Jul 312013

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.

Call # Official description What I saw
QD 911 General works 234 items on geometric crystallography (including aperiodicity), crystal analysis, supporting theory and science and math, tables, methods and software
QD 912 Fundamental systems; including tetragonal, orthorhombic, monoclinic systems One book on “Direct synthesis of silanes, chlorosilanes, methylchlorosilanes …” published in 1971
QD 913 Diagrams Sixteen books of tables and diagrams, mostly German, all published between 1866 and 1973, inclusive
QD 914 No description No items; the Library of Congress leaves blank spaces to give the system room to grow
QD 915 Goniometric Measurements Four items published between 1825 and 1934, inclusive
QD 916, 917, 918 No descriptions No items
QD 919 Statistical methods Uri Shmueli & George Weiss’s Introduction to crystallographic statistics

And that’s it.

Meanwhile, the general breakdown for Crystallography is:

Call # Number of items Overview
QD 901 – 908 676 Periodicals, anthologies, jargon, history & biography, early works, general works and texts, pictures etc., juvenilia, essays & lectures, special topics, lab stuff, handbooks
QD 911 – 919 256 Geometrical and mathematical crystallography
QD 921 – 926 1,203 Crystal structure and growth (921: General works; 923: Liquid crystals; 924: Photonic crystals; 925: Polycrystals; 925: Quasicrystals)
QD 931 – 947 846 Physical properties of crystals (931: General works; 933: Mechanical properties; 937: Thermal properties; 939: Electric properties; 940: Electric properties; 940: Magnetic properties; 941: Optical properties; 945: X-ray crystallography; 947: Other physical properties
QD 951 103 Chemical crystallography (including the four items in QD 954?)
QD 999 2 “Miscellany and curiosa”

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).

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:
MathSciNet on ACA and ZK
Meanwhile, zbMATH lists 209 publications from Acta Cryst. A, which it no longer covers:
and 238 from Z. Krist., which zbMATH no longer covers:
zbMATH on ZK
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…

  One Response to “Mapping the Community IV: What, if anything, is mathematical crystallography?”

  1. I have been informed that ZBL actually covers Acta Crystallographica, albeit sparingly, and that they are reviewing their coverage of that journal.

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