Abstract

Soon after the announcement of their discovery in 1984 [1], quasi-crystals hit the headlines. Here was a substance—an alloy of aluminum and manganese—whose electron diffraction patterns exhibited clear and unmistakable icosahedral symmetry (a view along a five-fold axis is shown in Figure 25-1). A clear and unmistakable diffraction pattern of any sort is evidence of “long-range order”: The diffraction pattern is a picture of a Fourier transform. Long-range order is usually synonymous with periodicity, and every periodic structure has a translation lattice. But a simple argument shows that five-fold rotational symmetry is incompatible with lattices in R 2 and R 3: every lattice has a minimum distance d between its points, but if two points at this distance are centers of five-fold rotation about parallel axes, the rotations will generate an orbit with smaller distances between them (Figure 25-2). By this chain of reasoning, it appeared that the impossible had occurred.

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