Wednesday, January 11, 2012

primes on a log spiral

Since looking again at Theodore Andrea Cook's The Curves of Life a few posts back I've been planning on playing with logarithmic spirals, which are identified in that book as the type of spiral that you often encounter in nature and in architecture. I was inspired to finally spend some time with them after reading a recent post on Math Hombre.

For fun I treated the curve like a number line and plotted prime numbers on it  using Processing. It seems to me that a nice thing about curling up the number line is that it allows you to take in more of the line at a glance. You can notice both the (seemingly) increasing gaps that occur between primes, as well as the (apparent) persistent occurrence of twin primes.

Tuesday, January 10, 2012

the best of 2011

Once in a while I get sent books to review and recommend - this is very nice, but unfortunately I haven't had the chance to post many reviews. It is not only in the book review department that I'm failing - I seem to be having a more general problem finding time to do any recreational mathematics (and then to post about it here).

So it was a treat to receive a copy of The Best Writing on Mathematics, 2011 (Mircea Pitici, ed.), a book that solves both problems: it is a book that I really have to recommend, and it is also certain to inspire me in more mathematical recreations.


The anthology gets off to a good start: In the forward, eminent physicist Freeman Dyson proclaims that "Recreational mathematics is a splendid hobby which young and old can equally enjoy... To enjoy recreational mathematics you do not need to be an expert." A great statement that I should probably take as the motto for this blog.

This anthology offers a lot for recreational mathematicians, mathematics educators, professional math practitioners, and hopefully others as well. A couple of the articles in the collection were "old favorites" that inspired posts on this blog when they appeared in their original contexts. Doris Schattschneider's article on Escher and Coxeter prompted this post, and Dana Mackenzie's article on Apollonian gaskets motivated this one and another. These articles remain among my favorites in the collection, but there are many others that make interesting reading, including others like these that focus on aesthetic aspects of mathematics (in ribbed sculptures, in bronze and stone, and in strange-attractors).

Some of the articles are against the grain of our prevailing zeitgeist - Melvyn B. Nathanson strikes a somewhat contrarian tone against the promises of polymath, and Martin Campbell-Kelly wistfully recalls the now obsolete numerical table. I particularly liked how Underwood Dudley asks "What is Mathematics For" and takes aim at an assumption that is now almost sacrosanct: that we teach mathematics because it is useful.

Dudley's thesis, that mathematics (particularly school algebra) may not be used very often but helps us learn to think and reason, although not currently popular, is actually one of the oldest arguments in favor of learning algebra. The very first English-language algebra textbook (published by Robert Recorde in 1557) was titled "The Whetstone of Whitte" precisely because algebra was considered by its author to be like a knife-sharpener for the brain. Of algebra, it said:

Its use is great, and more than one. 
Here if you lift your wits to wet, 
Much sharpness thereby shall you get. 
Dull wits hereby do greatly mend, 
Sharp wits are fined to their full end.


I think that many who appreciate the appeal to the aesthetics and cultural significance of mathematics in Lockhart's Lament will agree with Dudley's call for a more subtle (and accurate) understanding of what mathematics education gives us beyond the merely utilitarian.

Although there is a broad appeal to these articles, I'm guessing that the audience that will most appreciate this collection are those involved in mathematics education. Of particular interest to teachers are two career retrospectives by eminent math-education- theorists Alan Schoenfeld and John Mason, the previously mentioned paper by Underwood Dudley, two other papers specifically about mathematics education, as well as a paper on the cognitive aspects of perceiving numbers.

 Thinking about these kinds of articles, I was reminded that when Martin Gardner died in 2010, many wrote about how his columns inspired them to take up mathematics as a hobby and as a profession. With Gardner as an example, it is clear that the authors of these and other popular mathematics articles are doing something worthwhile.