Crystal Mathematician

The SIAM Mathematical Aspects of Material Science conference started today in Philadelphia, just a few blocks south of City Hall, and we are meeting in the Doubletree Hilton. The Mathematical Crystallography minisymposia are meeting in Maestro B on the fourth floor, and looking out we see a space frame:



Um, so what is this? It’s a slice, but if we expand it out it looks uninodal, 12-regular, with two orbits of edges (four to a vertex horizontally and eight to a vertex diagonally). This crystal net has been associated with both James Maxwell and R. Buckminster Fuller, so the fact that I can’t find it in RCSR (I don’t think that its space group is Im-3m) or EPINET is doubtless my incompetence. If some nice person would enlighten me…

Meanwhile, one thing we are thinking about is what this blog can be used for. Like asking questions. Further posts this week, hopefully daily…

In 2005, the 125th anniversary of Science magazine, that journal produced a list of the 125 questions on what we don’t know. Among the Top twenty-five was: How Far Can We Push Chemical Self-Assembly? Robert Service wrote that

… chemists thrive on finding creative new ways to assemble molecules. For the last 100 years, they have done that mostly by making and breaking the strong covalent bonds that form when atoms share electrons. Using that trick, they have learned to combine as many as 1000 atoms into essentially any molecular configuration they please.

But

… this level of complexity pales in comparison to what nature flaunts all around us. Everything from cells to cedar trees is knit together using a myriad of weaker links between small molecules. These weak interactions, such as hydrogen bonds, van der Waals forces, and p-p interactions, govern the assembly of everything from DNA in its famous double helix to the bonding of H₂O molecules in liquid water. More than just riding herd on molecules, such subtle forces make it possible for structures to assemble themselves into an ever more complex hierarchy. Lipids coalesce to form cell membranes. Cells organize to form tissues. Tissues combine to create organisms. Today, chemists can’t approach the complexity of what nature makes look routine.

He concluded by asking, “Will they ever learn to make complex structures that self-assemble?”

Taking “self-assembly” as a compound word, it sounds like it applies to structures that assemble themselves. But many scientists assume more than that: the phrase “shake and bake” applies to the process of adding the correct ingredients to a pot containing the correct media, along with the correct catalysts, heating it and cooling it by the correct sequence of temperatures over the correct time span, et voila! A nano-scale Swiss watch.

Shake and bake is what the alchemists did, and while it is quite powerful – that’s how we got those structures of up to a thousand atoms joined by covalent bonds – we shouldn’t expect to build a watch this way. In fact, an economist (!) addressed the watchmaking problem in a seminal paper on The Architecture of Complexity, and proposed an alternative process he observed in social systems – and others have observed in nature. Herbert Simon imagines two watchmakers. One assembles a watch from its components, one watch at a time from scratch. The other assembles basic components into modules, one module at a time, making umpteen modules of type A, umpteen of type B, and so on, then modules into super-modules, umpteen of type I, umpteen of type II, and so on, and then puts together the super-modules into umpteen watches. This hierarchical construction reduces the number of errors and increases efficiency. It is also the way Mother Nature does things in biology.

Contrary to the image of “protoplasm” entertained by biologists of a century ago, biologist now envision a cell as a highly organized structure, with partitions, communication links, supporting superstructure, all serviced by armies of protein molecules, each a specialist in a particular task. These protein molecules do not wander around in a crowd; they move around in compartments, corridors, and other spaces, encountering other entities that are supposed to be there. The cedar trees that Service describes as structures currently beyond our technology were the result of constrained if not managed assembly.

Since biological systems are the most familiar ones employing managed assembly, we should expect managed assembly in the literature to be heavily biological at first. For example, Bartosz Lewandowski et al report in Science that Sequence-Specific Peptide Synthesis by an Artificial Small-Molecule Machine. Here, they “report on the design, synthesis, and operation of an artificial small-molecule machine that travels along a molecular strand, picking up amino acids that block its path, to synthesize a peptide in a sequence.”

The trick will be to organize many of these little tasks within a controlled space to get … a factory. And factories will be able to build things way beyond anything we’ve made thus far.

When looking at (the mathematics of) symmetric structures in space, it doesn’t matter much whether those structures are ensembles of points (representing locations of atoms or molecular building blocks), or graphs embedded in space, or even continuous functions: the theories of each of these is roughly equivalent. The crystallographic restriction remains the crystallographic restriction.

The feature of the November 12 Notices of the American Mathematical Society – the glossy magazine the AMS sends to all 30,000+ members – looks at the crystallographic restriction applied to continuous functions. A little review for visitors a bit hazy on their Calculus III. If we represented a function f : R^d → R as an infinite sum of vectors

f(x) = ∑_{n ∈ Z^d} a_n e^{2 π i n ⋅ x},

where R is the set of reals and Z is the set of integers, we get a periodic function as follows. Write n = (n₁, …, n_d). Recall that in R^d, the standard basis is the set of vectors e_i = (0, 0, …, 0, 1, 0, …, 0), with the 1 in the ith position. Then for any term a_n e^{2 π i n ⋅ x},

a_n e^{2 π i n ⋅ (x + e_i)} = a_n e^{2 π i n ⋅ x} e^{2 π i n ⋅ e_i} = a_n e^{2 π i n ⋅ x} e^{2 π i n_i} = a_n e^{2 π i n ⋅ x}

as e^{2 π i} = 1. Thus f is periodic, with the standard basis as its geometric lattice. A symmetry of f is a function g : R^d → R^d such that fg = f, and if Π is some subspace of R^d, then a symmetry of Π is a function g : Π → Π such that fg = f on Π.

In this article, Frank Farris looks at rotations on planes Π in R^d, specifically rotations of order 3 and 5. The point is that you can always get a rotation of order d in a plane in R^d (use the rotation

(x₁, x₂, …, x_d) → (x_d, x₁, …, x_{d – 1}));

the problem is the tilt of the resulting plane.

• For d = 3, Farris obtains a plane parallel to a pair of lattice vectors, so f restricted to that plane has all of its translational symmetries – as well as the rotational symmetry of order 3.

• For d = 5, the plane is not parallel to any lattice vector, so it has no translational symmetries. But it does have a rotational symmetry of order 5 – and there are lattice vectors arbitrarily close to parallel to the plane, so there are translations that are “nearly” symmetries.

This is a variant of one of the standard procedures for obtaining “quasi-periodic” structures.

This is a fairly accessible introduction to basic issues and basic nomenclature.

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.

(Pause.)

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.

Welcome to the show!