You are currently browsing the tag archive for the ‘percolation’ tag.

The month of April has been designated as Mathematics Awareness Month by the major American mathematics organisations (the AMS, ASA, MAA, and SIAM).  I was approached to write a popular mathematics article for April 2011 (the theme for that month is “Mathematics and Complexity”).  While I have written a fair number of expository articles (including several on this blog) aimed at a mathematical audience, I actually have not had much experience writing articles at the popular mathematics level, and so I found this task to be remarkably difficult.  At this level of exposition, one not only needs to explain the facts, but also to tell a story; I have experience in the former but not in the latter.

I decided to write on the topic of universality – the phenomenon that the macroscopic behaviour of a dynamical system can be largely independent of the precise microscopic structure.   Below the fold is a first draft of the article; I would definitely welcome feedback and corrections.  It does not yet have any pictures, but I plan to rectify that in the final draft.  It also does not have a title, but this will be easy to address later.   But perhaps the biggest thing lacking right now is a narrative “hook”; I don’t yet have any good ideas as to how to make the story of universality compelling to a lay audience.  Any suggestions in this regard would be particularly appreciated.

I have not yet decided where I would try to publish this article; in fact, I might just publish it here on this blog (and eventually, in one of the blog book compilations).

Read the rest of this entry »

As you may already know, Danica McKellar, the actress and UCLA mathematics alumnus, has recently launched her book “Math Doesn’t Suck“, which is aimed at pre-teenage girls and is a friendly introduction to middle-school mathematics, such as the arithmetic of fractions. The book has received quite a bit of publicity, most of it rather favourable, and is selling quite well; at one point, it even made the Amazon top 20 bestseller list, which is a remarkable achievement for a mathematics book. (The current Amazon rank can be viewed in the product details of the Amazon page for this book.)

I’m very happy that the book is successful for a number of reasons. Firstly, I got to know Danica for a few months (she took my Introduction to Topology class way back in 1997, and in fact was the second-best student there; the class web page has long since disappeared, but you can at least see the midterm and final), and it is always very heartening to see a former student put her or his mathematical knowledge to good use :-) . Secondly, Danica is a wonderful role model and it seems that this book will encourage many school-age kids to give maths a chance. But the final reason is that the book is, in fact, rather good; the mathematical content is organised in a logical manner (for instance, it begins with prime factorisation, then covers least common multiples, then addition of fractions), well motivated, and interleaved with some entertaining, insightful, and slightly goofy digressions, anecdotes, and analogies. (To give one example: to motivate why dividing 6 by 1/2 should yield 12, she first discussed why 6 divided by 2 should give 3, by telling a story about having to serve lattes to a whole bunch of actors, where each actor demands two lattes each, but one could only carry the weight of six lattes at a time, so that only 6/2=3 actors could be served in one go; she then asked what would happen instead of each actor only wanted half a latte instead of two. Danica also gives a very clear explanation of the concept of a variable (such as x), by using the familiar concept of a nickname given to someone with a complicated real name as an analogy.)

While I am not exactly in the target audience for this book, I can relate to its pedagogical approach. When I was a kid myself, one of my favourite maths books was a very obscure (and now completely out of print) book called “Creating Calculus“, which introduced the basics of single-variable calculus via concocting a number of slightly silly and rather contrived stories which always involved one or more ants. For instance, to illustrate the concept of a derivative, in one of these stories one of the ants kept walking up a mathematician’s shin while he was relaxing against a tree, but started slipping down at a point where the slope of the shin reached a certain threshold; this got the mathematician interested enough to compute that slope from first principles. The humour in the book was rather corny, involving for instance some truly awful puns, but it was perfect for me when I was 11: it inspired me to play with calculus, which is an important step towards improving one’s understanding of the subject beyond a superficial level. (Two other books in a similarly playful spirit, yet still full of genuine scientific substance, are “Darwin for beginners” and “Mr. Tompkins in paperback“, both of which I also enjoyed very much as a kid. They are of course no substitute for a serious textbook on these subjects, but they complement such treatments excellently.)

Anyway, Danica’s book has already been reviewed in several places, and there’s not much more I can add to what has been said elsewhere. I thought however that I could talk about another of Danica’s contributions to mathematics, namely her paper “Percolation and Gibbs states multiplicity for ferromagnetic Ashkin-Teller models on {\Bbb Z}^2” (PDF available here), joint with Brandy Winn and my colleague Lincoln Chayes. (Brandy, incidentally, was the only student in my topology class who did better than Danica; she has recently obtained a PhD in mathematics from U. Chicago, with a thesis in PDE.) This paper is noted from time to time in the above-mentioned publicity, and its main result is sometimes referred to there as the “Chayes-McKellar-Winn theorem”, but as far as I know, no serious effort has been made to explain exactly what this theorem is, or the wider context the result is placed in :-) . So I’ll give it a shot; this allows me an opportunity to talk about some beautiful topics in mathematical physics, namely statistical mechanics, spontaneous magnetisation, and percolation.

[Update, Aug 23: I added a non-technical “executive summary” of what the Chayes-McKellar-Winn theorem is at the very end of this post.]

Read the rest of this entry »