Saturday, August 04, 2007

The Downward Spiral

In the Japanese horror movie 'Uzumaki', a small town is overtaken by a mysterious plague of spirals. The inhabitants start seeing them everywhere and are gradually driven to insanity - one woman suddenly realizes there are spirals in her inner ear with predictably grisly results.

You could be forgiven for thinking that something similar is going on in the Mandelbrot set and other fractals. There are spirals everywhere - is something sinister going on?

Well, sort of. The Mandelbrot is infested by spirals because they are the natural result when you repeatedly perform complex multiplication, as you do when constructing the M-set. This becomes more obvious when you represent complex numbers in polar coordinates.

Say you multiply z1 (= r1 e^(i a1)) by z2 (=r2 e^(i a2)). The result is  r1 * r2 * e ^(i a1+a2) - in other words, you've made it a bit bigger and rotated it a bit - and if you keep doing that, you'll create a spiral.

This is fine if you're a spiral fan. But what if, like the unlucky folks in Uzumaki, all these spirals are getting a bit much. Is there any escape?

Well, pretty much any fractal based on multiplying complex numbers is going to have them in there somewhere. But if you ensure there isn't any 'straight' multiplication in your formula, that will help.

For example one of my favorites, the 'Burning Ship' fractal, 'absolutifies' the value after each iteration (z = |z^2 + c|), which serves to break up the multiplication:

A zoom-in to the lower left of the Burning Ship fractal, showing a

This has some nice 'tapestry-like' detail if you zoom in:


Other functions which can help stamp out spirals include conj & flip.


Blogger Tim said...

I got to the point where I got excited over any formula or rendering method that had straight lines or squares in it and no spirals at all. Spirals can be interesting but I think they've got to be pretty special and have some rare attributes before they excite anyone anymore, otherwise they're a little tiring.

Your second image, the blue and black one, has a real west coast indian style to it, like the Haida.

8/04/2007 2:16 PM

Blogger Dzeni said...

I have a "thing" for spirals, but they do get "old" after a while. An interesting post :)

8/05/2007 1:22 AM


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