gumowski-mira attractor

Here's an unusual chaotic attractor: the web doesn't seem to have much information on this one, except that it was invented to model particle trajectories in physics. A google image search for 'mira fractals' does turn up some pretty results though.

The system seems to give interesting results only when b is close to one. It behaves less chaotically when b > 1 is fixed, so you can actually animate it - click the thumbnail to see.

128-bit AES electronic codebook

Rijndael (the core of AES) is an algorithm that might actually be shorter and simpler in C, but I was curious to see how natural I could make it look in Haskell. Thanks to Jeff Moser and Sam Trenholme for their clear elucidations.

Note that this code only does ECB mode; it computes rather than hard-codes the S-box; and it could be vulnerable to side-channel attacks. So enjoy reading it, but don't try to make a serious encryption app out of it. That kind of thing is best left to the professionals :)

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.

convex hulls

This code demonstrates the Graham scan, an O(n log n) method for finding the convex hull (smallest enclosing polygon) of a planar point set. It's a great example of exploiting order: it works by sorting the set by angle about a known extreme point, allowing the hull points to be found in linear time.

A variation on the algorithm, noted by A.M. Andrew, sorts the set lexicographically and finds the upper and lower hull chains separately. This 'monotone chain' version is often preferred, since it's easier to do robustly.

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!