Crystal Mathematician

I took the title of this post from Stephen Jay Gould’s What, if anything, is a zebra?, who in turn got it from Albert Wood’s What, if anything, is a rabbit? Since then, the web has asked similar questions about declarative languages, a mitochondrial Eve, a tilapia, Big Bird, a museum, Mahayana Buddhism, Byzantines, spinal shock, a wolf, Australopithecus sediba (this from one of this blog’s models), brackish-water fauna, and, of course, a bicyclist. So it’s only natural to ask this question of mathematical crystallography.

What, if anything, is mathematical crystallography? The field must exist: Google gave me “About 6,420 results” for “mathematical crystallography”. Nevertheless, Merriam-Webster doesn’t have the entry, Wikipedia redirects visitors to the “Crystallography” page, and Britannica offers me a platter of 885 alternative options – starting with the Swiss mineralogist Paul Niggli, who, Britannica informs us, “originated the idea of a systematic deduction of the space group (one of 230 possible three-dimensional patterns) of crystals by means of X-ray data and supplied a complete outline of methods that have since been used to determine the space groups.” Britannica’s reaction was typical: mathematical crystallography has something to do with mathematics and with crystallography, and the crystallographic groups are somehow in the middle of it.

I tend to look for books. I have a problem: I buy them far more rapidly than I read them, and as a result my library looks like something out of Dickens. But a book is a major investment to write or to read, so it can tell, better than articles (do you hear me, Thomson-Reuters?), what people value. So I went looking up books on “mathematical crystallography.”

The Library of Congress does not post much information about the subject headings of its call numbers online – this taxpayer-supported institution will sell you the information in books – but the USF Library has the books. Anyway, here are the call numbers, the official descriptions, and what I saw in the Library of Congress under those call numbers (by looking in its online catalog) for “Geometric and mathematical crystallographers. As one of our favorite Floridians, Dave Barry is prone to say, I am not making this up.

And that’s it.

Meanwhile, the general breakdown for Crystallography is:

And of course there is stuff in other parts of chemistry (QD), in physics (QC), in mathematics (QA), and even stranger places.

My university has repeatedly failed to get a Phi Beta Kappa chapter, and the two favorite reasons that that august society has for rejecting the University of South Florida are faculty-student ratios and the library. So of course, the USF Library does not have 234 books under the subject code QD 911, but we can look at the biased sample that is there.

The book that seems to come up at the top of the page is Harold Hilton’s Mathematical Crystallography, published by Clarendon in 1903. Perhaps this can give us an idea. In his preface, Hilton wrote, “the object of this book is to collect for the use of English readers those results of the mathematical theory of crystallography which are not yet proved in the modern textbooks on that subject in the English language.” Notice the use of the word “proved,” for the text omits “all practical applications” as “these are explained at length in the existing English textbooks.”

He goes on to write, “the theorems proved are in most cases to be found in the works of Schoenflies, Liebisch, &c.” and includes “a fairly full account of the geometrical theory of crystal-structure which the labours of Bravais, Jordan, Sohncke, Federow, Schoenflies and Barlow have now completed.” He says that much of his book is a shortened version of Schoenflies’ Krystallsysteme und Krystallstructur (Crystal Systems and Crystal Structure). And as a reflection of how times have changed, he writes that “I have not attempted to give a complete list of references …” And if you flip through the book, you find it is in two parts: point groups and space groups. This book is not a comprehensive or even focused introduction to a subject; for example, there is no mention of the crystallographic restriction. It looks like what it was advertised as: a shortened, cleaned, and accessible version of Schoenflies’ book – which, incidentally, USF does not have (although USF does have a copy of Federov’s book).

My three sources were: what was on the QD 911 shelf in the USF Library, what came up under “mathematical crystallography” (in quotes) in the USF Library, and what came up under “mathematical crystallography” (in quotes) on Amazon (the library had 199 hits and Amazon had 52, but few of these were what I wanted).

• Geometric crystallography seems to underlie this entire section.
  • Martin Buerger’s Introduction to crystal geometry is a warm, fuzzy, and gentle introduction to geometric crystallography. Maurice Jaswon’s An introduction to mathematical crystallography concerns crystal systems and structures, and the crystallographic groups.
  • And there are more theoretical works. Monte Boisen and Gerald Gibbs’ Mathematical crystallography: an introduction to the mathematical foundations of crystallography is a very detailed investigation of the lattices and the crystallographic groups using matrix representations. Peter Engel’s Geometric Crystallography : an Axiomatic Introduction to Crystallography covers the point groups, the space groups, and lattices.
  • David Hilbert’s Eighteenth Problem made crystallographic groups of higher dimensions into a priority for mathematicians. Harold Brown, Rolf Bulow, Joachim Neubuser, Hans Wondratscheck and Hans Zassenhaus’s Crystallographic groups of four-dimensional space is a catalog of the these groups, and Eric Whittaker’s An Atlas of Hyperstereograms of the Four-dimensional Crystal Classes has pictures of the four-dimensional crystal classes.
  • Adding combinatorics is a recent innovation. Michael O’Keeffe and Bruce Hyde’s Crystal Structures I : Patterns and Symmetry uses the point and space groups on polyhedral structures, including tilings and lattices. Toshikazu Sunada’s Topological Crystallography : With a View Towards Discrete Geometric Analysis applies some of the machinery of algebraic topology to (combinatorial) graph representations of crystals. Alexander Wells’ Three dimensional nets and polyhedra is a catalog with pictures and tables (and looks sort of like an application of his The third dimension in chemistry is an overview of polygons, polyhedra, and patterns, leading into geometric crystallography).
  • Hilbert’s question, whether there were finitely or infinitely many crystallographic groups for any given dimension, was answered by Ludwig Bieberbach. Since then his theory has been developed into a branch of Riemannian geometry, and turning to the QA (mathematics) section of the library we see books like Rolf Schwarzenberger’s N-dimensional crystallography (call number QD 911), and Leonard Charlap’s Bieberbach groups and flat manifolds and Andrzej Szczepanski’s Geometry of Crystallographic Groups (the former having call number QA 333 and the latter not listed in the Library of Congress (!).
• … and Crystal Structure Analysis is the great Twentieth Century application of geometric crystallography (and Fourier analysis).
  • Carmelo Giacovazzo edited the IUCr text / anthology Fundamentals of Crystallography, which has two chapters on crystal structures, three on crystal structure analysis, and then four on ionic crystals, molecular crystals, proteins, and physical properties. Maureen Julian’s Foundations of Crystallography with Computer Applications is a sort of cookbook for mathematical crystallography. Julian starts with crystal structures and ends with Fourier analysis, but she doesn’t go into the scary mathematics or the icky nitty-gritty. Edward Prince’s Mathematical techniques in crystallography and materials science concerns matrix and tensor techniques on symmetries, and uncertainty estimation and statistical inference on data generated by diffraction. John Rollett’s Computing methods in crystallography is essentially that part of advanced “applied” mathematics one would want to use in algorithms for crystallography programs.
  • Then there are more theoretical works. Herbert Hauptman’s Crystal Structure Determination : The Role of the Cosine Seminvariants is one take on using Fourier analysis to determine crystal structure. Aleksandr Kitaigorodskii’s Theory of Crystal Structure Analysis is largely about the Fourier transform methods used to derive a crystal structure from its diffraction pattern.
• But also … physical properties using Albert Einstein’s favorite toy, the tensor. Arthur Nowick’s Crystal Properties Via Group Theory applies tensors and tensor (matrix) group representations to understand physical properties. William Wooster’s Tensors and group theory for the physical properties of crystals applies tensors to teh analysis of stress, strain, prezoelectricity and elasticity, and then group theory to waves, vibrations, and physical constants.

Of course, this list omits more topical books like Jawson & Rose’s Crystal Symmetry : Theory of Colour Crystallography, which is on the color groups, and Marjorie Senechal’s Quasicrystals and Geometry, which is largely about quasicrystals, and heavy on the geometry of tilings. It also omits foundational books like Paul Yale’s Geometry and Symmetry, which is about the geometric foundations of much of the crystallography discussed in these books, and Grunbaum and Shephard’s Tilings and Patterns, which goes into mathematical details that most other books skip.

It appears that Hilton’s concentration on geometric crystallography still holds – one could look at Massimo Nespolo’s recent manifesto, Does mathematical crystallography still have a role in the XXI century? and see geometric crystallography, complete with the recent combinatorial tendencies. But the Fourier analysis (and statistics) for crystal structure determination and the tensor algebra (and analysis) for crystal properties have a noticeable presence. Even so, crystal structure lies at the center of it all.

Time to turn to the Mathematical Subject Classifications. There are about 6,000 areas listed, which may be misleading considering the old metaphor (which I cannot trace) of mathematics being like a tuber: no area is that far from any other. But the areas that would seem to be strongly related to the mathematical crystallography above would seem to be:

• 05 Combinatorics
  • 05B Designs and configurations, especially matroids & geometric lattices, packing & covering, tessellation and tiling problems.
  • 05C Graph theory, especially graphs and abstract algebra, and infinite graphs.
• 15 Linear and multilinear algebra; matrix theory
  • 15A Basic linear algebra, especially eigenvalues, singular values, eigenvectors & quadratic and bilinear forms, inner products & multilinear algebra, tensor products & vector and tensor algebra, theory of invariants.
• 20 Group theory and generalizations
  • 20F Special aspects of infinite or finite groups, especially word problems and other decision problems, connections to logic and automata & reflection and coxeter groups & geometric group theory
  • 20G Linear algebraic groups and related topics, especially cohomology theory and applications to physics
  • 20H Other geometric groups of matrices, especially crystallographic groups.
• 22 Topological groups; Lie groups
  • 22E Lie groups, especially discrete subgroups
• 42 Harmonic analysis on Euclidean spaces
  • 42B Harmonic analysis in one variable, especially Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
• 51 Geometry
  • 51N Analytic and descriptive geometry, especially Euclidean analytic geometry
• 52 Convex and discrete geometry
  • 52B Polytopes and polyhedra!!
  • 52C Discrete geometry, which includes lattices, packings, tilings, and rigidity.
• 55 Algebraic topology
• 57 Manifolds and cell complexes
  • Low-dimensional topology, especially geometric structures on low-dimensional manifolds
• 74 Mechanics of deformable solids
  • 74E Material properties given special treatment, especially anisotropy and crystalline structure
  • 74H Dynamical problems, especially vibrations
  • 74J Waves
  • 74N Phase transformations in solids, especially crystals
• 82 Statistical mechanics, structure of matter
  • Applications to specific types of physical systems, especially crystals

And that doesn’t count field equations.

And now for a reality check. I have been told that the two natural homes for mathematical crystallography are the IUCr’s Acta Crystallographica A : Foundations of Crystallography and Oldenbourg-Verlag’s Zeitschrift fur Kristallographie : Crystalline Materials. So how much coverage do these two journals get from the two big databases, the American Mathematical Society’s MathSciNet (covering 2,729 journals as of July 24) and Zentralblatt fur Mathematik’s zbMATH (covering 5,827 journals as of July 24)?

As of July 31, MathSciNet has records for 430 articles in Acta Cryst. A, and 110 for Z. Krist, which it no longer covers. Here is a picture:

Meanwhile, zbMATH lists 209 publications from Acta Cryst. A, which it no longer covers:

and 238 from Z. Krist., which zbMATH no longer covers:

Evidently some warm and fuzzy feeling that these two services had towards mathematical crystallography in the 1970s has recently diminished.

It would be interesting to know what happened…

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.