Of Primes, Fractals and Interactive Websites

spiral_primes1Warning: This blog post deals with mathematics, but not the math you may remember from school. But by virtue of it being mathematics, some people may be tempted to skip over this post. Don’t let this happen to you – there’s too much wonder here to miss out on.

Warning 2: The websites mentioned here use WebGL interactive 3D graphics: They only display correctly on browsers that support WebGL, such as Google’s Chrome browser. If you haven’t already done so, consider losing yourself for a few hours in the chrome experiments website.

Back when I was a mathematics graduate student in the early 1990’s, I felt that I had to sift through tomes of tedious formalism and obtuse notation to get at the few rare jewels of genuine mathematical insight. Or so my memory tells me. Then again, I also remember having to trudge uphill both ways through monstrous snow drifts to get to classes, so I can’t quite vouch for my memory (actually, that last part may have been somewhat true of the math department at the UW Madison, in the winter at least). In any case, my sense of mathematical wonder ebbed, and I eventually turned to the pursuit of more tractable goals, like finding a decent job and starting a family. In effect, I had built a mental wall between me and mathematics.

This wall only began to crack earlier this summer, when I saw a short but amazing presentation at the Eyeo festival in Minnesota. The talk concerned prime numbers – you know, numbers like 2, 3, 5, 7, 11 and so on that can’t be written out as a product of other numbers besides themselves and 1. If you’re bored one day at a meeting and start writing numbers starting with 1 out in a giant grid, then circle the prime numbers from that list, you’ll just get a bunch of randomly circled numbers. But if you were to start writing these same numbers out as a _spiral_, starting with 1 at the center, and then circle the primes, you’d notice weird patterns beginning to emerge from the apparent randomness. That the discovery of these patterns did indeed happen pretty much exactly as described here to a mathematician named Stanisław Ulam in the middle of the 20th century was stunning to me: Prime numbers and their distribution in the number line have been studied extensively and exhaustively for several hundred years. Doodling to pass the time at a boring meeting shouldn’t turn up completely new, surprising results like that!

Now the talk wasn’t really about the math. Rather, the math was an excuse for showing off http://primes.io, which features beautiful visualizations of the above concepts, and plenty more, done with the help of WebGL As you explore the various controls on the top right of the primes.io website, take a look also at the directions for self-guided study on the left. Yes, the pictures are pretty and weirdly intriguing – choosing the Euclidean spiral view in particular creates some lovely images – but they refer to phenomena in mathematics that have stumped and motivated mathematicians for centuries. Unlike those historical sages, though, you can actually see the patterns, and explore the effect of changing formulas and substituting different spiral layouts by just clicking a few buttons and dragging some sliders.

The effect of all this is that you’re getting an intuitive sense of the concept of prime number distribution through what is effectively play. You can even formulate some hypotheses of your own and then quickly test them through the interactive interface. Now that’s 21st century style learning! But a site like primes.io is just a gateway to a land of greater wonder, perhaps not as interactively fun or artistically compelling, but full of deep, historically resonant math ideas. Following one of the links from the website, I ended up reading about the Riemann Zeta function and the Riemann conjecture on Wikipedia. While these pages look more like a traditional math textbook, thick with formulae and seemingly obtuse notation, they’re actually much better written than the average math textbook of yore. I even found myself reading a detailed proof of the Euler product formula and thoroughly enjoying it, something I hadn’t experienced in ages.

Prime number distributions are sort of esoteric, but everybody loves fractals, right? These beautiful mathematical beasts are now being mined for pure artistic value but there’s one website out there that starts from the basics – adding and multiplying numbers – and leads you all the way to constructing a genuine, all-out fractal. This article on acko.net accomplishes this amazing feat once again with the help of WebGL. The page is divided into several tutorials, each one carefully building the geometric intuitions needed to follow the mathematical construction. Even if you don’t follow every detail, you get a sense of the process, and can see, right before your eyes, a fractal beginning to form as a result of repeatedly applying simple geometric rules.

While I hope to see the above two websites start a larger trend of WebGL based mathematics learning resources, there are some great videos to enjoy in the meantime. For an elephant dose of hyperactive, breathless inspiration, you need go no further than the excellent Kahn Academy _Doodling in math_ series. A good example is this video, concerning the prevalence of Fibonacci spirals in nature.

The web is full of excellent educational resources for mathematics – the quality and quantity of materials freely available makes me wonder if I would ever have lost my math bug had this been the case when I was in graduate school. Wonder is out there – all you have to do is seek it out, read up on it, and perhaps if you’re feeling old-fashioned, pull out some scratch paper and pencil to work some details out for yourself.

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Maciek Smuga-Otto

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