Julia

The Mathematics

Fix a complex number c and iterate z ↦ z² + c; the points whose orbits never escape to infinity form the Julia set of that c. This piece gathers many such sets together. Asking, for each c, whether its Julia set holds together in one connected piece traces out the Mandelbrot set — so the collective portrait of every Julia set reveals this singularly famous fractal moduli space.

Technique

Every pixel is computed in parallel on the GPU. We first use the equilateral-triangle reflection group to divide the plane into hexagonal tiles. Taking the center of each tile as the parameter c, a small GPU program then runs across the interior of that tile to compute the Julia set for that c. Running this across thousands of tiles at once assembles the final image.