The Source of all Good Bits

This is a followup to my What is a Buddhabrot anyway?-post where I only described the mathematical backgrounds of the Mandelbrot Set and its relationship to Buddhabrot renderings. This post will describe how I actually created the images.

As described in the linked article above one needs to find such points for which the Mandelbrot iteration becomes unbounded. Then one draws each point of such an unbounded iteration, repeats that for lots of other points and gets a Buddhabrot.

While the words sound easy enough, there are certain problems: how does one know that an iteration becomes unbounded when unbounded basically means that “it never ends up somewhere”. And then how do we find those interesting points and moreover lots of them within a lifetime?

Luckily it has been found out that if the continued iteration results in points that are farther than 2 units away from the origin of the coordinate system, that this iteration is unbounded. That means that if we started with a point that leads to an unbounded iteration we will sooner or later know it and can stop iterating.
However, if we started with a point that leads to a bounded iteration it will never lead to points farther than 2 units away from the origin and if we don’t stop using some other criterion we will have an infinite loop. This other criterion is quite simple: set some maximum iteration count and whenever you exceed it consider the iteration bounded and thus not interesting for the Buddhabrot rendering.
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