Friday, June 28, 2013

more friday fractals

Here are some more variations on Mandelbrot-like fractals, similar to those from a few weeks back, and all made by playing around with the Processing program described here.

The recursive formula used to generate the Mandelbrot set is quadratic - here are some variations that use different powers in a formula that is otherwise the same as the Mandelbrot formula.

A strange variation on the Mandelbrot theme is the burning ship fractal. It is generated in a similar way, but the first step in calculating each term is to take the "absolute value" of the previous term. Actually abs(z) is a complex number whose coefficients are the absolute values of the corresponding coefficients of z. Although not apparent in the picture below (see hpdz for deeper images) , just as in the fractals above there are small copies of the larger image (off to the left there, in the corner)

These variations on the burning ship use different powers with surprising results. Personally, I find these a bit sinister looking - each roughly polygonal form has strange organic looking out-growths and hidden small scale replicas of itself.

The last Mandelbrot variant shown here is sometimes known as the Tricorn or Mandelbar - so named because the first step in computing a term in the z_n sequence is to take the complex conjugate of the previous term (bar-z). Surprisingly you get three smeared copies of the Mandelbrot set radiating out of the vertices of what looks like a hypocycloid.