The 23rd Congress and General Assembly of the IUCr this August

 Miscellany  Comments Off on The 23rd Congress and General Assembly of the IUCr this August
Jun 012014
 


Every three years, the International Union of Crystallography holds a Congress and General Assembly. The first was in 1948 at Harvard University in Cambridge, Massachusetts, and the 23rd will be held this summer from August 5 to August 12 in Montreal, Canada.


The entire meeting looks rather large – at the 2011 meeting in Madrid, there were 2,655 “scientific participants” from 73 countries, resulting in 490 oral presentations and 1,550 posters – and this meeting will probably be comparable.

  • Registration opens on August 4.
  • The workshops are on August 5, and the Ewald Prize will be awarded to Aloysio Janner and Ted Janssen for their work on aperiodic crystals.
  • The keynote and plenary speeches, microsymposia, software fayres, poster sessions, and various commissions will be held from August 6 to August 12.
  • The Gjønnes Medal in Electron Crystallography will be awarded to John Steeds and Michiyoshi Tanaka on August 12.

Here are some events that may be of interest to the mathematically inclined:

  • As of May 31, there are four plenary lectures listed.
  • As of May 31, I counted 31 keynote speeches, not counting the two Gøonnes Prize lectures. A few heads-up for the mathematically inclined or the crystal engineer:
    • Crystal Engineering and Applications of Functional Metal-Organic Frameworks by Xiao-Ming Chen on August 7.
    • Mathematical Crystallography in the 21st Century by Marjorie Senechal on August 8.
  • Workshops consist of sessions of extended presentations by experts, and cost $ 25 to $ 60 … and space is limited, so make reservations asap. They are all being held on Tuesday, August 5. These seem to be largely on software.
  • There are several ancillary “commission meetings” during the conference, including an Open (i.e., everybody is invited) Commission Meeting on Mathematical and Theoretical Crystallography, which is scheduled to meet on August 11 during lunchtime – 12:15 – 13:45 on the fifth floor in room 441. Come and bring a colleague – or even better, a student!
  • Recent developments are presented in 112 microsymposia and in poster sessions. Here are some microsymposia whose descriptions suggest substantial mathematical content or opportunities.
    • 2. Recent Advances in Quasicrystal Research, organized by An Pang Tsai and Janusz Wolny, on August 6.
    • 31. In-situ XRD: Parametric and Symmetry Constrained Refinement, organized by Robert Dinnebier and John Evans, on August 7.
    • 33. Symmetry Constraints in Magnetic Structure Determination: Experiment and Theory, organized by Branton Campbell and Mois Ilia Aroyo, on August 8.
    • 34. Crystals and Beyond, organized by S. I. Ben-Abraham and Jeong-Yup Lee, on August 8.
    • 62. Symmetry and Isomorphism in Material Design and Crystal Growth, organized by Tatyana Bekker and Antoni Dabkowski, on August 9.
    • 72. Methods, Algorithms and Software for Powder Diffraction, organized by Ryoko Oishi-Tomiyasu and Jon Wright, on August 10.
    • 95. Symmetry and its Generalisations in Science and Art, organized by M.A. Louise de la Penas and Emil Makovicky, on August 11.
    • 96. New Computational Approaches to Structure Solution and Refinement, organized by Richard Cooper and Lukáš Palatinus, on August 11.
    • 104. Crystal Structure Prediction and Materials Design, organized by Roman Martonak and Tian Cui, on August 12.
    • 112. New Approaches to Crystal Structure Prediction, organized by Graeme Day, on August 12.

    Of course, in addition to the microsymposia, there will be approximately five million posters, including mine. Drop by and browse.

  • There will also be a Software Fayre where software developers can demonstrate their software. While mathematical crystallography may influence crystallography in developing theory, it seems likely that the most impact will be in software.


About Montreal…


Montreal is an island in the St. Lawrence River with a population of about two million people. It was inhabited as early as 4,000 years ago, but during the Sixteenth century, all the people disappeared (!). The French started settling the place in the Seventeenth century, but it was surrendered to the British in 1760. It lies in the province of Quebec, and is now the second largest city in Canada.


Major academies in Montreal include Concordia University, Université Laval, McGill University, the Université de Montréal (including its affiliate, the École Polytechnique de Montréal), the Université du Québec à Montréal (the Ecole de technologie superieure) is affiliated with the university), and the Université de Sherbrooke.


The deadline for reserving accommodations is “[u]ntil blocks are sold out or June 25, 2014, whichever comes first,” and since this is a tourist season, blocks may not last until June 25 (and affordable airplane tickets may be difficult to find).


Summers are, according to Wikipedia, hot and humid, with temperatures ranging from 63 F to 77 F (17 C to 25 C), with mean temperature 70 F (21 C), 4 inches (100 mm) of rain a month, and relative humidity of 79 %. In general, sunny with occasional storms. Despite last year’s story on Montreal in July, High heat and humidity warnings bring serious health threat, to a Florida resident like me it sounds comparatively pleasant.


For future reference …


The 24th Congress and General Assembly will be held in 2017 in Hyderabad, India, capitol of Andhra Pradesh, with a population of nearly eight million people, and home to several academic institutions, including the Birla Institute of Technology and Science, Pilani – Hyderabad, the English and Foreign Languages University, the University of Hyderabad, the Indian Institute of Technology Hyderabad, the Jawaharlal Nehru Technological University of Hyderabad, and Osmania University.

Mapping the Community VIII: Infrastructure

 Miscellany  Comments Off on Mapping the Community VIII: Infrastructure
May 072014
 


I am starting up this blog roughly where I left off. I am still working on material that showed up in the paper, Prospects for Mathematical Crystallography. Despite the fact that the paper discusses infrastructure and access, I think I will go ahead and post on those two subjects anyway: they are important for developing an emerging or re-emerging field. I will then move on to crystal prediction, which I just started in the paper, but which I think should be reviewed in detail, primarily because that is my own field.


Science and mathematics history books tend to devote little attention to the infrastructure of mathematics and science. The traditional view is that what is important are the great discoveries, and the great discoverers who made them. Lately, science and mathematics historians have grown more interested in the process of discovery, and that includes a close look at how discoveries take place. For some of the natural sciences, this entails obvious infrastructure. For example, even before the invention of the telescope, astronomers used an array of observational devices, which cost money. These devices ranged from little astrolabes to immense sextants, not to mention observational towers. And while the telescope put a relatively cheap instrument into the hands of the masses, Isaac Newton’s reflecting telescope was an omen of the incredibly expensive telescopes to come.


Even mathematics could create a demand for this kind of infrastructure: just about everybody nowadays uses computers for something.


But there was a need for another kind of infrastructure. Philosophers and scientists, as well as artists and writers, want to go where things are happening because that’s where people who want to do things are going. In Medieval Europe, that tended to be the big monasteries; now, it’s the big universities and big corporations, which are concentrated in cultural centers.

  • Scientists and engineers tend to be employees of institutions, especially corporations and educational institutions. Scientists and engineers not only need the expensive equipment required by modern science – and engineers have always needed expensive equipment, ever since Imhotep built the Djoser pyramid – but they also need libraries and colleagues. Also, many scientists and engineers support themselves by marketing their results, which means publishers, often integrated with academic institutions.
  • Artists, writers, and philosophers often view themselves as freelancers, but considered as an economic activity, most visible art these days (on products from book covers to billboards) is commercial art, and much of the text read these days (on products from greeting cards to computer manuals) is commercial text. Poetry is a multi-billion dollar industry: just turn on the radio (most rock stars are independent contractors associated with a small number of record labels), and philosophers follow ancient traditions when they hire themselves out to publicity machines or join senior management. Some artists (e.g. sculptors and architects) need materials, and all need to interact with colleagues. And all need someone to market their works.

All fields need infrastructure to build and maintain lines of communication and collaboration. Recalling from the 17 August 2013 post that recruitment is critical for new fields, the infrastructure must make a field attractive and accessible. That means introductory books and expository articles as well as workshops and tutorials, not to mention special topics courses and course modules. And in the Twenty-first century, software.


Beyond the colleges and universities themselves, there are several academic organizations with a particular interest in crystallography. These organizations can provide forums for events like workshops and tutorials, as well as assist in finding resources for developing and / or marketing books, articles, course materials, and software. Some of them also provide mechanisms for people seeking jobs (always a concern). Such organizations include:

There are many other organizations, most notably the American Association for the Advancement of Science, which publishes one of the two likely most prestigious science journals in the world, Science (the other being Nature).


The scientific community is huge, and just reading journals and attending meetings can create a narrow view of what is going on in one’s own subject. Sometimes one needs a broader perspective of one’s own field. And occasionally, when working on a project, it is more convenient to use someone else’s invention rather than reinventing the wheel. For these and other reasons, it is often convenient to have a database of publications at hand. All these databases have quirks. These databases are expensive to maintain, so some are available only by subscription (and occasionally very expensive subscription at that). If you go to the public links of some of these databases, you may find only limited or no functionality for non-subscribers. Anyway, here are some of the major ones:

Starting up Again

 Miscellany  Comments Off on Starting up Again
Feb 262014
 

Probably the worst thing that can happen to a blog is that it stops. Blogs are supposed to move forward. So I apologize for the hiatus; things happened.
Two things:

  • The idea for mapping this community came from writing an article on the mathematical crystallography community. This is one article of a “virtual issue” of Acta Crystallographic A co-edited by Massimo Nespolo and myself. This virtual issue will be distributed over several issues, starting with the March 2014 issue. My paper on Prospects for Mathematical Crystallography has just appeared, and it can be regarded as a distillation of the blog posts thus far.
  • I am starting up the blog again. It will not move as rapidly as earlier last year, but it should keep moving. I will continue the project of mapping the mathematical crystallography community, starting with crystal prediction – which was featured in the article as one area where mathematical crystallography had considerable potential.

I apologize for the hiatus.

Mapping the Community VII: Mathematical Crystallography versus Mathematics in Crystallography

 Miscellany  Comments Off on Mapping the Community VII: Mathematical Crystallography versus Mathematics in Crystallography
Dec 132013
 


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.”
  • 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.”
  • 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.”

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:

zbMATH

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

QD 911

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.

Comments Shut Down … For Now

 Miscellany  Comments Off on Comments Shut Down … For Now
Oct 122013
 

Very few people post comments on this blog, but I have been getting about 40 spams, pings, and other nonsense daily. I just came back from a 2-day trip and found 616 posts awaiting moderation.

Until I figure out how to control the spam, I am shutting down comments. Many apologies for the inconvenience.

More about open access journals

 Miscellany  Comments Off on More about open access journals
Oct 042013
 

I’m afraid that because of a lot of recent business I have to attend to, I haven’t kept up with the Mapping the Community project. I will return to it asap. Meanwhile …


The latest kerfluffle in the journal biz is an expose by John Bohannon in Science magazine of open access journals. Bohannon composed a spurious manuscript with obvious flaws, sent it to 304 open access journals, and by the time Science had gone to press, 157 had accepted it, 98 had rejected, 20 were still working on it, and … 29 of the journals seemed not to exist, consisting only of zombie websites receiving papers. Among those 255 journals that had acted, acceptance came on average in 40 days, while rejection took 24.


The “open access” model means that anyone can go online and get a posted article; the awkward question is how an open access journal gets funded. Some are supported by foundations or universities, but many are vanity presses, charging authors for printing charges. Since the rather extravagant subscription costs of many journals is a hot topic these days, open access has many defenders. And some of them rose to defend open access against Bohannon.


Inside Higher Ed (the big open access higher education newspaper) reported that many open access journals haven’t established their editorial processes and that is how they got stung. Meanwhile, The Chronicle of Higher Education (full access to subscribers only), reported that defenders of open access noted that Science is a subscription journal and therefore presumably suspect, and besides the same problem occurs in print journals.


There have been complaints like this about journals for eons. I remember in the 1980s, a mathematician complaining about what he called “write-only journals.” But if the problem is getting worse, there are two things to consider.

  • If an open access journal is supported by an institution that pays the bills, then that institution has an interest in quality control for its own reputation. But if it relies on page charges, submission fees, or publication charges, then its paying customers are its authors. If it is to survive, it has to have a sufficient number of happy authors. On the other hand, a subscription journal survives by keeping subscribers (readers and libraries) happy, and they are happy only if the journal seems to be worth the money. So we would expect that an open access journal that acted as a vanity press would have the same kind of quality problems that the old vanity press had. Meanwhile, journals that lived off subscriptions would be more careful.
  • Notice that acceptance took an average six weeks while rejection took less than four. Shortening the reviewing time is all the rage these days, and complaints about the difficulty of finding reviewers are more common. It would be interesting to know what the correlation between reviewing time and quality control are.

At any rate, it looks as if any journal that promises to rush your article into print in return for a fee should be regarded with suspicion.

Mapping the Community VI: The Better Mousetrap

 Miscellany  Comments Off on Mapping the Community VI: The Better Mousetrap
Aug 172013
 


Many of us in mathematical crystallography, or theoretical crystallography, or the foundations of crystallography, or whatever you want to call it, have the impression that this is a field, that it is emerging (or re-emerging), and perhaps it would be a good thing if this field became more active and popular.


But what is entailed by an emerging field? And how would we encourage its growth? It turns out that there are experts on this kind of thing: the sociologists and philosophers of science. Let’s begin at the beginning (philosophers, like mathematicians, like to define their terms): what do we mean by a field? This is not a silly question: in fact, I claim that it is central to what we want to do.


Lindley Darden wrote that a field as an area of science with “a central problem, a domain of items taken to be facts related to that problem, general explanatory factors and goals providing expectations as to how the problem is to be solved, techniques and methods, and concepts, laws and theories related to the problem which attempt to realize the explanatory goals.” A field is about a problem. Darden outlined the emergence of cytology and biochemistry, and claimed that what they had in common was the discovery of something new, the introduction of new techniques for studying it, and the successful explanation of the new thing.


This is the underlying economic reality: people are not interested in we are interested in; they are interested in what they are interested in. They have their own agenda. Getting involved in a new field – even paying attention to a new field – is a lot of work. Why should they do that? An emerging (or re-emerging) field will gain attention and support if it provides something useful. Here is Ralph Waldo Emerson’s famous mousetrap quote (the original quote didn’t mention mousetraps):


… If a man has good corn, or wood, or boards, or pigs, to sell, or can make better chairs or knives, crucibles or church or gans, than anybody else, you will find a broad hard-beaten road to his house, though it be in the woods. And if a man knows the law, people find it out, though he live in a pine shanty, and resort to him. And if a man can pipe or sing, so as to wrap the prisoned soul in an elysium; or can paint landscape, and convey into oils and ochres all the enchantments of Spring or Autumn; or can liberate or intoxicate all people who hear him with delicious songs and verses; tis certain that the secret cannot be kept: the first witness tells it to a second, and men go by fives and tens and fifties to his door. …

It may not be clear that there is something entirely new. For example, the notion of a group took several decades in the late Eighteenth century and early Nineteenth century to emerge: mathematicians from Leonhard Euler (in 1761) to Evariste Galois (in 1831) toyed with groups of permutations, but it was Galois who launched the word group and only later, in 1846, that Augustin Cauchy announced that for over eighty years, mathematicians had been exploring (drum-roll, please), groups of permutations (see Hans Wussing’s The Genesis of the Abstract Group Concept for details). And through those eighty years, and ever since, groups held mathematicians’ attention because they were helpful.


Here is an example I have been foisting on passersby. How did a growing knowledge of physics, and a growing sophistication in engineering drawing, transform architecture? There are many ways, and here is one. Before the Renaissance, domes were either shallow or small or buttressed.

Image posted on Wikimedia Commons by Anthony M. Image posted on Wikimedia Commons by idobi Image posted on Wikimedia Commons by Milos Radevic

Three famous pre-Renaissance domes. The Pantheon (left) is shallow, the Dome of the Rock (center) is a small wood dome, and the Hagia Sophia (right) is intermediate- sized and buttressed on all sides. (Images hotlinked from Wikimedia Commons.)

The problem was hoop force, the outwards force on the lower part of the dome: a big, tall, masonry dome was likely to collapse as the lower part burst outwards. So before the Renaissance, architects made the dome shallow (no lower part to burst outwards), or small (much smaller hoop force), or made of wood (which has greater tensile strength than stone), or buttressed, etc.


In the early Renaissance, the City of Florence decided to build a cathedral dedicated to St. Mary of the Flowers, with an immense octagonal dome 43 meters wide – wider than any dome since the Pantheon itself. They built the cathedral, but prepared no buttressing for the dome, and then … they had a church with no roof. After several decades, they were persuaded to let Filippo Brunelleschi solve the problem, which he did. The solution was a pair of concentric domes: the light outside one we see from afar, and a sturdy inner dome reinforcing the outer one (and itself reinforced with several bands) and helping support the lantern on top. Shortly afterwards, Michelangelo employed the same design when building the dome of St. Peter’s Basilica. And then came Christopher Wren‘s triple dome, with light inner and outer shells giving the hemispherical shape, and a sturdy catenary-shaped cone in between. Wren’s variant of Brunelleschi’s innovation was anticipated in Asia: the multiple dome / shell design makes onion domes (like the Taj Mahal, built before St. Paul’s cathedral) possible.

Image posted on Wikimedia Commons by Bob Tubbs Image posted on Wikimedia Commons by Amandajm Image posted on Wikimedia Commons of engraving produced in 1755 Image posted on Wikimedia Commons by Dhirad

From left to right, Brunelleschi’s double dome in Florence, then Michelangelo’s double dome in Rome, then cross section of Wren’s triple dome in London, and to the right, (possibly) Ustad Ahmad Lahauri‘s double dome in Agra. Images hotlinked from Wikimedia Commons.

This was also an era when the drawing techniques of secretive medieval guildsmen were gradually displaced by the more precise drawings of (often geometry-oriented) artists of the Renaissance realist movement – including Brunelleschi and Michelangelo’s successor for St. Peter’s, Jacopo Barozzi da Vignola, who took over Michelangelo’s project when he died, and completed it using Michelangelo’s drawings. The engineers of the fifteenth, sixteenth, and seventeenth centuries were able to build unprecedented structures not only because of the growing understanding of physics, but also because of the more careful and comprehensive design. (See Henry Cowan, The Master Builders : A History of Structural and Environmental Design from Ancient Egypt to the Nineteenth Century and Peter Jeffrey Booker’s A History of Engineering Drawing for more details.)


Science and technological progress is a field of study in itself these days. One of the major recent studies presents European technological innovation during the last millennium as a culture of improvement, in which we do not have one inventor suddenly inventing the steam engine, but instead several centuries of inventors successively improving on their predecessors’ steam engines (which, in turn, had been rediscovered from antiquity). The two most popular models of this kind of cultural process probably are:

  • Donald Campbell’s notion of evolutionary epistemology, which was an extension of his psychological theory of Blind Variation and Selective Retention (BVSR). A creative process – either individual or social – relies on an idea generator, and ideas are selected based on the needs of the moment. The selected ideas are remembered, and gradually a structure of knowledge (or of culture) is assembled. Campbell was motivated by Darwin’s theory of natural selection, and subsequently by the theory of punctuated equilibrium advanced by Niles Eldredge and Stephen Jay Gould: during the evolutionary process, there are long periods of relatively little change in a species, “punctuated” by brief periods of rapid change when new species emerged from the old.
  • Thomas Kuhn’s division of scientific research into two categories. Normal science, which is the “puzzle solving” that occupies most of industrial and academic science, and which operates under the auspices of a consensus “paradigm” accepted by most practitioners. On occasion, internal and external stresses on a paradigm leads to a transition from one paradigm to another, which we call a scientific revolution.

Notice that both models encompass scientific revolutions very well, but do not particularly address the problem of stretching to reach a notion just outside of the scientific community’s grasp.


Several years ago, I advanced a sort of merged metaphor. Imagine scientific truth (whatever that is) being an invisible edifice, and imagine that scientists are gardeners growing a vine up the edifice. We can’t see the edifice, but we can see the vine on the edifice creating an outline of the turrets, balconies, gargoyles, and other (invisible) features of the edifice. This metaphor has two important features for us, both based on the notion that an emerging field needs recruits, and thus the vine’s growth must be guided not only for exploring the edifice, but to make it navigable by novices.

  • The metaphor distinguishes Kuhn’s normal science – which consists of the vine filling in the gaps to reveal the reliefs on the wall (which is very important to industry, which needs to get things right) – from non-revolutionary frontier science, where gardeners try to persuade the vine to grope towards … something that they can’t see but that a few of them guess is there. This image was used to describe the difficulties some precocious mathematicians had in getting anyone to understand their work.
  • The metaphor does not address revolution as a research activity, but instead the result of revolution as a pedagogical one. The pedagogical problem at the base of the vine is how to get novices going; gardeners solve this problem by tending a few vast trunks, with numerous low branches to climb on and explore. The pedagogical problem near the frontier is to find an accessible path to the frontier, which means developing a bough or thick rope of tendrils to the desired location.

    In his history of group theory, Wussing (above) points to Galois’ almost wilful opacity as one of the reasons why his work took over a decade after his death to have an impact, and in my paper, I followed Michael Crowe’s account of Hermann Grassman burying his predecessor of vector algebra in difficult tomes, as opposed to Willard Gibbs’ more strategic marketing (with preprints). And this brings us to where all this sociology and philosophy can help us: if we are in an emerging field, what are we supposed to do?


    Let’s first look at how an innovation happens. The popular vision is of some hermit who invents a better mousetrap, and the world beats a path to his door. Probably the most popular example of this in animal behavior was that of a low-ranking female in a Japanese monkey troop that twice discovered more efficient food processing techniques that subsequently percolated up the (male-dominated) hierarchy. Unfortunately, the human innovators of this type that come to mind are often people like Galois, Grassman, Vincent van Gogh, and Gregor Mendel: while every once and a while, such a lonely genius (like Srinivasa Ramanujan) is discovered Hollywood-style by a perceptive master (like Godfrey Hardy), but very few masters seem as perceptive and receptive as Hardy.


    Kuhn thought that the social and institutional obstacles to a marginal figure making a major innovation and having the community respond positively were so difficult that most successful innovations would arise closer to the center of the action. Indeed, Bernard Barber enumerated a whole list of different reasons and ways that the scientific community resists innovation: some scientists cling to their old models and methodologies, many have a variety of attitudes towards mathematics, some are wary of anything outside of their immediate specialty, and following the stereotype, “… sometimes men of higher professional standing sit in judgment on lesser figures before publication and prevent a discovery’s getting into print,” and “That the older resist the younger in science is another pattern that has often been noted by scientists themselves and by those who study science as a social phenomenon.” An even gloomier view was advanced by Donald Hambrick and Ming-Jer Chen, who wrote that practitioners in neighboring fields may see a new field as a competitor for scarce resources.


    Dean Simonton additionally argued that a successful innovation would likely be made by someone who had already so mastered the subject that they were already known (I seem to remember one writer trotting out Isaac Newton, Charles Darwin or Maynard Keynes as examples).


    But there is an intermediate possibility suggested by the explosion of Cubism a century ago. Stoyan V. Sgourev notes that in 1907, when Pablo Picasso produced Les Demoiselles d’Avignon below, he was a minor figure but not marginal: he was already known for his Blue Period paintings. In addition, the growth of the middle class and the appearance of mass art had ironically produced niche markets for exotic art, and Picasso had connections to one of the niche marketers. After four years, Cubism was probably “the dominant avant-garde idiom in Paris,” but as Cubist art circulated in private markets (and not the conservative salons), the movement fragmented with no real leaders, but instead a growing cohort of practitioners. In essence, what Cubism needed was infrastructure, an accessible economic demand, and a network to exploit, but no stellar leaders (although Picasso and Georges Braque would be proclaimed the leaders later, after the critics recovered from the shock).

    Posted on Wikimedia Commons by Olpl as a work in the public domain Image posted on the Artchive

    Les Demoiselles d’Avignon by Picasso (hotlinked from Wikimedia Commons) and Houses iin L’Estaque, (hotlinked from the Artchives) the work described by art critic Louis Vauxcelles as “Bizarreries Cubiques,” thus giving “Cubism” its name.


    But mathematical crystallography already has senior and prominent participants and friends. So where do we go from here? Well, we are feeling our way, and considering the far-flung nature of the field (both as an interdisciplinary field and one whose practitioners are geographically scattered), one issue is communication: Diana Crane claimed that practitioners who are clustered together can support each other, while those lacking contacts in ancillary or outlying areas that they develop interest in will have as their primary resource journals, books, and conferences, but that can be difficult.


    So we are talking about infrastructure (and economic demand, which will be the subject of later posts). Conferences, journals, workshops, and so on. There is another consideration: Jan Fagerberg and Bart Verspagen noted several authors who argued that in order to established credibility among other scientific fields, a new field has to construct visible quality controls – although I suspect that a snooty exclusivity could be unhelpful, especially in this field, which is ancillary to so many amateurs and enthusiasts, whose presence can be felt by the success of conferences like the one described in Marjorie Senechal and George Fleck’s Shaping Space: a Polyhedral Approach and books like Herman Weyl’s Symmetry and John Conway, Heidi Burgiel and Chaim Goodman-Strauss’ Symmetries of Things; indeed, young amateurs and enthusiasts will be among our recruits (and older amateurs and enthusiasts will be among our community supporters – and in an era of intelligent design, climate denial, and ancient aliens, science can use all the community support it can get).


    One priority for infrastructure is recruitment. Luis Bettencourt et al constructed a model of new scientific fields and looking at cosmological inflation, cosmic strings, prions, H5N1 influenza, carbon nanotubes, and quantum computing, they found that the following formula is a good fit. Let ΔP be the number of new publications in a year, ΔA the number of new authors, and α a “scaling exponent”, then ΔP / (ΔA)α is nearly constant – and they call this the normalization constant. In other words, the number of publications corresponded well to the number of recruits (although there is also an α which reflects how well the field is doing).


    Notice that this is the number of new authors, not the number of young practitioners. But either way, we are talking about recruits, and we are back to the pedagogical issue. The field will need expository materials and tutorial programs – not unlike the workshops organized by the IUCr Commission on Mathematical and Theoretical Crystallography – and we need to think about students. How would a student learn mathematical crystallography? How would they become prepared to learn mathematical crystallography? One of the issues raised in the National Research Council’s Mathematical Challenges from Theoretical / Computational Chemistry is that mathematics students don’t learn chemistry and chemistry students don’t learn (much) mathematics, so neither cohort is prepared to do much in the intersection of chemistry and mathematics. This will have to change if mathematical crystallography is to progress.


    At the SIAM Mathematical Crystallography mini-symposiums, one of the participants worried that if we recruited graduate students, they might have difficulty getting employment. As someone who concentrated on mathematical logic as a student – the joke there was that one had to wait until someone died before one could get a job – I am sensitive to the issue. There seem to be two reactions.

    • One of my colleagues would tell his students not to go into mathematics unless they were committed to it. It was a calling, not a career. One went into it because it was beautiful. And if one would be practical, learning mathematics prepared one for many careers (like the humanities, which seems just as good – or possibly better, according to some studies – than business school in preparing students for business careers).
    • If we believe that mathematical crystallography is good for something, that it is useful in solving important problems, then we believe that our work will create an economic demand for our students. That puts the ball squarely in our court: it’s up to us to create a market for our students.

    If we are going to recruit people, we have an obligation to know what we are doing.

  • Aug 102013
     


    Every April, the International Society for Nanoscale Science, Computation and Engineering meets in Snowbird, Utah, for the annual Foundations of Nanoscience (FNANO) conference – and some skiing during the last week of the season. In 2007, Omar Yaghi was invited to talk about chemistry’s “change from ‘shake and bake’ to rational design.” Rational design leads inevitably to mathematics, and Yaghi spent a few minutes complaining about mathematicians who were unwilling to work on problems that were not “interesting”.


    Chemists are not alone: physicists have been complaining for centuries. But the reality is that mathematicians are pretty much like any other group of academics: they went into their own field because that was what they were interested in. Just as chemists and physicists are often more interested in chemical or physical problems, mathematicians are often more interested in mathematical problems. One difference is the physicists – and more recently chemists – depend on mathematics for their toolkit, but before the Twentieth century, mathematics did not need much help from other disciplines for solving problems. The rise of computer-aided proofs and experimental mathematics suggests that that may be changing, but even now mathematicians can be fairly self-reliant.


    Except when it comes to looking for dragons to slay: mathematics has a long history of getting its problems from outside. So mathematical crystallography should be interesting to mathematicians – after all, crystallography provided mathematicians with the crystallographic groups. But still, the asymmetry resulted in cultural differences that may create difficulties for collaboration.


    Despite several years of collaboration, I don’t really know enough about chemists to comment on them, so I will concentrate on the problems with collaborating with mathematicians.

    1. There is an agenda problem. Non-mathematicians are interested in the goodies that mathematics can provide, while mathematics has a long tradition of being an enterprise undertaken for its own sake.
    2. Mathematics is notoriously difficult. In fact, many students abandon more mathematical sciences (like physics) in favor of less mathematical ones (like chemistry) out of mathematics anxiety.
    3. Abstraction. During the last two centuries, mathematics has grown increasingly abstract, and that impulse can make mathematics and mathematicians less accessible.
    4. The fragmentation of the field. Mathematics is now fragmented into about six thousand fields, and few mathematicians have mastery over more than a few of these (for example, back in the 1990s when I was pretty much a normal mathematician, I was quite happy in my four fields).

    Three things to keep in mind.

    • In those societies where academic mathematics was substantial (and where surviving documentation is substantial) – like ancient Greece, China, India, the Middle East) – mathematics either had religious roots or at least religious entanglements, and there were strong connections and sympathies between mathematicians and metaphysical philosophers.
    • Ever since Egypt and Mesopotamia, mathematics has been a middle class activity. Every urban society has had an enormous demand for mathematics and logic (considered broadly), so there was always a certain status accorded to the Queen of the Sciences.
    • Mathematics has gone through several crises when it wasn’t clear what the standard should be for accepting a mathematical fact as true. A sequence of Greeks from Thales of Miletus and Hippocrates of Chios to Aristotle and Euclid developed and popularized the Axiomatic Method: one starts with a set of clear definitions, unequivocal assumptions, and Rules of Inference and, starting from the assumptions, one draws conclusions, one by one, using the rules. Such a sequence, from assumptions through successive conclusions up to the desired fact, is called a proof. This form of verification is extremely labor-intensive, but mathematicians rely on it because of many unhappy episodes when mathematicians evaded or fudged it. Much of the mathematical literature, and much of intermediate and upper level mathematical courses, are exercises in the axiomatic method.

    Now let’s look at the four issues more closely.

    1. The Agenda. This is probably the issue that bothers non-mathematicians the most: they complain that mathematicians do not find their work “interesting.” But again, as murder mystery writer Emma Lathen observed, “People … [are] basically not interested in [other people’s] problems; they [are] interested in their own.” The National Research Council, in its 1995 report on Mathematical Challenges from Theoretical / Computational Chemistry, recognized that chemists are interested in chemical problems while mathematicians are interested in mathematical problems:
      • From page 110, “… interdisciplinary work may be regarded … as “not real mathematics …” and most “Most academic mathematicians would agree that it is difficult to [evaluate] ‘interdisciplinary’ work …,” and at any rate, it is unlike to constitute “new mathematics.” “Such issues are particularly worrying for junior mathematicians” because of the tenure and promotion processes.
      • From page 111, “… analogous principles of departmental autonomy can affect chemists seeking to work with mathematicians.” While the report does not worry about the status of theoretical chemists – in computer science at least, theoreticians are often marginalized, and that is true to some extent in physics as well (it is the experimentalists Arno Penzias and Robert Wilson, not the theoreticians George LeMaitre, George Gamow and Ralph Alpher, who are went to Stockholm for the Big Bang) – it does remark that “Because theoretical/ computational chemists must often demonstrate the applications of their work to experimental areas of chemistry [something not even computer science requires!], fundamental work of a mathematical nature … may be undervalued.” On the other hand, “… chemistry departments have more experience evaluating multidisciplinary research …”

      There is a sort of political spectrum in mathematics, ranging from a mathematical Right that values mathematics for mathematics’ sake to a mathematical Left that values mathematics for its contributions to society.

      To the Right. Pythagoras and Plato associated numbers and geometry, respectively, with the overarching metaphysical reality of the universe. Leading Twentieth century advocates of the Right include the great mathematician Godfrey Hardy, whose Mathematician’s Apology can seem like a description of a recreational activity (“I have done nothing useful,” wrote the founder of mathematical genetics), and the mathematician (and science fiction writer) Eric Temple Bell, whose Development of Mathematics describes the history of the subject as if it was an art.

      To the Left. Despite their occasional Rightwing rhetoric, both Isaac Newton and Albert Einstein were to the mathematical Left, and their primary interest in mathematics was what they could do with it in physics. Leading Twentieth century advocates of the Left include the biologist (and political Leftist) Lancelot Hogben, whose Mathematics for the Million is an attempt to bring the first semester of calculus to the public, and mathematical educator Morris Kline, whose Mathematical Though from Ancient to Modern Times presents a more mathematical Left wing view of history (and whose mathematical Left-wing diatribe Why the Professor Can’t Teach: Mathematics and the Dilemma of American Undergraduate Education condemns modern mathematics – and modern art).

      Most mathematicians are moderates, but during the last century, the incentive structure for academic mathematics was Right-wing, although this may be changing.

    2. Math is hard. Barbie got in trouble for saying that math class is tough, but it seems to be true. In fact, perennial mathematics education reform efforts are motivated (at least in part) by the perception that students are not learning adequately learning mathematics. Exactly what makes mathematics hard is unclear – it could be that students are taught that math is either easy or impossible and thus do not work hard at it, or it could be that there is something intrinsically unnatural about mathematical thinking that many people cannot master, or even (as Frank Smith argues in The Glass Wall: Why Mathematics Can Seem Difficult) the abstract language of mathematics. The former theory was the most popular until recently (and even now, the math gene is a favorite alibi for students who don’t associate homework scores with TV watching habits); nowadays, educators like Sheila Tobias have advanced the idea that aversion to mathematics, and not lack of math genes, is the problem. (There is also the theory that the problem is an aversion to hard work – for mathematics is hard work.) Whether the issue is the difficulty of mathematics, or an aversion to mathematics, many people who work with mathematicians have issues with the subject in itself, and that limits their ability to collaborate effectively.
    3. Abstract Mathematics. Mathematics has a tendency to drift towards general statements of as universal applicability as possible. In other words, mathematics has a tendency towards the abstract. One can see this in numbers:
      • By repeated encounters with queues, one gets the notion of the first, the second, the third, and so on: from these one abstracts the notion of ordinals, giving the position of a bird on a pecking order, or landmarks on a route.
      • By repeated encounters with collections, one gets a notion of one object, two objects, three objects, and so on: from these one abstracts the notion of cardinals, giving the size of one’s herd of cows or collection of cowry shells.
      • Somehow these two notions were merged together to produce the notion of a (natural) number. Experiences with pieces of pies and shares of spoils led someone to the notion of a ratio, which was soon seen as another kind of number – which mathematicians call rational. (Indeed, the Greeks wrestled with a menagerie of kinds of numbers.)

      During the last two centuries, the rise of abstract algebra has put wheels on abstraction. For example, groups were originally actions that permuted solutions of polynomials, but these solutions were numbers so we had groups of numbers, but then there were groups of symmetries acting on polyhedra so we had groups of actions, and so on. By the time the crystallographic groups were enumerated, a group was any collection G of objects (together with an identity e) and a “binary operation” * such that the following are true:

      • For any x and y in G, x * y was in G. (Think of the integers with * being addition, or the positive rationals with * being multiplication, or crystallographic groups with * being composition.)
      • For any x, y, and z in G, (x * y) * z = x * (y * z). This is called associativity; notice that on numbers, addition and multiplication are associative while subtraction and division are not.
      • There is an element in G, call it e, such that for any x in G, x * e = e * x = x. If * is addition on numbers, then e is 0, while if * is multiplication on numbers, then e is 1.
      • For each x in G, there exists exactly one x -1 in G (called the inverse of G) such that x * x -1 = x -1 * x. On numbers, if * is addition then x -1 is –x, while if * is multiplication then x -1 is 1/x.

      The advantage of this level of abstraction is that once you verify a fact about groups, you verify it for all applications. The disadvantage is that abstractions are not as readily apprehended as concrete examples. The aesthetics and metaphysics suggested that abstraction allowed a mathematician to get at the mathematical content of the issue. Mathematics is now so abstract that a book on crystallographic groups can come out and not even mention screws, glides, or inversions at all.

    4. The Fragmentation of Mathematics. As mentioned in a previous posting, there are about six thousand fields listed by the joint committees of the American Mathematical Society and Zentralblatt fur Mathematik. Since mastering a field entails dealing with the proofs of major results in the field (!), it is extraordinarily difficult to master more than a few of them. Then looking at the number of fields listed in that posting as being relevant to crystal structures, and how scattered those fields are, one cannot expect a given mathematician to be able to handle any crystal structure problem that comes up. (I have a unique alibi: I am not particularly competent in any of the listed fields on that posting.)

    So there are likely to be challenges in collaborating with mathematicians. And as one might guess, mathematicians face challenges dealing with chemists. The most obvious is the language barrier.

    • If you look at a mathematical article or text, you will see bricks – definitions and propositions, often with big block-like proofs – held together by motivating mortar. One can scan such a paper, but actually reading it is a Zen-like experience, since (unlike a chemist) the reader is supposed to actually verify the work by recapitulating the proof. (Certainly, this is what is expected of referees – which is why the refereeing process in mathematics journals can take months, or even years.)
    • Non-mathematicians do not define their terms, so it’s hard to tell what they are talking about. While it is true that chemists will define terms like “phenol”, “aromatic”, and “organic”, they will tend to be mushy when talking about, say, polyhedra (which are defined very carefully in math books). This means that mathematicians have to work to nail down what the non-mathematicians are talking about.
    • Non-mathematicians do not know what they want. An industrial mathematician (speaking at a seminar at USF) once said that a mathematical consultant should not just solve the problem that some industrialist asks about, for that solution often isn’t that helpful. The first thing to do, said this mathematician, is to figure out what the real problem is.
    • Non-mathematicians have lower standards of verification. Mathematicians are so concerned with verification that they actually have fields devoted to verification in itself. Compare that to natural scientists, who are prone to optimistic generalization. This is partly Galileo’s fault – he was the one who said that one starts with a simple model and then adapt it to experimental results – and experimental results can establish a theory only within a limited domain. Lord Kelvin’s teapot on a burner is not the Earth warmed by the Sun, so it was perhaps unwise of him to use his teapot to model the cooling of the Earth with the cooling of his teapot – and worse, rejecting empirical evidence that his calculations were off.

    So in comparison with natural scientists, mathematicians can be obsessive-compulsive. But we’ve learned to be careful the hard way, and we have the scars to prove it.

    There is an additional problem. On page 112, the National Research Council wrote that “the mathematics curriculum is structured like a tree, with courses of potential interest to chemists at the end of a very long branch of prerequisites; the effect is to discourage chemists from obtaining any knowledge of advanced topics.” As we shall see in a later post, the emergence of a new field depends critically on recruitment, so curriculum is a very important matter. But there is a lot more to it than the Council suggests.

    • Unlike other science departments – in fact, to an extent matched only by English – mathematics is a service department. Many science and engineering departments require that their students take then entire calculus sequence (up through multiple integration and elementary vector calculus), linear algebra, and differential equations. Many universities then offer a year course on “applied mathematics”: this course is usually generic, satisfying the needs of many science and engineering departments with quite different needs. This common approach is a reflection of limited resources.
      • Mathematics departments have heavy teaching duties, especially at the lower division (calculus and below). Resources are limited for upper division and graduate courses.
      • Students who want to complete their degrees in four years have only so many courses that they can put in their schedules. This problem is compounded by engineering schools with overstuffed major requirements.

      So a mathematics department would not have the resources to offer a panoply of advanced courses, one for each discipline. Chemistry students would have to make do with the applied mathematics course, and that means that a chemistry department would have to lobby for the content it wants.

    • Mathematics is not the only curriculum shaped like a tree. For example, at USF, a student who wants to take practically any advanced topic must take the three semesters of calculus, and then an indoctrination course called Bridge to Abstract Mathematics, and only the Elementary Abstract Algebra. Chemistry, meanwhile, has two semesters of chemistry (plus lab) and two semesters of organic chemistry (plus lab), and some advanced topics do not require the latter: the tree is shallower, but it is there. As a result, a mathematics student who wanted to learn crystallography would also face a lot of courses.

      This becomes an issue when mathematicians and non-mathematicians work on interdisciplinary programs, especially on graduate interdisciplinary programs, which have to contend with college-level restrictions on how many undergraduate courses graduate students may take for credit.

    There are many levels of collaboration, from researchers working on a common problem to departments articulating their curricula. The National Research Council makes a number of recommendations, like setting up joint seminars. But notice what this recommendation entails: people from both departments would have to organize the thing, and then faculty in both departments would have to find the time to attend it. Collaboration will require labor as well as stretching.

    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:
    zbMATH on ACA
    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…

    About Comments

     Miscellany  Comments Off on About Comments
    Jul 222013
     


    There seem to be more spectators than participants in this blog, so once again, I invite comments.


    Comments are assigned to individual posts or individual pages, and are not visible on the home page. But if you go to another page, or click on a post, you will see at the bottom of the page all the comments of that page, together with a form for entering your own comments. Comments are invited.


    Unfortunately, there are software robots – call them spiders – that roam the web, looking for forms, and when they encounter one, they submit comments on Viagra, hair styling, advertising one’s blog, the brilliance of my postings (I get long generic essays of obvious arachnid origins), and abandoned Nigerian bank accounts. So comments are moderated as follows. Every time a comment from an unfamiliar machine comes in, it waits in the pending queue until I approve or reject it. I approve all comments that recognizably come from human beings.


    So far, only a few people have posted comments. I have gotten as yet unfulfilled promises to post comments, and email responses from people who do not want to post comments. But I recommend posting comments. If you rely on me to compose a post or a comment, I will put my spin on it; if you want a comment done right, do it yourself.