterns of point densities in the plane. This is still a rich subject with many open questions.

Senechal is currently writing a book called "Quasicrystals and Geometry" and is at the Geometry Center for a month to prepare the illustrations. One chapter of the book will consist of an atlas of diffraction patterns of tilings of the plane. Some are Penrose tilings, but others are more general. The atlas is meant to supplement a book called "Atlas of Optical Transforms."

To help Senechal prepare these illustrations, Center staff have developed software to compute the Fourier transform of an arbitrary set of point densities and to project portions of lattices in high dimensions onto the plane. For more information on the software to compute projections of high dimensional lattices, see Quasitiler. Senechal's book should be available in about a year from Cambridge University Press.


G. Harburn, C.A. Taylor, and T.R. Welberry, Atlas of Optical Transforms, G. Bell & Sons, Ltd., London, 1975.

Marjorie Senechal and Jean Taylor, "Quasicrystals: The View from Les Houches," The Mathematical Intelligencer, Vol. 12, No. 2, 1990.

The original article on quasicrystals:

D. Schechtman, I. Blech, D. Gratias, J. Cahn, "Metallic phase with long range orientational order and no translational symmetry," Physical Review Letters, Vol. 53, 1984, p. 1951-1954.

Also see:

Richard Kenyon, "Self-Similar Tilings," Geometry Center Preprint 21, 1990.

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