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).

**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…