Built on Facts

The brilliant folk-rock musician Jonathan Coulton has written a song about the Mandelbrot set. The set is a quite neat and famous mathematical object, and that it’s crossed over into immortalization in song is one of those beautifully weird things we are privileged to see in the internet age. No, it’s not really physics but I don’t think a purely mathematical diversion ever hurt anyone.

Any number is either in the set or not. Here’s how you tell. Pick a number. Square it. Add the original number to the result. Repeat.

If the magnitude of the result you get becomes larger and larger without bound, it’s not in the set. If it stays bounded, it is. Take the number 1, for instance. Square it, you get 1. Add 1, you get 2. Square it, you get 4. Add 1, you get 5. Square it, you get 25. Add 1…

You get the picture. 1 is not in the Mandelbrot set. Try the same thing with 0, and of course you just get 0 every time. So 0 is in the Mandelbrot set. Those are just examples, the set itself is extremely complicated, and in fact its boundary is fractal when you include the complex numbers. Here’s a picture of the set so you can get an idea of where the cutoffs end up being for boundedness. You can see a huge version if you click on it [image credit Wikipedia, by the way]

You can see that along the real line, the Mandelbrot set is [-2, 1/4]. But strictly speaking the fact that a number is in the set doesn’t tell us much about what happens as we continue to iterate forever. Maybe it does nothing (like 0), or skitters around crazily in the complex plane, or oscillates between values, or converges gradually to a single value. But just for fun, let’s figure out what happens to the right endpoint (1/4) as we iterate forever. Departing a tiny bit from Mr. Coulton’s notation in his song, let’s denote the iteration procedure as

Iteration a hundred thousand times gives a value of 0.4999900013, which is suggestive but not definitive. We’re trying to find the limit as n approaches infinity, and brute-force computation won’t do it. Let’s a assume such a limit exists: if so, then the successive change from one z to the next will be smaller and smaller. Thus the requirement for convergence is just to say that

So what we’re after is the value z eventually approaches as n gets really huge. So take the limit of the defining iteration for the Mandelbrot set.

But we can substitute that convergence criterion we found above into this.

And this is just a quadratic equation.

The initial z value we’re interested in is 1/4 endpoint, so solving the quadratic gives a limit of 1/2 exactly as we might have suspected from the numerical computation.

There is a HUGE hole in this proof however. We have completely failed to prove that the sequence actually converges. If it doesn’t, our convergence criterion fails and the result is worthless. Though the general case of proving convergence or divergence for an arbitrary point in the set is impossible, we can in fact do it for the case of 1/4. It does converge. In fact, the proof is not difficult. But I’ll leave it for you as an exercise in case you’re bored this weekend!