Crystal Mathematician

My primary project in crystallography is an exercise in cartography. There is (literally) a “space” of crystal structures that I intend to map. Some time ago, I wrote a program that hung around a lamppost and enumerated crystal nets that it found; I am now working on a program (current version at sourceforge) that gingerly extends a toe and ….

But since I was but a mere mathematical logician who knew nothing of crystals, or geometry, chemistry, or topology, and precious little about group theory or physics, I thought it might be a good idea to find out who else is in this neck of the woods, and what they are up to. This was partly a resurrected version of a similar project I had planned for my corner in mathematical logic (long story), and partly a result of my longstanding interest in the philosophy of science.

So what is my neck of the woods in crystallography?

Among my assigned reading was a Nature article by Omar Yaghi, Mike O’Keeffe, and friends on something that they called “reticular synthesis”. “Reticular” because they were building vast net-like structures out of building units, but the critical point was that they hoped to design in advance the crystal structures that they intended to synthesize. What struck me was that they were proposing a step analogous to, but actually more revolutionary than, the step church architecture took in the Renaissance. During the Middle Ages, cathedrals were built on a somewhat ad hoc basis (and unsurprisingly, acquired structural problems en route). The more ambitious domes of the Basilica of St. Mary of the Flower and of St. Peter’s Basilica required a more intense planning by Filippo Brunelleschi and Michelangelo di Lodovico Buonarroti Simoni, respectively.

So I wrote and posted a manifesto of my own. And it seemed to me – and still seems to me – that the subject was crystal design. For perhaps a quarter century, there has grown a sentiment that crystal engineering should work sort of like building a building. An architect, in concert with an engineer, composes a set of blueprints, from which the building is constructed – and that building is what you wanted to get built. And just as ordinary architects can design and oversee the construction of shopping centers, ordinary chemical engineers should be able to design novel crystals and oversee the successful synthesis of novel crystals.

We are talking about crystal design, and for it we are talking about mathematics analogous to the descriptive geometry that helped make the Industrial Revolution. We are talking about geometric crystallography. So for this series of posts, we are interested in a snapshot of the crystal-design-and-geometric-crystallography community.

What does it mean to map a community? There actually was a historical “map”, the Historical Atlas of Crystallography, which was (to a limited extent) able to identify major players and where they were. Of course, this kind of exercise is misleading: the heroic model of science is popular not only because it is consistent with our popular mythology but also because it is easier to follow. But Marjorie Senechal‘s article on Geometric Crystallography was a good place to begin.

And what did that article tell us? That geometric crystallography made crystallography into a science, and then was largely sidelined when physicist and chemists found that functional analysis found the answers from X-ray diffraction patterns. And geometric crystallography was primarily concerned with analyzing crystals, for few people thought about designing crystals until recently.

This is a project to find out how much crystal design and geometry have to do with each other – as a sociological phenomenon. So we look at networks. In this first installment, let’s ask some basic questions.

Let’s get out some basic tools, to which I was introduced by some experts. The one much beloved of deans is the Thomson-Reuters Web of Science, which has (for us) at its heart the Science Citation Index. We immediately have a problem. Web of Science proudly states that it “… covers over 12,000 of the highest-impact journals worldwide.” Not impressively helpful, for two reasons:

• This is an emerging field, and ethologists and sociologists agree that many movements do not emerge from the top of the hierarchy. The Web of Science may be slow to register an emerging field.
• The search is by “topics”, a kind of keyword. So the problem is whether a phrase like “geometric crystrallography” actually appears somewhere in the textual description.

With these caveats in mind, let’s proceed.

We start with “crystal engineering”. Web of Science reports a steady growth in citations (left) over the last two decades, but more problematic growth in actual publications (right) over the last few years (that may just be because “crystal engineering” no longer sounds as edgy as it used to, but who knows?):

What about crystal design? Again, citations to the left, publications to the right.

And in the entire index, there are eight entries under “geometric crystallography”. Incidentally, all searches were conducted with quotes.

One last search in the Web of Science, arising from a historical accident. For reasons that I will go into at some later date, there are a bunch of crystal design people who use graphs and call them “topologies”; this is because some mathematicians regard graphs as spaces composed of two-manifolds glued together … well, anyway, Web of Science won’t draw a graph for “crystal structure” because there are 303,364 entries, but if you restrict the search with the word “topology”, you get only publications, for Web of Science freaked out over citations:

So that’s what’s happening in the highest-impact journals. But what about further down the ladder, which is (according to ethologists and sociologists) where new things emerge?

WorldCat advertizes itself as “The World’s Largest Library Catalog”. Unfortunately, I haven’t figured out how to get it to draw pictures. But anyway, here are a few keyword searches:

• The second item in “crystal design” without quotes was Indiana Jones and the Crystal Skulls, so we need quotes, and we get 472 entries (including 352 articles, 251 of them peer-reviewed, and including 80 books) going back to 1912.
• For “crystal engineering”, there were 5,449 entries, going back to 1900.
• There were 227 entries for “mathematical crystallography” and 23 for “geometric crystallography”. For ‘Full Record=”geometric crystallagraphy”, there were no results (!). For ‘Full Record=”crystal design”‘, there were 85 hits, from 1921 (!) to last year.

And there were 391,755 entries on “crystal structure”, not too many short of the Web of Science number.

Google Scholar is free. Of course, there is a diversity of things that are “hits”, but anyway … “about 1,720,000” hits for “crystal structure” (“about 217,000” if “topology” is added), “about 33,700” for “crystal engineering”, “about 4,740” for “crystal design” and “about 235” for “geometric crystallography”.

Mathematical Crystallography sprawls across several academic disciplines, and every discipline has one or more organizations. Some of them have their own databases.

• The American Chemical Society publishes 51 journals, including some of the highest impact journals of any field, and has an engine for searching all of them. For ‘Anywhere=”geometric crystallography”‘, there were ten hits, dates ranging from 1938 to 1970. For ‘Anywhere=”crystal design”‘, there were 241 hits, ranging from 1980 to last week.
• The American Physical Societry publishes 17 journals. For ‘Full Record=”geometric crystallagraphy”, there were no results (!). For ‘Full Record=”crystal design”‘, there were 85 hits, from 1921 (!) to last year.
• Excuse my parochialism, but mathematics is the most organized of fields. It’s a virtue born of necessity: mathematics is the most fragmented of fields. There are two massive databases, the old Zentralblatt MATH (also known as ZBL), which currently covers 3500 journals and over a thousand serials and goes back to 1826, and the Mathematical Reviews (also known as MR) launched by the American Mathematical Society in 1940.
  I do not have paid access to ZBL, but I can get basics: ZBL has eight hits for “geometric crystallography” and one (my paper in JGAA) for “crystal design”.
  MR has twelve hits for ‘Anywhere=”geometric crystallography”‘ (from 1986 to 2007), 372 for ‘Anywhere=”crystal structure”‘ (from 1941 to this year), and none for ‘Anywhere=”crystal design”‘.

These are some of the basic tools that I’ll be using, although there are some others. More next weekend.

The third Mathematical Crystallography minisymposium was entitled Structure-Building Principles. Unfortunately, I forgot to recharge my camera battery, so I didn’t take pictures. Fortunately, Massimo Nespolo did.

Jean-Guillaume Eon talked about Imprimitivity in Non-Crystallographic Nets. The automorphism group of a periodic net graph might not be crystallographic, and if it is not, we call the net non-crystallographic. Some descriptive machinery is introduced, and using such machinery, one can dissect non-crystallographic nets via their (labeled) quotient graphs.

Henrik van Lengerich talked about Self Assembly and the Structure of Matter. An objective structure can be constructed by applying a finite subgroup of O(3) to an object, and obtain a structure consisting of copies of that object. Experiments show that some structures are unlikely to assemble properly because during assembly, they get trapped in an incomplete metastable state. Thus successful freely assembled structures require design, and Langevin dynamics are presented as a candidate design regime.

Mike Zaworotko talked about Why Topology Matters to Crystal Engineers. Crystal engineering is aimed at applications like gas storage and capture, and in these applications, “the material is the application.” Metal-organic materials are designed at the molecular or atomic level using as building blocks highly symmetric polygonal or polyhedral molecular building blocks.

It was the last day of the conference, an appropriate day for slumming, so we went to a local Olive Garden.

The second Mathematical Crystallography minisymposium was entitled Beyond Classical Crystal Symmetry.

Massimo Nespolo talked about A Stroll Along Structural Paths: Symmetry, Pseudo-Symmetry and Their Exploitation to Understand and Design Structures. Equivalence relations among crystal structures, e.g. isostructure or isomorphous crystals define properties that may be preserved in phase transitions. This results in a classification using archetypic aristotypes that represent families of lower symmetry hettotypes.

Uwe Grimm talked about Recent Advances in Mathematical Diffraction Theory. A diffraction pattern will be a sum of discrete Bragg peaks, a diffuse scattering, and an absolutely continuous background. (For a crystal with atoms at lattice points, the diffraction consists solely of the reciprocal lattice of the given crystal lattice, perhaps represented as a Dirac comb.) But for quasicrystals, one may use a salami slice from a higher dimensional lattice, and a more complex structure. Less orderly structures will have diffusion and background components in their diffraction.

Marjorie Senechal talked about Periodicity, Aperiodicity and Prehistory. 270 years since Rene Just Hauy’s birth, his model of crystals as periodic structures of arrayed “nucleal molecules” is no longer the consensus. Quasicrystals – some of which were portrayed as aperiodic tilings, some of which are systems of clusters (even icosahedral clusters) – made Hauy’s model “wobble”. With the International Year of Crystallography upon us, what is the role of mathematics in crystallography? What are the major problems in mathematical crystallography? E.g., infinite structures, diffuse diffractino, self-assembly, folding, dense packing.

Peter Zeiner talked about Coincidence Site Lattices and Well-rounded Sublattices in the Plane. There are several kinds of nice lattices. For example, a lattice in d-space is well-rounded if its vectors of minimal length span d-space; dense sphere packings induce well-rounded lattices. So what lattices have well-rounded sub-lattices? There is a necessary and sufficient condition for a lattice in the plane to have a well-rounded sub-lattice, and there are methods for enumerating them. Other issues and other kinds of niceness were described.

Then we went to lunch at Imperial Inn.

I appear to have fallen into a trap. Stephen Hyde says that the structure is what RCSR calls fcu and what EPINET calls sqc19. So how did I goof?

When I looked up at the space frame (photo below), I assumed that the edges parallel to the window were in a different orbit than the diagonal ones, and hence that there were two orbits of edges. So I looked for crystal structures with two orbits of edges:

• In RCSR, I went to the net search page and made the following entries: “1” for lower and upper bounds on the number of kinds of vertex (only one orbit of vertex), “2” for the lower and upper bounds on the number of kinds of edges (two orbits of edges),”3″ for the lower and upper bounds on the smallest ring (a graph theorist would call this a cycle), and “12” for the lower and upper bounds on the coordination (a graph theorist would call this the degree or valency). RCSR found five nets, four of which could be readily excluded from their pictures, and one with no picture (!) with space group Im-3m.
• In EPINET, I went to the net search page and entered “1” node per asymmetric unit (uninodal), “2” for edge transitivity (two orbits of edges), and “1” for nodes per unit cell (clearly from picture of space frame), and vertex degree exactly 12, not chiral, and I got two unlikely candidates.

But in fact, I did not realize that while the space frame looked as if the faces of the polygons (triangles and squares) chopped space into pyramids and tetrahedra, it was actually chopping space into octohedra and tetrahedra, all regular, and hence there was only one orbit of edges.

• In RCSR, if I entered one orbit of edges instead, I would get two candidates, one of which was fcu.
• In EPINET, if I entered one orbit of edges instead, I would get one candidate, sqc19.

So that’s how they work. You have to look at the crystal structure with the correct squint.

The SIAM conference on Mathematical Aspects of Material Science starts tomorrow, Sunday, on June 9, and will run for four days. Participants are already arriving and picking up their registration packets.

The minisymposia on Mathematical Crystallography will be held in Maestro B, up on the fourth floor. The schedule is:

• Mathematical Crystallography I: Geometric Foundations on Sunday, June 9, from 10:15 to 12:15 pm. The presentations will be:
  • 10:15 am, Frank Morgan on Minimal Interface Structures.
  • 10:45 am, Egon Schulte on Polyhedral Geometries and Symmetry.
  • 11:15 am, Ma. Louise N. de las Penas on Nanostructures Arising from Crystallographic Tilings.
  • 11:45 am, Bernd Souvignier on Capturing the Essence of Infinite Graphs in Quotient Graphs.
• Mathematical Crystallography II: Beyond Classical Crystal Symmetry on Monday, June 10, from 10:15 to 12:15 pm. The presentations will be:
  • 10:15 am, Massimo Nespolo on A Stroll Along Structural Paths: Symmetry, Pseudo-Symmetry and Their Exploitation to Understand and Design Structures.
  • 10:45 am, Uwe Grimm on Recent Advances in Mathematical Diffraction Theory.
  • 11:15 am, Marjorie Senechal on Periodicity, Aperiodicity and Prehistory.
  • 11:45 am, Peter Zeiner on Coincidence Site Lattices and Well-rounded Sublattices in the Plane.
• Mathematical Crystallography III: Structure-Building Principles on Wednesday, June 12, from 10:15 – 12:15 pm. The presentations will be:
  • 10:15 am, Jean-Guillaume Eon on Imprimitivity in Non-Crystallographic Nets.
  • 10:45 am, Greg McColm on Next Year is the International Year of Crystallography : How are We Going to Celebrate?.
  • 11:15 am, Henrik van Lengerich and Richard James on Self Assembly and the Structure of Matter.
  • 11:45 am, Mike Zaworotko on Why Topology Matters to Crystal Engineers

Notice that on Wednesday there will be discussion, open to the floor, on how to continue this campaign to popularize mathematical crystallography. This is the third gathering I’ve organized – after the Crystal Design Using Discrete Structures in Geometry minisymposium held in 2010 and the Special Session on Modeling Crystalline and Quasi-Crystalline Materials held last year. Next year, the International Union of Crystallography will meet in Montreal from August 5 to 12; something to think about.

If you cannot come to the discussion Wednesday at 10:45 am, feel free to email suggestions to me.