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Quasicrystals

From tilings to coverings

Nature volume 396, pages 14–15 (1998)Cite this article

Quasicrystals occur in a great number of alloys, most of which consist of aluminium and transition metals. They were discovered1 in 1984, being revealed by a rotational symmetry of X-ray or electron diffraction patterns (for instance, five-fold or ten-fold) which is impossible for true periodic crystals. Since then, physicists have wondered why atoms form in these complex patterns rather than in a regularly repeating periodic crystal. On page 55 of this issue2 Paul Steinhardt and colleagues present an analysis of new electron microscope data that supports a simple answer to this question.

Although a quasicrystal is non-periodic, its structure still follows a subtle construction plan. Mathematically, this can be described with reference to a higher-dimensional analogue of a cubic lattice: the atom arrangement of an icosahedralquasicrystal (which is quasiperiodic in three dimensions) can be constructed starting from six-dimensional space; the decagonal quasicrystal (whose lattice is quasiperiodic in a plane but periodic along the third dimension) requires reference to five-dimensional space. But why should atoms care about higher-dimensional spaces? The mathematical recipe to describe the lattice does not give us any hint as to how the atoms manage to create it.

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Figure 1: Penrose's planar quasiperiodic tiling.
Figure 2: Gummelt's planar quasiperiodic coverage.

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  1. Institute of Solid State Physics, Helmholtz Research Centre Jülich, D-52425, Jülich, Germany

    Knut W. Urban

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  1. Knut W. Urban
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Urban, K. From tilings to coverings. Nature 396, 14–15 (1998). https://doi.org/10.1038/23806

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