During the last four years, a protracted debate has been evolving in an effort to explain the existence of solids with “forbidden symmetries.” The debate ensued following the 1984 discovery (D.Shechtmanetal., Physical Review Letters, 53, 195 1-3, 12 November 1984) of an aluminum-magnesium alloy whose atomic structure appears to have an icosahedral symmetry—a symmetry that is supposed to be impossible, according to a century-old dictum of solid-state physics.

The icosahedron is a regular polyhedron with fivefold symmetry axes that, according to the well-known theorems of crystallography, are geometrically incompatible with crystalline. (periodic) structure.

In just the few years since the initial discovery, more than one hundred other alloys with icosahedral symmetry have been found, adding further incentive for finding an explanation for the solids. Finally, in just the past few months, two break- throughs have taken place that appear to settle the issue. These advances were presented at a...

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