It’s been a century since Walter Friedrich and Paul Knipping performed an experiment proposed by Max (von) Laue: diffracting an X-ray through a crystal lattice. And it’s been about half a century since Robert Berger’s aperiodic tiling of the plane which, together with concerns about phase transitions in crystals from one configuration to incompatible ones, revived the issue of what crystals looked like at Mr. Feynman’s bottom.
The two questions – What do crystals look like? and What is a crystal anyway? – are entangled in the algal mat of crystallographic definitions. For about a century, we knew that a crystal was lump of matter with molecules or atoms lined up like soldiers on parade. Since then, we’ve encountered a variety of materials with atoms or molecules in very parade-like arrays, and amidst a sort of Kuhnian paradigm shift, it was decided that a crystal has a parade-like diffraction pattern.
But how well do we understand crystals?
The big event this year, according to our colleagues in the spires of physics, is the (virtual) confirmation of the existence of the Higgs boson. Using the high-falutin’ contraption that is the Standard Model, Peter Higgs postulated this rather bizarre creature, and physicists persuaded governments to pour billions of euros into a machine to find this thing. And find it they did.
There is a particular credibility to a theory that makes a prediction – especially a prediction of the existence of something – and someone following the theory finds it. Quasicrystals were (sort of) predicted before Dan Schechtman presented his not-entirely-welcome exhibit, and a growing number of crystal structures have been predicted prior to synthesis. But does this mean that mathematical crystallography compares in coherence, scope, and predictive power with the Standard model?
Of course, the mathematics may be delusory. We should not forget the collision between Lord Kelvin and Alfred Wallace on the age of the Earth: Kelvin had the math, but Wallace didn’t care, for Kelvin must be wrong somewhere. And it turned out that Kelvin was wrong – but that’s another story. The point is that we should not get too entranced with physicists and their equations.
But mathematics is a powerful tool, when used properly, so the question is: what sort of tool is it for crystallography and materials science? What sort of predictions (and design regimes) can we expect from it? And what do we want from it?