The Buddhabrot is a beautiful fractal shape, generated through similar iterations to Mandelbrot set images. A brief description can be found below and more details can be found at https://en.wikipedia.org/wiki/Buddhabrot.
Buddhabrot visualises a probability distribution over the complex numbers. We start by paritioning the complex plane into bins. Complex numbers are sampled uniformly from a circular region around the origin of the complex plane, then iterated according to the same rule as the Mandelbrot set: Z(n) = Z(n-1)^2 + c, where c is the sampled point and Z(0) = 0. If the point remains close to the origin after a predefined number of iterations k, we discard the sample, however if the point diverges then we increment each bin associated with a point in {Z(0),...,Z(k)}. By sampling many points this gives us a tally accross all the bins. Then, by normalising these bin frequencies to values between 0 and 1, we can turn this into a greyscale image.
Further, by creating 3 different tallies with a different number of iterations, we can create a false colour image whose colours come from the 3 tallies (one for red, green and blue, giving an RGB pixel). The image at the top of the page is composed of the following three greyscale images: red with k = 5000, green with k = 500 and blue with k = 50
These differences mean that the only real parameters to the images are the samplesize, the max number of iteraions and the resolution of the image. As each pixel is a bin, resolution doesn't actually increase compute time (unlike for Mandelbrot images) but the samplesize and max number of iterations do. The following gifs demonstrate how varying each of these parameters affects the result:
Iteration limit
Sample size
Resolution
To generate your own images, you can run generate_greyscale.py or generate_rgb.py. Adding a -h flag will show options for customising the output image. Note that these scrips will look for existing tallies stored in the csvs directory with the specified resolution and iterlimit. The number of new points specified will then be sampled and either added to an existing tally (if it exists) or to a zero-matrix. You can force a fresh start with the flag --fresh.