Tenfold Symmetry and a New Definition of Crystal: The Enduring Impact of Alan Mackay

In 1975, Professor Alan L. Mackay, Birkbeck College, London, was a plenary lecturer at the 10th Yugoslav crystallographic meeting in Kumrovec, Croatia, with a rather controversial lecture, "Generalised Crystallography", in which he strongly attacked classical crystallography and The International Tables of X-ray Crystallography (Mackay, 1975) striving to see crystallography as a general science of structure. He thought that 'modern crystallography' should explain and control the behaviour of more complicated systems that are not crystals in the classical sense. Furthermore, Mackay's special interest was in pentagonal symmetry – for some classical crystallographers, 'a forbidden symmetry'. However, there are natural crystals in the form of pentagonal dodecahedra like common pyrites (fool's gold). He used the Penrose pattern, which is a tiling of two-dimensional and three-dimensional space using identical tiles of two kinds (acute and obtuse rhombi) in 2D and acute and obtuse rhombohedra in 3D. These patterns were unique and non-periodic (Mackay, 1981, 1982). Mackay even produced an optical transform of the Penrose pattern that exhibited tenfold symmetry, which is very similar to the diffraction pattern of D. Shechtman (Nobel laureate) obtained under an electron microscope for a so-called quasi-crystal. The terms quasi-identity and quasi-equivalence were already defined by Mackay in 1981, and, somehow, he forecast the discovery of materials with these properties.

However, as so often happens in science, many excellent and hard-working scientists are not getting proper rewards for their work. Despite this, they can live with the feeling that they did something good for humanity. At this point, I, as a chemist, should mention Dmitrij I. Mendeleev (2019 was the International Year of the Periodic Table), who established the periodic table of elements (now in general use), and who predicted three new elements and their properties but never received the Nobel prize.

Needless to say that the definition of 'crystal' has changed. In 1992 the IUCr altered the definition of a crystal – it should have the ability to produce a clear-cut diffraction pattern (see https://www.iucr.org/news/newsletter/volume-29/number-2/change-to-the-definition-of-crystal-in-the-iucr-online-dictionary-of-crystallography for the latest definition of a crystal) with the possibility of ordering being either periodic or aperiodic.

References

Mackay, A. L. (1975). lzvestija Jugoslav. Centr. za Kristallogr. 10, 25-36.

Mackay, A. L. (1981). Sov. Phys. Crystallogr. 26, 517–522.

Mackay, A. L. (1982). Physica A 114, 609–613.