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Where the escaping points go
The Mandelbrot set is made by taking a point c of the plane, starting from z = 0, and repeating z → z² + c. For some points the numbers stay small forever: they are in the set, the black shape of the famous pictures. For the others the numbers grow and run off to infinity, and the usual picture colors each one by how fast it escapes.
In 1993 Melinda Green asked a different question: not how fast a point escapes, but where it goes on the way out. Take millions of random points, follow the path, the orbit, of each one that escapes, and count how many times those paths pass through every pixel. Shown as a long exposure, the counts reveal a figure seated in meditation, with a head, a halo and folded hands. Lori Gardi named it the Buddhabrot. Move the pointer over the picture to see the orbit of the point under it: each dot is one step.
Max iterations sets how long an orbit may wander before it counts as never escaping. With a few dozen you get a soft ghost; with thousands, the long orbits that start close to the edge of the set draw fine threads and filaments. The Nebulabrot renders three Buddhabrots at once, with different limits, in red, green and blue, the way astronomers color the filters of a Hubble photograph. The Anti-Buddhabrot does the opposite and draws only the orbits that never escape: they settle into loops around the points of the set.
The picture is computed in your browser by as many workers as your processor has cores, and stops by itself once its brightest parts have collected a few thousand points per pixel, when more would not show. The figure is symmetric, so each orbit is drawn twice, mirrored, and the points inside the main cardioid and the big disk, which are known never to escape, are skipped. A first quick pass also maps where the useful points are, so the random points are taken only there, and an orbit that starts going round in a cycle is dropped at once: it will never escape.