A couple months ago, I used my Mandelbrot generator to create several animations. The purpose of these animations was to gain a better understanding of the paths that an iterated point takes before escaping. I knew that they didn’t travel very far around the number plane – but besides that, I didn’t know much. This project shows how the Mandelbrot set behaves when the escape threshold is not a circle.

The ideal tool to understand how the points move would be a Buddhabrot generator that can limit displayed paths to show only the points around the mouse pointer. I’m still interested in doing that, but this provides other useful information.

In all of these image, yellow is a low escape angle, magenta is a high escape angle, and higher iteration counts shift the color through blue, green, and red. This is very faint around the edges of the set. Whenever I would need to divide by zero to find the escape angle, I instead used magenta – this resulted in the dark lines through the middle of yellow regions.

In the rotating parabola experiment, I mainly wanted to see how far a point could travel in a straight line. If I used two planes as the escape shape, certain points would never escape (any point where i=0 for example.) The parabola guarantees that points will escape eventually. In this test I actually rotate the points before testing, while the parabola stands still – but for the same effect.

The parts of the set that are near the center of the parabola are the most defined, taking the longest to escape. Each shape within the parabola has a purple strip which is offset from the purple strip in the next shape. To me this implies that the longer a point is iterated, the larger the step that finally takes it out of the parabola.

I created several other animations with different escape shapes. They demonstrate another property of the Mandelbrot set – No matter how you define “escaped,” as long as it is outside of a radius 2, no new region added will join the opposite sides of the set; The set remains connected.

The set contains the same points no matter what the escape shape is – so using a square escape is actually a decent shortcut to remove the distance equation from each iteration.