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Symmetry

36 37 penrose tilinGs QuAsicrystAls surprising fivefold symmetries In the mid1980s the world of crystallography was taken aback by the announcement of an entirely new kind of material, midway between the crystalline and amorphous states. What was particularly surprising about this new state of matter was that it appeared to be based on a 5fold symmetry, apparently violating the basic laws of crystallography. Until this time the conventional understanding was that only 2, 3, 4 and 6fold symmetries could create the lattice structure on which crystals were formed. The new material, Shechtmanite 3 named after its discoverer, soon became classified as a quasicrystal, and other examples of these materials which, on the scale of solids, lie somewhere between crystals proper and glass gradually appeared. Naturally, new uses soon began to be found for these exotic materials. High magnification microscopic images and Xray diffraction patterns of quasicrystalline structures reveal unusual dodecahedral symmetries, and the appearance of the phi ratio below. Interestingly, the loose symmetries on which they are based had been prefigured by the Oxford mathematician Roger Penrose in the early 70s. Penrose had produced a pair of nonperiodic tilings, based on approximate pentagonal symmetry 4,5,6. As with quasicrystals, these patterns have elements of a longrange order despite their 5fold symmetryand they can fill the plane in an infinite number of ways 7. The rhombic triacontahedron. the 3D analogue of a Penrose tiling, and the building block of a quasicrystal. 8. Schechtmanite snowflakes form when an aluminium manganese alloy is cooled rapidly. 1. A flow pattern showing 5fold symmetry 4. Penrose tiling no. 1, using two golden diamonds 5. Penrose tiling no.2, using golden darts and arrows. It is impossible to fill the plane using only pentagons, but Penrose tiles 6. can be laid out in many ways 2. An unusual 5fold Islamic decorative mosaic 3. A microphoto of schechtmanite, showing its 5fold structure 6. 7. 8.
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