A few days ago, I was mucking about in the vast swampland that the Internet has become, and I stumbled upon yet another reference to a programming language I’d heard about a few times before: “Processing.” My interest piqued, I went in search of a compiler, and found one at Processing’s website (www.processing.org), and immediately started learning the language.
Given my many previous failures trying to learn Java, upon which Processing is based, I didn’t think I’d have much chance of learning the language, but I tried anyway, and actually found it just about as intuitive and elegant as Python, which remains my favorite programming language of all times. For a while, I cobbled together various tiny programs to do things like graph functions in one and two variables, graph parametric equations by replacing x, y, and z with rgb color values (producing a rather strange-looking wave of color that wasn’t nearly as interesting as I’d hoped), and visualizing one-dimensional celluar automata (which, by the way, was a complete failure, because Python is the only language I’ve found whose array-handling I can both tolerate and understand). Then, since I’m always a mathematician at heart, I thought I’d do something that mathematicians love to do: visualize the primes.
Before I go on, I should re-iterate just how much of a godsend Processing is. I’ve been trying to write methods in Python to visualize various kinds of functions and data for many moons, and my results have never been much more than mediocre. The only graphics module I’ve learned in Python (Tkinter, in case you were interested) is clumsy and runs slowly, and really isn’t meant to handle the kind of pixel-by-pixel manipulations I’d had in mind. Processing, though, exists solely for this purpose, which is the reason for my gushing for the last three paragraphs.
Anyway, the primes. I put together a simple program that computes the gap between the current prime number and the last prime number (using the standard Sieve of Eratosthenes method), and draws a circle at the location of the current prime whose size is based on the gap between the two primes.
I suspect I could have saved these thousand words by doing as the cliché says, and just giving you the picture:
(You can see a higher-resolution version here).
There’s a great deal of hidden beauty in this picture, most of which I can’t claim responsibility for. There’s a certain order to it, even though the primes seem to be quite random. Really, the beauty comes from the delicate, elegant structure of mathematics. The structure of the primes, as the structure of pi, is an expression of the deep structure of numbers, and thus, of the deep structure of the universe itself. It can be an almost religious experience, a sort of holy communion with the Numbers, to be given a glimpse of that structure.
I don’t know why I’ve been so sentimental lately…either way, the point I’m driving at is this: visualization is a really powerful tool for understanding mathematics. And Processing is a great programming language for visualization. (And, once again, I sound like I’m on somebody’s payroll, but I’m not. As far as you know.).