More Challenges

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Jun 262013

At the Open Floor session on the future, I mentioned that one way to attract attention to a field was to pose major open questions. I forgot to mention a list of Big Problems presented by Marjorie Senechal, to which she has added another item. Here is the list, of Big Problems, with editorial asides..

  • Infinite structures. Crystals are among the best-behaved of these, so they are a natural place to begin. Crystalline structures are beguiling, in the sense that they look (to mathematicians) like something we already understand perfectly well, but as chemists point out, we don’t really understand them all that well, do we?
  • Diffuse scattering. Uwe Grimm talked about a rough classification of three types of scattering, the sharp Bragg peaks, the diffuse scattering, and the absolutely continuous background.
  • Self-assembly. How does it work (the subject of Henrik van Lengerich’s talk) and, is it actually the way we want to go (see the Free versus Managed Assembly posting of January 28.
  • Folding. This is what proteins do (for an example, see the internet game This may be connected to the basilisk of the secret chamber of chemistry: Mr. Schrodinger’s equation.
  • Crystal design. If we really understood crystals we should be able to make the crystals we want to make, just as we build the buildings we want to build.

So here is another set of challenges, which are tricky things. After all, I distinctly remember Mike Treacy throwing down the gauntlet several years ago and saying that we should design a zeolite and synthesize it from the design…

Jun 242013

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.

Jun 122013

This morning, we spent 25 minutes talking about plans for the future, using the Future Mathematical Crystallography posting as a starting point. Here are some of the points that were made.

  • The 20-minute format is difficult. Perhaps we should try round-table discussions following a presentation, or just a panel discussion. Either way, the panelists would be chosen from a wide range of fields, and there could be a Q &amp A component for the audience to participate. This was done at Madrid and it got the audience involved.
  • We are a small group, and if we spread ourselves over a large number of meetings, we would be spread thin and lose focus. On the other hand, there is no reason why we need some central authority to decide what meetings we have; we could have meeting organization be an emergent phenomenon. (At this point, I said that while I plan to go to one meeting a year, I am willing to help organize other meetings).
  • There are several places that offer support for more focused meetings (like Leiden) and places that offer support for longer programs, extending for weeks or even longer (like Cambridge).
  • We are starting discussions with SIAM about doing something jointly. We could also try the Materials Research Society or the American Crystallographic Society, as well as other societies.
  • There was a general feeling that we have to have titles of minisymposia and special sections that sell mathematical crystallography to people who may have jaundiced views of mathematics. Perhaps using adjectives like “applied”, or phrases like “crystal design” or “crystal structure”.

One point came up later, during Mike’s talk. One device for attracting attention is to list major open problems. David Hilbert did this in 1900, when he proposed 23 major problems for mathematicians (and crystallographers should all know about Hilbert’s Eighteenth Problem). Perhaps we should think about problems we might want to pose, perhaps like the list Mike Zaworotko proposed:

  1. Some nets appear to be special cases of others. For example, mjz is what a mathematician might call a homomorphic pre-image of pcu. How could the relationship between such pairs of nets be defined, and what does such a relationship between them imply?
  2. Crystal designers have engineered from design a number of uninodal and binodal nets. They are starting to get new trinodal nets. Is there a systematic way to do this?
  3. Mathematical crystallography spans several sciences and engineering fields. How can we develop a common language comprehensible and tolerable to (most) all participants?
  4. A number of polyhedra (often “complexes” like “faceted polyhedra”) seem particularly useful in crystal design. What (properties of) polyhedra should we expect to be useful?
  5. For porous crystals composed of molecular building blocks and linkers, we could try to build crystals with linkers that are not straight; i.e., they are angular. How should this be done?

We might think about composing a list and publicizing it.

And now, the floor is open to further discussion. To make a comment, click on “X Responses” at upper right, and leave a comment. To look at comments, click on “X Responses” to look at comments. Comments are somewhat moderated because of the spiders that want to leave viagra ads; the first comment you make, I have to approve (I will check daily to approve comments made by real human beings). But after that, the IUCr server should know you and your computer and email address and accept postings from you without moderation provided you used the same computer and email address.

Jun 122013

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 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 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 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.

Jun 112013

At the third Mathematical Crystallography session, there will be a discussion of future mathematical crystallography plans. Here are some things to think about.

First of all, I don’t see a real “mathematical crystallography” community. There seem to be a number of groups scattered amidst a much larger community of crystallographers who are quite happy with their software packages but otherwise might ask what mathematics is for. And there are several branches of mathematics (itself the most fragmented of the natural sciences – if you can call mathematics a natural science) that would be involved, from discrete geometry and graph theory to geometric group theory and Riemann manifolds to cohomology theory and algebraic topology, etc., many of whom may regard mathematical crystallography as an antiquarian activity.

And I haven’t even mentioned the physicists, biologists, nanoscientists and assorted engineers. Mathematical crystallography seems to be a panoply of villages scattered across several empires. From this panoply, how do we create a community?

One of the initiatives is this blog, where I am hoping to start a discussion. Blogs live or die by participation, which means that we need people to join the discussion. Hopefully, this posting is a start: how do we want to build the community?

Face-to-face meetings, attracting colleagues and students would be a likely mechanism. So conferences are a possibility. Next year is the International Year of Crystallography, so that is a good time for an event or two. Possible events include:

Two things about the longer term.

  • One very likely consumer of mathematical crystallography is metal-organic frameworks, and the biggest MOF community is in China. In 2015, SIAM will not hold its regular annual meeting, and instead it will set up shop in the gigantic 8th International Congress on Industrial and Applied Mathematics in Beijing. I have no idea what the ICIAM is like, but if it is at all like the International Congress of Mathematicians (and the ICIAM is modeled after the ICM), it will require a serious proposal to get in.
  • At the business meeting today, the SIAM SIAG on Mathematical Aspects of Materials Science decided that their next meeting will be in 2016 – in Europe.

One last point, following up on the business meeting. The SIAG is itself trying to get entry into the Materials Research Society, which I understand is a vast group of scientists and engineers who regard mathematicians as aliens from the planet Vulcan. During the discussion of what kind of minisymposia the SIAG might organize in a MRS meeting, one person said, “If you don’t have a hook to an existing MRS community, you will have an empty room.” Another said, “No one is going to come to a meeting on ‘Mathematical Methods’.” Since attendance at our minisymposia is smaller than attendance at the 2010 minisymposium (where the title was ‘Crystal Design using Discrete Structures in Geometry’ rather than ‘Mathematical Crystallography’), the fellow at the business meeting may have a point.

So now, comments, suggestions, criticism…?

Monday, June 10

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Jun 112013

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

Massimo Nespolo 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 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 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 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.

Sunday, June 9

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Jun 102013

The first Mathematical Crystallography minisymposium was entitled “Geometric Foundations”.

Frank Morgan Frank Morgan talked about Minimal Interface Structures.

Just as soap bubbles in 3-space assume the shape of minimal area enclosing the fixed volume of air they contain (subject to constraints), so a soap bubble in a 3-torus will assue such a shape of minimal area within that torus. Using that 3-torus, one kind generate periodic structures minimizing surface area (subject to constraints).
Egon Schulte Egon Schulte talked about Polyhedral Geometries and Symmetry.

Branko Grunbaum’s “skeletal” approach to polyhedra presents a polyhedron as a graph embedded in 3-space: vertices are vertices and edges are edges and a distinguished family of cyclic subgraphs (“rings” to chemists) are faces. This approach includes several infinite (periodic) polyhedra.
M. L. N. de las Penas Ma. Louise N. de las Penas talked about Nanostructures Arising from Crystallographic Tilings.

Given a graphene sheet and two appropriate vectors lying on its plane, one to define a unit and one to define a gluing identity, one can construct a graphene tube. Similarly, one may construct a torus out of the tube. Other manifolds may be constructed similarly, and in the ensuing discussion, it was proposed that algebraic and other mathematical criteria might determine when a graphene (or other layer) sheet may be shaped into a given manifold.
Bernd Souvignier Bernd Souvignier talked about Capturing the Essence of Infinite Graphs in Quotient Graphs.

Given a graph G, and a group of automorphisms A, one may define a quotient graph G/A whose vertices and edges are the orbits of the vertices and edges of G. Given sufficient information about how G/A articulates with G, one may go the reverse direction, constructing G out of G/A (and the articulation information).

Then we went to lunch at Reading Terminal Market.

Jun 102013

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.

Jun 092013

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:

Scene from fourth floor

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…

SIAM in Philadelphia

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Jun 082013

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:

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.