I’ve written here before about the series of columns on mathematical topics by Prof. Steven Strogatz of Cornell, in the “Opinionator” blog at the New York Times. (If you want to go back to have a look my previous notes, there’s a list at the bottom of this post.) I’ve gotten a bit distracted by software patches and other things, but I do want to come back to Prof. Strogatz’s series, because it seems to me a very nice overview of how mathematics fits together; moreover, it illustrates a general principle of development in math that I’ll come back to a bit later.
Three of Prof. Strogatz’s columns have appeared since I last wrote about them. The first two of these, “Square Dancing” and “Think Globally”, are about geometry. In my schooling, geometry was the first “real” math course I took; by that, I mean that it was not just about learning the rules of arithmetic and solving simple problems, but introduced the idea of the theorem, something that could be proved to be true in an absolutely watertight, good for all time kind of way. And yet, as Prof. Strogatz says, geometry, especially in its first development, is one of the most visual parts of math. I suspect that he is right when he hypothesizes that the combination of logic and visual immediacy is the key reason many people find geometry appealing.
In “Square Dancing”, Strogatz goes on to discuss the Pythagorean Theorem, and outlines two proofs of it. The first is visual, and can even be worked out by using little square cookies or crackers, so that once you have enjoyed completing the proof, you can enjoy eating it. It is a proof very much in the spirit of the old Greek geometers, who saw the Theorem as being primarily about areas. The second proof is more algebraic; in a way, it seems less likely to contain hidden assumptions (we were always warned, back in geometry class, not to let our drawings mislead us), but it lacks the immediacy of the first proof, in which one can see that the Theorem is true.
In “Think Globally”, Prof. Strogatz introduces the ideas of non-Euclidean geometry, first using the example of “great circle” routes around the Earth. Our ordinary Euclidean geometry, in which Pythagoras’s Theorem is true and the angles of all triangles sum to 180°, describes a world that is flat. Once we try to generalize these ideas to a world that is round, or shaped like a torus (a doughnut), we find that they still work, but need some modifications. (For example, on the surface of a sphere, the angles of all triangles add up to more than 180°.) The shortest distance between two points is no longer necessarily a straight line, but a path called a geodesic, which is a sort of generalization of the idea of a straight line adapted to a curved world. The column has links to a couple of amusing videos that illustrate how these paths behave.
Once again, we are using a set of observations that originated with the real physical world (the word “geometry”comes from the Greek words for “earth measurement”), and trying to generalize them. This is in much the same spirit that led us to rational, irrational, and imaginary numbers when we tried to generalize the rules of arithmetic. There is a saying among artists that “Form is liberating”. Working within a framework of rules can help us see things that we might otherwise have overlooked.
Previous Posts on this Topic
Finally, although it was not a part of this series (either mine or Prof. Strogatz’s in the New York Times), I have to mention a considerably earlier note about an amazing analog computer.