Jul 042013

I’ve started reading Crystal Engineering : A Textbook by Gautam R Desiraju, Jagadese J Vittal and Arunachalam Ramanan. It was published by World Scientific, a relatively new press (only three decades old) that has yet to reach the radar of Thomson Reuters’ Science Citation Index (a reminder that the standard metrics are better at measuring established fields – like crystal X-ray diffraction – than emerging ones – like crystal engineering).

My impression is that there will be a big demand for geometric crystallography from crystal designers, and today here is one strategy to find out what a community looks like. Follow your nose. In 1929, Firgyes Karinthy published a short story, Chain-Links, a dialogue that included…

A fascinating game grew out of this discussion. One of us suggested performing the following experiment to prove that the population of the Earth is closer together now than they have ever been before. We should select any person from the 1.5 billion inhabitants of the Earth – anyone, anywhere at all. He bet us that, using no more than five individuals, one of whom is a personal acquaintance, he could contact the selected individual using nothing except the network of personal acquaintances. For example, “Look, you know Mr. X.Y., please ask him to contact his friend Mr. Q.Z., whom he knows, and so forth.”

Social network people claim that five links is an underestimate, but not by much. So one way to explore a community is to follow links. Which brings us back to Crystal Engineering. In the back of the book is a page of links to websites (and another page of links to articles in the Journal of Chemical Education that students might find interesting). I had seen some of those websites before, but most of them I hadn’t, so I added them to the links I already had on this site: see the tabs above on mathematics, crystallography, science & technology, and education & enthusiasm. Each of these tab pages lists books, journals, organizations, and websites & software. Most of Crystal Engineering’s links were to crystallography websites, but a few were to journals, and I put those links on those pages.

Incidentally, the book was published just last year, and already a few urls had changed, and one or two sites had simply disappeared. The world wide web is not a library.

Feel free to browse around the links on this site. That’s what they’re for. I would like these pages to be useful, so recommendations are welcome: you can post a comment or send an email to me at mccolm@usf.edu.

But links are not the only thing on my mind as I read this book. One thing that I find striking is that my formal chemistry background consists of a high school chemistry course, and yet I am following it reasonably well. Of course, when I hit one of those dreadful nouns chemists are fond of – like “nitrobenzoic acid” – I have to scurry off to Wikipedia and ask what these are …

… and I conclude that I take carboxylic acid, which consists of a carbon atom double-bonded to an oxygen and an OH and something else, and if that something else is a benzene ring I have benzoic acid. If I add a nitrogen atom bonded to two oxygen atoms, I get nitrobenzoic acid.

But let’s be honest. A substantial fraction of the time, when I encounter something like “nitrobenzoic acid” I think of it as one of those nouns Professor Snape applies to hydrogenated newts’ tongues, and let it go at that. Interestingly, I do not get lost.

Compare this to a book I recently read on the crystallographic groups of arbitrary dimensions. I did not like the definition of “holonomy group” of a crystallographic group, so I went to Wikipedia and found that the notion was probably related to the notion of holonomy in Riemann manifold theory, but in going through my other books on Riemann manifolds, I was left no more enlightened. Fortunately, I knew that “holonomy group” usually (but not always) in crystallography refers to the point group, so I was able to reverse engineer a bit. But if I hadn’t already known the holonomy – point group connection, I would have been at sea. And you cannot follow the text unless you know what a “holonomy group” is …

I am not the only one to observe that mathematics is harder to read than, say, sociology. One reason may be the way definitions work in mathematics. Anyone familiar with Dr. Seuss will recall the dialogue in Fox in Socks between a crystallographer named Mr. Knox and a foxy geometer in blue socks. The climax concerned tweedle beetles (on You-tube, start at 6:15). When tweedle beetles fight, it’s called a “tweedle beetle battle”, and when they fight in a puddle, it’s called a “tweedle beetle puddle battle”, and when they fight in a puddle with paddles, it’s called … and if you miss any one of those definitions, abandon all hope …

Which is why I was pawing around, looking up holonomy groups. It did not help that there are several incompatible definitions.

If you asked Mr. Fox why he creates these mountains of definitions only to inflict these proofs on us – he will tell us the hard road is the only one that doesn’t lead into the swamp. The axiomatic method developed in Greece and popularized by Aristotle and Euclid is the way to avoid mistakes. Let’s look an example.

Suppose, like Leonard Euler, you look at a lot of polyhedra and you notice that if you take a polyhedron and count the vertices (let V be the number of vertices), edges (let E be the number of edges), and faces (let F be the number of faces), then

VE + F = 2.

For example, consider the slightly squished octahedron below:

There are V = 6 vertices, E = 12 edges, and F = 8 faces, and indeed VE + F = 6 – 12 + 8 = 2. Anyway, Euler had an argument that it was true (it involved thinking of the polyhedron has a rubber shell which you open up), but later he retracted the proof. Why? Consider the following polyhedron:

Here, V = 13, E = 20, and F = 10, and 13 – 20 + 10 = 3. Oops. It turns out that the problem was a mushy definition of the word “polyhedron”.

In fact, getting to the bottom of Euler’s formula took a century of failed definitions, broken proofs, and counterexamples. (For an accessible and entertaining account of that century, see Imre Lakatos’ Proofs and Refutations.)

But there’s the problem. In chemistry, one conducts the experiments to confirm what is reported in published accounts of previous experiments. In mathematics, the verification is itself the published proof, and the confirmation is the act of checking the proof. Reading mathematics carefully is like reading chemistry and redoing the reported experiments.

So there’s the conundrum facing anyone attempting to address Mike Zaworotko’s third problem: how do we build a common vocabulary for this community? Since different members of the community use their text for different things, the communication problem goes beyond vocabulary.

Anyway, returning to finding links, here are the articles that Desiraju, Vittal, and Ramanan recommended:

As far as my reading of their book goes, so far, so good. It’s an interesting book, and I recommend it to fellow mathematicians who want to get an idea of what the major issues are. WARNING: this is a book by chemists about chemistry, so of course, there is some nitrobenzoic acid in it…

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