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Oct 222012

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?

Oct 102012

Whatever happened to mathematical crystallography?

That’s an odd question, because there is a lot of mathematical crystallography out there. On the recent 60th anniversary of the International Union of Crystallography, Massimo Nespolo, Chairman of the IUCr Commission on Mathematical and Theoretical Crystallography, enumerated several active areas in mathematical crystallography. The field would seem to be alive and well. Still, there is something slightly defensive about the title, Does mathematical crystallography still have a role in the XXI century?. After all, no one asks if mathematical ecology has a role in the XXI century.

A rational economist – of the University of Chicago variety – would say that of course, mathematical crystallography has a major role in this century. Our society faces enormous technological challenges, and many of these have components in materials science. There is a real need and an economic demand for scientific and technological work in crystal design, synthesis, and analysis. That includes developing new mathematical tools. So there is a real need and an economic demand for mathematical crystallography. So grant-awarding government agencies will fund mathematical crystallography, so students will be attracted to the field, and so there will be a lot of mathematical crystallography in coming decades. In fact, one can hear mathematics graduate students knocking down the doors at this very moment.


Well, if we learned anything from the financial meltdown of 2008, it’s that Chicago is all wet. As Chicago’s increasingly caustic critics observe, people have limited information, they have their own perceptions and preconceptions, and they have their own interests. People will respond to what they see, in the context of where they are. And where they are nowadays is in mathematics, physics, or chemistry departments (each with varying amount of mathematics), each with their own curricula, major requirements, comprehensive exams, and overworked researchers (according to a 2006 study, principal investigators spend half their time pursuing grants). Especially in times of limited resources, people will tend to keep doing what they are doing, never mind golden opportunities.

In addition, mathematical (and theoretical) crystallography are inherently interdisciplinary fields. That would seem to be an attractive feature, for interdisciplinary studies are all the rage. But the reason why they are all the rage is that major agencies are playing them up – and they are playing them up because interdisciplinary fields (which require collaboration and dealing with unfamiliar things) are unfamiliar and intimidating. What scholars and students need, perhaps, is some way of seeing interdisciplinary fields in a more attractive light.

Hence this site on mathematical and theoretical crystallography, maintained (if I may paraphrase Jane Austen here) by a partial, prejudiced & ignorant mathematician. The weblog is the most prominent feature: I will display my unsupported opinions, and invite everyone to join in. I make the usual requests: people be civil, obey copyright laws, and no spam. The site also contains resources: lists of books, sites, organizations, grant awarding agencies, events (including deadlines), conferences, etc. I am not omniscient – in fact, I tend to be oblivious – so I welcome items to link to.

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