So what is ‘mathematical crystallography’? One place to look is in the prefaces, introductions, and descriptive comments of expository works. This is where the author tries to distill what is going on. So I went to the USF library’s somewhat Spartan mathematical crystallography shelf, and I checked what the authors had to say for themselves.

- In her
*Brief history of geometrical crystallography*(in J. Lima-de-Faria’s*Historical Atlas of Crystallography*), Marjorie Senechal wrote that “Geometrical crystallography includes the study of crystal form, the mathematical representation of crystal structure, and the relations between them.” Senechal was interested in crystallography from the neolithic to the present, but the definition appears fairly stable except for the notion of*crystal form*, which was originally the macroscopic shape of the crystal itself but now is the nanoscopic structure of the crystal. Many books take this view.- In their
*Crystal Symmetry: Theory of Colour Crystallography*, M. A. Jaswon and M. A. Rose wrote that “The central problem of mathematical crystallography is to determine the independent microscopic symmetries consistent with every macroscopic crystal symmetry.” - In her
*Foundations of Crystallography with Computer Applications*, Maureen Julian wrote that “Crystallography is the science of finding the locations of atoms in crystals,” and that “The goal of this book is to describe the tools used to interpret the x-ray reflections from a crystal.”

- In their
- The nanoscopic view of crystal structure goes back at least to Kepler, but the group theoretic machinery used to address this approach is less than two centuries old. Looking inside Harold Hilton’s Mathematical crystallography and the theory of groups of movements, we can see that a century ago, mathematical crystallography had become almost a branch of applied group theory. Jaswon and Rose were overstating the spectrum when they wrote that “Expositions of the theory vary in emphasis from the early geometric arguments of Hilton to the more recent algebraic apparatus of Schwarzenberger.”
- In their preface of
*Induced Representations in Crystals and Molecules: Point, space and nonrigid molecule groups*Simon Altman noted that there were already many books on group theory, and “I have chosen for this reason to construct a text around the theory and application of induced representations in finite groups….” - In their
*Similarity Submodules and Semigroups*(in Jiri Patera’s*Quasicrystals and Discrete Geometry*), Michael Baake and Robert Moody begin with “The symmetries of crystals or, more generally, of geometric objects with translational degrees of freedom, have been studied for ages and are well understood.” - In their
*Mathematical Crystallography: An Introduction to the Mathematical Foundations of Crystallography*, M. B. Boisen and G. V. Gibbs are even more specific: “This book is written with two goals in mind. The first is to derive the 32 crystallographic point groups, the 14 Bravais lattice groups, and the 230 space groups. The second is to develop the mathematical tools necessary for these derivations in such a manner as to lay the mathematical foundation needed to solve numerous basic problems in crystallography and to avoid extraneous discourses.” - Although Julian’s
*Foundations of Crystallography with Computer Applications*covers a range of subjects, she stressed the group theoretic aspects in her preface: “The material in the book is given in a logical order with the goal of understanding not only how atoms are arranged in crystals but also how crystal systems are related to each other. Examples of this are the point group and space group trees.” - In his
*Crystal Properties via Group Theory*, Arthur Nowick extended the reach of group theory: “This book deals with the effect of crystal symmetry in determining the tensor properties of crystals.” And in his*Tensors and group theory for the physical properties of crystals*, W. A. Wooster wrote that “While ‘classical’ crystallography may be called ‘static’, the group-theoretical treatment of the same symmetry may be called ‘dynamic’, since it brings vibrations of atoms and molecules into the range of what may be included.” - Our library actually doesn’t have a copy of Rolf Schwarzenberger’s
*N-dimensional crystallography*, but a kindly friend gave me a copy, and Schwarzenberger wrote that his book concerned “… the study of Euclidean groups in n-dimensional euclidean space.”

- In their preface of
- Some authors stress the geometric aspect of crystallography, with the algebra playing a supporting role.
- In her
*Quasicrystals and Geometry*, Marjorie Senechal wrote that “Crystal geometry is the resultant of chemical and physical forces and processes and no account of it that does not treat its dynamical aspects can be considered complete. But geometry – together with a generalized notion of symmetry – will probably continue to be the basis for crystal classification, and it is the classification problem with which this book is chiefly concerned.” And specifically, “… the book deals primarily with the problem of relating the geometry of discrete point sets to the diffraction spectra of functions associated to them, and with the emerging theory of aperiodic tilings.”

- In her

But there was one literalist. Mathematical crystallography is … the mathematics developed for crystallography. In his *Mathematical Techniques in Crystallography and Materials Science*, Edward Prince describes himself as a “practicing crystallographer” and “… I have used two criteria in choosing the material to be included in this book. Either they are things that I have had to learn, or look up frequently because I didn’t use them enough to retain the details in ready memory, or they are things that I have frequently explained to other colleagues.” The topics in the book: are matrices, symmetries of finite and infinitely repeating objects, vectors and tensors, data fitting, uncertainty estimates and statistical significance, data fitting in crystal structure determination, and the fast Fourier transform.

Prince gives us a reality check. Ever since Hilton’s book, mathematical crystallography has been about the structures embedded in spaces acted upon by the crystallographic groups. That is, after all, what is in QD 911 – QD 919 of the Library of Congress, and that is how this blog answered the question, What is Mathematical Crystallography? But following the famously circular definition that “mathematics is what mathematicians do” (the sentiment is due to David Hilbert), we should ask what mathematics is being done for, by, and in the name of, crystallography. The answer turns out to be very revealing.

If we look at the Mathematical Subject Classification, we can see that the string “crystal” occurs (occasionally as part of other words) eight times. Here is where it occurs:

Field | Subfield | sub-subfield |

14 Algebraic Geometry | 14F (Co)homology theory | 14F30 p-adic cohomology,crystalline cohomology |

20 Group Theory and Generalizations | 20H Other groups of matrices | 20H15 Other geometric groups, including crystallographic groups |

52 Convex and Discrete Geometry | 52C Discrete geometry | 52C23 Quasicrystals, aperiodic tilings |

74 Mechanics of Deformable Solids | 74E Material properties given special treatment, especially anisotropy and crystalline structure |
74E15 Crystalline structure |

74 Mechanics of Deformable Solids | 74N Phase transformations in solids |
74N05 Crystals |

76 Fluid Mechanics | 76A Foundations, constitutive equations, rheology |
76A15 Liquid crystals |

82 Statistical Mechanics, Structure of Matter |
82D Applications to specific types of physical systems |
82D30 Random media, disordered materials (including liquid crystals and spin glasses) |

These are not subfields of crystallography. These are those sub-subfields of mathematics whose development were influenced or inspired by crystallography, especially (in some cases) by the crystallographic groups. Some of these sub-subfields have substantially diverged from crystallography, so we want to take a closer look. Perhaps the most parsimonious first approximation is to look at the works classified under 74E15 Crystalline structure, 74N05 Crystals, and 82D25 Crystals.

Because of its data mining features (more on these in some future post), we will use zbMATH to get a picture of the literature. zbMATH’s website says that it covers “over 3,000” journals. The way that zbMATH works, they receive journal articles and keywords and MSC codes are somehow tentatively assigned to them; many articles are then sent to reviewers, who write micro-reviews, complete with keywords and MSC codes, which are then posted online. This takes time, so it is not clear how up-to-date zbMATH is.

On August 23, I queried zbMATH for all items classified under 74E15, 74N05, and 82D25. There were 4,486 items listed. Since zbMATH lists the number of items published each year, we can get a picture of how publication rate has changed through time:

Compare this with the publication dates of books in the QD 911 section in the Library of Congress (as of August 24):

During the same period that publication of items in 74E15, 74N05, and 82D25 were going up, publication in QD 911 was going down. This suggests that these are not quite the same groups of researchers – and not quite the same field of research. But zbMATH’s data mining tools permit a closer look.

zbMATH assigns multiple codes to publications, which allows us to look at publications with codes 74E15, 74N05, and 82D25, and determining what other codes were assigned to those publications. Among these publications, the codes (besides 74 and 82) appearing at least one hundred times were:

Field | Number of items | Percentage |

20 Group theory and generalizations | 493 | 11 % |

35 Partial differential equations | 367 | 8.2 % |

52 Convex and discrete geometry | 342 | 7.6 % |

81Quantum theory | 336 | 7.5 % |

65 Numerical analysis | 210 | 4.9 % |

78 Optics, electromagnetic theory | 179 | 4 % |

80 Classical thermodynamics, heat transfer | 168 | 3.7 % |

51 Geometry | 137 | 3.1 % |

It turns out that in many of these fields, many or most of the mathematical crystallography publications intersecting that field actually intersect one particular sub-subfield. Here are the concentrations of mathematical crystallography publications in each of these eight fields:

Subfield (and sub-subfield, if there is a concentration) | Number of items in subfield or sub-subfield | Percentage of the field |

20H15 Other groups of matrices > Other geometric groups, including crystallographic groups | 451 | 91.5 % |

35Q Equations of mathematical physics and other areas of application | 194 | 52.8 % |

52C23 Discrete geometry > Quasicrystals, aperiodic tilings | 119 | 34.8 % |

81Q General mathematical topics and methods in quantum theory and 81V Applications to specific physical systems | 273 | 81.3 % |

65C Probabilistic methods, simulation and stochastic differential equations and 65M Partial differential equations, initial value and time-dependent initial-boundary value problems and 65N Partial differential equations, boundary value problems and 65Z Applications to physics | 142 | 67.6 % |

78A General | 174 | 97.2 % |

80A Thermodynamics and heat transfer > 80A22 Stefan problems, phase changes, etc. and 80A23 Inverse problems | 137 | 81.5 % |

51F15 Metric geometry > Reflection groups, reflection geometries and 51M20 Real and complex geometry > Polyhedra and polytopes; regular figures, division of spaces | 113 | 82.5 |

If we looked at *where* these articles were published, we find that that the journals with the highest number of mathematical crystallography publications were physics journals or theoretical crystallography journals. Here is a table of the journals that published the most articles in the sub-subfields 74E15, 74N05, and 82D25, with the number of articles published in each field: we list the top five journals in the three sub-subfields, and for each field, the top three journals in that field:

Journal | All | 20 | 35 | 51 | 52 | 65 | 78 | 80 | 81 |

Acta Crystallogr., Sect. A | 165 | 88 | 10 | 34 | |||||

Archives of Mechanics | |||||||||

Dokl. Akad. Nauk, Ross. Akad. Nauk | 10 | ||||||||

Int. J. Solids Struct. | 167 | ||||||||

International J. of Heat and Mass Transfer | 6 | ||||||||

Int. J. Mod. Phys. A | 10 | ||||||||

International J. of Plasticity | 218 | ||||||||

J. Math. Phys. | 10 | ||||||||

J. of Computational Physics | 19 | 27 | 8 | ||||||

J. of Mechanics and Physics of Solids | 286 | ||||||||

J. of Physics A: Math. and Theoretical (formerly J. of Physics A: Math. and General) | 224 | 40 | 17 | 7 | 59 | 9 | 51 | ||

Math. Models & Methods in App. Sci. | 7 | ||||||||

Modern Physics Letters B | 8 | ||||||||

Physica D | 16 | 6 | 41 | ||||||

Physics Letters. A | 10 | 27 | |||||||

Zeitschrift für Kristallographie | 23 | 7 | 19 |

No chemistry journals. It is as if mathematical crystallography is, as far as the journals are concerned, a branch of physics.

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