The Mathematics of Impossible Crystals

By Kim Becker, The University of Adelaide

In 1982, a chemist by the name of Dan Shechtman had a curious problem in front of him. He was staring at a chemical structure that mathematicians had long ago deemed impossible. Shechtman had discovered quasicrystals.

But what exactly is a quasicrystal – and why was their discovery met with such controversy? How could mathematicians have got it so wrong? As we know, mathematics is the study of definitions. The conflict that followed Shechtman’s discovery can be attributed to one thing – an incorrect definition.

Mathematicians defined a crystal to be a structure built from a repeating unit, stacked indefinitely in space. We call this type of structure periodic. This definition should sound reasonable to those of us who have studied lattice structures of chemical compounds, which usually look like repeated cubes. What mathematicians failed to anticipate, however, was the possibility of crystals that were not built from a repeating unit – which we know today as quasicrystals. To understand what such a crystal would look like, we turn our attention to the Penrose tiling, depicted below.

At first glance, the Penrose tiling looks like any other repetitious pattern. I employ the reader to find a small segment of the Penrose tiling that can be repeatedly translated to create the pattern above. The reader may find this difficult, because this actually is impossible! There is no finite segment of tiling that can be repeated to create the whole tiling. This property is what distinguishes quasicrystals from our traditional definition of a crystal.

Although mathematicians were aware of aperiodic tilings like the Penrose tiling, it was not incorporated into their definition of a crystal, since it was doubted that this behaviour was possible in chemicals. Using their definition, mathematicians were able to find restrictions on the types of symmetry patterns crystals could display. Shechtman had observed a chemical that contained a `forbidden’ symmetry. For this reason, his discovery was discredited for many years. It did not occur to scientists that Shechtman had discovered a chemical that was not a crystal by their definition – and therefore did not obey the patterns they predicted.

Shechtman’s evidence was, however, indisputable – and eventually scientists accepted his claim. In 2011, twenty-three years after his original discovery, Shechtman was rightfully awarded a Nobel prize in chemistry, and thus quasicrystals were born. Their relatively recent discovery makes quasicrystals a contemporary area of interest for mathematicians and chemists alike. Further advancements in this field could lead to a more unified theory in crystallography that blends our traditional definition of crystals with our new theory of quasicrystals.

Kim Becker was one of the recipients of a 2016/17 AMSI Vacation Research Scholarship.

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