Crystal Mathematician

Mapping an academic field is not something new. In fact, mapping academic fields is itself an academic field, largely associated with library and information science, but scattered about. So if one wants to map a field, there is an established way to do it.

Let’s take a look. Alan Porter & Scott Cunningham‘s Tech Mining is primarily about how to use large online databases to find out what academics and fellow travelers are publishing, patenting, or otherwise producing. I haven’t gotten my copy yet – the library only has an ebook, and these are difficult to deal with – but here are some observations.

1. First, Porter and Cunningham make it clear that their techniques are designed to survey a mature science, in the following sense. In 1962, Thomas Kuhn published a book, The Structure of Scientific Revolutions, in which he proposed that science does not develop smoothly. Instead, he claimed that a scientific field can develop somewhat steadily as practitioners explore what is at hand, under a consensus paradigm. That paradigm is the consensus of the basic facts, common perspective, and general agenda of what’s important. Every once and a while, a paradigm shift radically alters the basic facts, perspective and agenda of the practitioners as they replace an old paradigm with a new one; we are familiar with shifts like the revolution in quantum mechanics in the early Twentieth century, or of plate tectonics in the latter. We could consider the long steady periods as mature science.

   Another way of looking at this is via David Hilbert’s analogy of a science as a tree. Following my exploration of a similar metaphor, one could consider a scientific field as a vine or tree, slowly exploring the features of some edifice. Every once and a while, there is a dramatic rearrangement, but then things return to normal. What Porter and Cunningham outline is a methodology for mapping out the structure of this tree / vine, under the assumption that it is at some level of maturity.

2. Second, this is not a philosophy book; it is a book describing how to mine data. The data consists of records in vast and idiosyncratic databases like Thomson-Reuter’s somewhat snooty Web of Science (which includes the Science Citation Index), The Online Computer Library Center‘s World Cat (catalog) (which supposedly contains practically everything), and Google Scholar, among others. The book is about how to get meaningful data, which is the rub. Suppose, for example, you wanted to find the articles published last year on “Mathematical Crystallography.” Unfortunately, nothing requires scholars to classify their papers as “Mathematical Crystallography” if that is one of the primary or secondary subjects of the paper. Instead, one relies on authors, editors, and reviewers to insert keywords, codes, and review text that will somehow indicate whether or not this is a paper on “Mathematical Crystallography.” Consequently, when searching for scholarly activity in “Mathematical Crystallography,” there is the old problem of what statisticians call “false positives” (falsely identifying a book on crystal gazing as a work in “Mathematical Crystallography”) versus “false negatives” (falsely failing to recognize an article on symmetries of unbounded polytopes as a work in “Mathematical Crystallography). There is always a tradeoff between the two, and Porter and Cunningham provide techniques for minimizing both errors.
3. One strategy that should be useful in a (let’s be honest) scattered and marginal field like “Mathematical Crystallography” arises from Porter & Cunningham’s observation that “Most papers are rarely read; few are heavily cited (the most common number of citations to a paper is zero). A few papers and authors in any specialty are cited repeatedly. Those papers already cited become easier to find, and more attractive to scientists looking for key references. As a result ‘the rich get richer.’ This is known as ‘the Matthew Effect.” One starts with a few highly cited works in the subject, or a few works that seem central to the subject, look at the works that they cite or that cite them, and pick up the keywords and codes of these primary and secondary works, and then use them in searches.

I decided to look at some papers on this kind of thing, and thanks to some kindly assistance from Mike Grienesen at UC Davis, I decided to look at a few publications on nano-stuff. That is also the route that led me from logic to crystallography (which, you must admit, is a strange transition), and it happens to be all the rage. In fact, the papers I looked agreed that nano-stuff is all the rage. So here are a few papers, in chronological order, which should give us an idea of what this sort of investigation might consist of, and what sort of light it might shed.

• Plenty of room, plenty of history by Chris Toumey, Nature Nanotechnology 4 (2009), 783 – 784. Toumey explores the effect of Richard Feynman’s 1959 talk There’s Plenty of Room at the Bottom (which was followed up by a 1983 talk on Infinitesimal Machinery; see also Feynman on Tiny Machines on You-Tube). Legend has it that nano-stuff arose out of the first talk, but Toumey found only seven citations as of 1980 (although Toumey reports that nano-stuff-ologists recall talking about it at coffee shops; this brings us to the ugly subject of the reliability of memory, so moving right along…). But then the scanning (tunneling) microscope was invented in 1981, Eric Drexler published Engines of Creation in 1986, and in 1991, Don Eigler and Erhard Schweizer announced that they have written “IBM” in Xenon atoms on a nickel surface. Perhaps the moment that things happened was around 1980, not 1960. Toumey does not address the usual history of science question – whether Feynman was simply prescient in anticipating something that would be in the air decades down the road – but he does ask what, exactly, was the effect of Feynman’s speech.
• An empirical analysis of nanotechnology research domains by Nazrul Islam and Kumiko Miyazaki, Technovation 30:4 (2010), 229 – 237. “This paper attempts to answer the questions: (1) Which areas of nanotechnologies are currently state of the art and how mature are they? (2) How is the involvement of organizations, regions or countries in the development of nanotechnology knowledge? (3) Which areas of research are most important for specific types of organizations and for specific regions?” One technique described here is to define “research domains” (e.g. “nanomaterials”), which one then defines (e.g. ” Nanomaterials concern [the] control of the structure of materials at nanoscale with great potential to create a range of advanced materials with novel characteristics, functions and applications.” This paper inadvertently introduces a problem for researchers: a lot of this research into academic research involves expensive databases and software that your university might not subscribe to.
• Refining search terms for nanotechnology by Alan Porter, Jan Youtie, Philip Shapira & David Schoeneck, J. Nanopart. Res 10 (2008), 715 – 728. This paper concentrates on a “bootstrapping approach” for searches analogous to (3) above in Porter & Cunningham, which Porter et al describe as more effective than just tracking codes and keywords, but also more labor intensive. They outline a sort of sociological survey – complete with questionnaires sent to samples of the relevant population – for getting keywords that maximizes “recall” (i.e., minimizes false negatives) while maximizing “precision” (i.e., minimizes false positives). One problem with nano-stuff is the use of “nano-anything” as a keyword: it is a trendy keyword that increases funding prospects.
• How interdisciplinary is nanotechnology? by Alan Porter & Jan Youtie, J Nanopart Res (2009) 11:1023–1041. This article starts with a substantial literature review addressing the question: is nanoscience a field or a collection of somewhat disconnected “mono-disciplinary” fields? What I found most interesting in this article were two recommendations. One is that researchers in a multidisciplinary field learn how to conduct data mining in that field – if only to get some idea of the area they are in. The other recommendation is worth quoting in full:
  We suggest two additional paths to nurture crossdisciplinary research. First, to enhance understanding of findings in other disciplines, we encourage attention be given to the language used to present essential findings. Authors and editors should strive to assure that the essential findings of nano-relevant research are presented so as to be as accessible as possible to researchers from other disciplines. For instance, work presented in a materials science journal may well hold high value for a nano-bio researcher. Minimizing jargon and acronyms (and we know that we use them here!), and checking understandability by researchers from other disciplines, should reduce the barriers to nano research knowledge transfer.
  This advice should remind readers of Mike Zaworotko’s third challenge.
• Nanoscience and Nanotechnology: Evolving Definitions and Growing Footprint on the Scientific Landscape by Michael Grienesen & Minghua Zhang, Small 7:20 (2011), 2836 – 2839. Recalling that Islam & Miyazaki followed Socrates’ advice to first define your terms, Grienesen and Zhang discuss the difficulties in defining one’s terms, beginning with Porter et al’s bugaboo, the keyword “nano-*”. They developed a narrow query, a boolean combination of 270 terms beginning with “nano-“, and conducted a survey using Web of Science (snooty, remember). While the narrow query found 80 % of the “nano*” records for the year 2010, it found only 22 % of the “nano*” records for the year 1991, suggesting that it took a while for researchers to settle on common keywords. They also looked at the proportion of the nation’s entire scientific output that is falls under their narrow query. For 2010 in science, there were 475,745 records in the European Union, 404,226 records in the USA, and 131,742 in China. In nanoscience, using their query, 5.24 % of the European Union’s output was in nanoscience, 4.57 % of the USA’s output was in nanoscience, and 15.32 % of China’s (China was surpassed only by Singapore, at 16.41 %).

Of course, this only gives us an idea of what this kind of study looks like. There are several differences between an investigation of “nano-science / technology” and “mathematical crystallography”.

• Scale. Nano-stuff is a vast field – or vast array of fields – and it might be more accurate to compare nano-stuff with all of crystallography. Mathematical crystallography is merely the theoretical support for crystallographic design and analysis (and hence computation), with tendrils extending into mathematics, physics, chemistry, and other areas. Our project has a smaller, if comparably diffuse, subject.
• Activity. Nano-stuff is taking off. Meanwhile, at the SIAM Mathematical Aspects of Materials Science conference (below), only two of the eleven presentations in the Mathematical Crystallography minisymposia had an audience of over twenty. I strongly believe that mathematical crystallography is desperately needed for crystallographic design and analysis to achieve their goals, but at the moment, that need has not translated into economic demand.

But at least, this gives us an idea of how to proceed.

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 fold.it. 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…

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.

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:

• While the schedule of the XXIIIrd Congress of the International Union of Crystallography (in July, 2014) has been set, we may be able to set up a little something just outside the gates.
• SIAM has two possibilities, the SIAM Conference on Discrete Mathematics in June, 2014 and the much larger Annual Meeting in July, 2014. But the problem with SIAM minisymposia is that they are smaller: this year, we arranged for only twelve speakers, which is half as many as at last year’s AMS meeting, which suggests another possibility.
• The AMS has eight sectional meetings next year, during weekends during the academic year. We could organize a special session at one of them, and that would give us space for twenty speakers or so.
• We could try descending on an area besides mathematics. The American Chemical Society is having two national meetings next year as is the American Physical Society; we could find out how to crash their parties.

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…?

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

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 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.

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 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.