Sketch · 2026-08-05
Julia set, picked from the Mandelbrot.
One rule: z becomes z squared plus c. Fix c and vary the starting z across the plane, and you get a Julia set. Fix z at zero and vary c instead, and you get the Mandelbrot set. Every point of the Mandelbrot is the address of one Julia set. Click anywhere on the left canvas to pick a c; the Julia set for that c redraws on the right.
Click or drag on the Mandelbrot to steer c
Escape counts
Each pixel is a starting z. The iteration runs until the modulus of z passes two, or a cap is hit. Points that never escape are drawn as the set itself. Points that escape are shaded by how long they took: the fastest to leave are lightest, the slowest are darkest. The threshold of two is not decorative. Once the modulus of z exceeds two, one more square already grows faster than c can pull back, and the sequence is guaranteed to run to infinity.
Why the Mandelbrot indexes the Julias
The Mandelbrot set is exactly the set of c values for which the orbit starting at z equals zero stays bounded. A theorem of Fatou and Julia says the same c also determines a Julia set that is connected. Pick c outside the Mandelbrot and the Julia set shatters into Cantor dust; pick c on the boundary and the two sets rhyme in a way you can see. The little bulbs of the Mandelbrot each correspond to Julia sets built from a specific attracting cycle: the round bulb on the left of the main cardioid gives period-two Julias like the Basilica; the bulbs up top and below give period-three, period-four, and so on.
Beyond the square
The exponent slider swaps the iteration for z to the power n plus c. At n equals three the Mandelbrot grows a mirrored second lobe and the Julia sets pick up threefold symmetry. Higher n gives an n-lobed set with n minus one symmetry in the Julias. The same escape argument still holds, only the escape radius grows a bit.
Sources
- Julia, G. (1918). Mémoire sur l'itération des fonctions rationnelles. Journal de Mathématiques Pures et Appliquées, 8, 47–245. The paper that gave the sets their name.
- Fatou, P. (1919–1920). Sur les équations fonctionnelles. Bulletin de la Société Mathématique de France, 47–48. The parallel treatment that fixed the vocabulary of orbits and basins.
- Mandelbrot, B. (1980). Fractal aspects of the iteration of z → λz(1−z) for complex λ and z. Annals of the New York Academy of Sciences, 357(1), 249–259. The first published picture of what is now his set.
- Douady, A. and Hubbard, J. H. (1984). Étude dynamique des polynômes complexes. Publications Mathématiques d'Orsay. The proof that the Mandelbrot set is connected, and the framework the bulbs live in.