README.md

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Fractalbot

A bot that generates a random fractal and posts it to Mastodon.

How it works

The bot renders a Julia set via a distance-estimation method. Each Julia set is determined by a complex function $f_c(z) = z^2 + c$ depending on parameter $c$: it is the set of complex numbers $z$ for which the sequence

$$z_0 = z,\ \ z_{n+1} = f_c(z_n) = z_n^2 + c$$

stays bounded.

In order, the bot does the following:

1. Read the complex parameter $c$ from the command line (-c), or sample it randomly.
2. Approximate the bounding box of the set via inverse iteration. This method yields a sequence of points on the boundary of the set from which the bounding box is computed.
3. Scale the bounding box to 120% to add a slight margin.
4. Select a color palette, depending on whether the set is connected or not. A Julia set is connected if and only if iteration for $z_0 = 0+0i$ stays bounded (i.e. $c$ is contained in the Mandelbrot set).
5. For each pixel in the final image, estimate the distance of the corresponding complex point in the bounding box to the set. For distances $\leq 0$, the point is assumed to lie inside the set and is colored black, otherwise a color is sampled from the palette. Color palettes are generated following Inigo Quilez' amazing tutorial. Rendering is parallelized using rayon.
6. Depending on the mode, save the image to disk or post it to Mastodon. To interact with Mastodon, megalodon is used.