ramanujan's Π summation

When G.H. Hardy first began his correspondence with Srinivasa Ramanujan, he was amazed by the prodigy's elegant and surprising series for irrational and transcendental values. Although Ramanujan hadn't included proofs of the theorems, Hardy told his colleagues, "they must be true, because, if they were not true, no one would have the imagination to invent them."

For example, here's Ramanujan's sum for Π which, after just 3 terms, overwhelms IEEE 754 double precision. Modern discoveries, like spigot algorithms on non-decimal bases, can find a given individual digit in constant time. But for fully expanding Π, no one has surpassed Ramanujan's expression: both the Chudnovsky brothers and Yasumasa Kanada adapted it to achieve their milestone calculations.

butterfly curve

Temple Fay discovered this complicated curve in 1989. It can be defined either parametrically or as a polar equation; I did it the former way.

One application I thought of for this is object motion in games: I tried it out by writing this little Canvas game, where the comets follow the curve's trajectory. The differences in plot density along the curve create natural-looking comet tails.

bézier curves

De Casteljau's algorithm is a fast, numerically stable way to rasterize Bézier curves. This code implements an interactive demo: click to append nodes to the curve's defining polygon, and drag any node to alter the curve.

more Π obscurantism

Here's an illustration of an approach to Π pointed out by Kevin Brown. Define f(n) as the nearest greater or equal multiple of n-1, then of n-2, etc (yielding OEIS sequence 2491). Then, inverting a result found by Duane Broline and Daniel Loeb, Π = n² / f(n).

But as you can see from the comment, the series converges very slowly!

today in oblique approaches

If your standard library offers complex numbers and you're not in any hurry, you can't ask for much simpler (or more obscure!) ways to compute Π than this.

hénon attractor

French astronomer Michel Hénon reported on this strange, fractal attractor in 1976. Since then, it has been among the most studied examples of chaotic dynamical systems.

lyapunov fractals

It took me some experimentation to figure out how to color this derivation of the logistic map; I'm still not quite sure how the hues should scale as you zoom. But the bi-tonal method shown below works well enough to produce the image at left - click it for more detail.

sqrt 2, visualized

This code, inspired by my earlier post, animates the digits of √2 in an unsuual way. Each digit advances the curve in the direction given by a numeric keypad, with 5 and 0 both considered (0,0). The thumbnail at left links to a 4000x4000 window on 9856041 digits of the curve, which originates at the image center.

The program makes for an interesting screensaver, but does it follow any pattern, or illustrate any special properties of √2 ? Though I'm no mathematician, from what I've read this seems unlikely. √2 is suspected (but not proved) to be a normal number. This would mean the digits of its expansion (respective to some number base) follow a uniform distribution; no digit would be more likely to appear than any other.

Nonetheless, I'd be curious to see the results of a really long run!