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!

segment intersection search

Here's a simple method I learned from Gareth Rees for finding the intersection (or parallel / collinear status) of two line segments. Rees credits Ronald Goldman, though I'd imagine the technique goes further back.

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.

ikeda map

Continuing the theme of strange attractors, here's the well-known one embedded in the Ikeda map. The thumbnail at left shows the central 'vortex' of the attractor, and links to a larger viewport.

In these images, I've plotted the real and imaginary components along the x and y axes respectively. But the more popular way to visualize this attractor adds an extra parameter to the system and is expressed in trigonometric functions. Such adaptation of the code below yields these results.

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.

langton's ant

For a round of code golf, I wrote this spare implementation of Chris Langton's remarkably simple universal computer. If you want amenities like pause, random starting pattern or even quit, check out this more complete version.

connett circles

Like Barry Martin's 'Hopalong' fractal, this dynamical system from John Connett was first published in Scientific American in 1986. This demo is interactive: successively clicking two points specifies a rectangle to zoom into. Doing so, you'll see that the system isn't actually a fractal. Instead of self-similarity, deep zooms reveal peacock-like images.

martin attractor

This pattern generator, discovered by Barry Martin, was nicknamed 'Hopalong' when Scientific American introduced it in their September '86 issue.

Clicking the window adjusts the viewport position; there is also an alternate version with color and animation.

Also, for a certain Rubyist friend, I wrote another lazily-evaluated, colored and animated implementation in Ruby.

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.

the mandelbrot set

What programmer hasn't at some point written an implementation of Benoit Mandelbrot's great discovery, the most famous fractal in the world? Here's my own minimal version, with the simplest possible coloring scheme. To interact with it, just click any two points: the window will zoom in on the rectangle they define.

a lava lamp

This 'lava lamp' is actually a simple cyclic cellular automaton; the CA rule is courtesy of Jason Rampe. In keeping with the spirit of Haskell, I chose to implement it with a hashmap of points rather than an array. Needless to say, that's not practical; but this is just a demonstration.

random undirected graphs

This little program generates random, undirected graphs without loops or isolated vertices (though the graph is not necessarily connected).

I have found it useful to generate random inputs for testing graph algorithms.

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!

triangle geometry

Here's a trivial post, mostly just to show off how nice mathematical code can look in Haskell. It illustrates the centroid, incircle, and circumcircle of a random triangle, with everything defined in terms of vertex triples. The viewport centers on the circumcenter, and the triangle's interior is filled using barycentric coordinates.

primitive totalistic automata

This code renders any of the 2187 possible 3-colored, 1-dimensional, totalistic cellular automata. I was charmed by these and many other beautiful demonstrations in Stephen Wolfram's notorious compendium, though I regret I can't say the same for its tendentious style.

The program input is an integer representing the intended CA rule in base 3.