the spaces between lines are short and some are long. The long-range spacing of short and long lines in each set of Ammann lines forms a Fibonacci sequence in which the ratio of long to short spacings approaches the golden ratio.

"It is not periodic, but it is ordered and predictable," Steinhardt said. "You can have both ordering and nonperiodicity."

Quasicrystals may have practical applications. There are indications that they may be highly resistant to deformation, which would make them valuable for use in heavy-duty bearings. The task of exploring their physical, as well as their mathematical, properties is just beginning.

"We're in the same boat as physicists were with crystals a hundred years ago," Steinhardt said: "We have the structure of quasicrystals. Now we must predict their electronic and physical properties. This is a mathematical challenge, because the mathematics for crystals doesn't work for quasicrystals."


Onoda, George Y., Paul J. Steinhardt, David P. DiVincenzo, and Joshua E.S. Socolar. 1988. Growing perfect quasicrystals. Physical Review Letters 60:2653–2656.


Asimov, Isaac. 1988. Understanding Physics. Dorset Press Reprint Series. Hippocrene Books, New York.

DiVincenzo, D., and P.J. Steinhardt (eds.). 1991. Quasicrystals: The State of the Art. World Scientific Publishing Company, Singapore.

Steinhardt, P.J. 1990. Quasicrystals: A New Form of Matter. Endeavour 14(3):112–116.

Steinhardt, P.J., and S. Ostlund (eds.). 1987. The Physics of Quasicrystals. World Scientific Publishing Company, Singapore.

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