Quasicrystals

The story of Penrose's tilings was popularized in Martin Gardner's Mathematical Games column in Scientific American [Gar77, Gar78]. From that account, Robert Ammann (``a brilliant young mathematician working at low-level computer jobs in Massachusetts'' - M. Gardner) began his own investigations leading to many fundamental discoveries of his own. The most dramatic was his discovery of a pair of marked ``rhombohedra'' (i.e. every face is a rhombus) that tile only non-periodically. Nothing like this kind of structure had previously been suspected. What started as purely recreational mathematics was poised to explode on the deadly serious field of solid-state physics and materials science.

The molecular structure of crystals are usually analyzed by bombarding them with x-rays and capturing the results on film. The result is a pattern of smaller and larger bright dots called an x-ray diffraction pattern. The pattern reveals the symmetries of the crystal. It was an axiom of crystallography that only 2-fold, 3-fold, 4-fold, or 6-fold symmetries could occur.

In 1984,

• P. Steinhardt and D. Levine wrote a computer program to generate an x-ray diffraction pattern for Ammann's tiling by rhombohedra, if it existed in nature.
• A team of experimentalists headed by D. Shechtman at the National Bureau of Standards created an alloy of aluminum and manganese whose x-ray pattern has 5-fold symmetry.

By coincidence, the two parties ran across each other and compared the experimental pattern against the theoretical pattern. They matched, and a new kind of material had been discovered in nature: quasicrystals.

Traditional chemists and crystallographers (including Linus Pauling, a Nobel prize winner) doubted the quasicrystal model at first. The problem is that there is no known model of Local Growth for Penrose tilings. This would be a method of laying the tiles down after inspecting their nearest neighbors. Also, despite the coincidence of the diffraction patterns, the molecular structure of known quasicrystals do not seem to be arranged in rhombohedra shapes. The nature of the actual molecular structure remains a mystery.

Penrose and Conway don't believe a method is possible because of

Recurrence Property:

Any finite arrangement of tiles occurring in any valid Penrose tiling occurs infinitely often in that and every other Penrose tiling.

Forcing Property:

The location of two tiles arbitrarily far apart determine the position of infinitely many other tiles in the tiling.

In The Emperor's New Mind [Pen89], Penrose speculates that a quantum-mechanical process must be at work in the growth of quasicrystals. This process takes account of all previously laid tiles.

Several physicists have advanced ``local growth'' laws. None have gained universal acceptance. The efforts on the analysis and creation of quasicrystals have exploded so dramatically that hundreds of papers have been written in the few short years since their discovery. There is a Journal of Non-Crystalline Solids dedicated to these materials; a special issue on quasicrystals appeared in 1993. See [Bae90, Hof90] for some expository accounts of the subject.

Some Web resources on quasicrystals are:

• ZomeTool molecular structure system based on icosahedral symmetry. This is a sophisticated ``tinker toys'' set which may be used to construct 3-dimensional quasicrystals.
• Thesis on quasicrystals, in dvi format. This is the doctoral thesis of Petra Sonneborn (Ph.D. Cornell University, 1992) which contains a great deal of expository information on quasicrystals.
• Quick article by Senechal. This is a Geometry Center report.