Mandelbrot & Julia
Iterate z ← z² + c starting from z = 0; if the orbit stays bounded, c belongs to the Mandelbrot set. Move the cursor over the Mandelbrot to set c, and the right panel draws the corresponding Julia set. Connected Julia sets correspond to c inside the Mandelbrot.
Mandelbrot set (hover to set c)
Julia set for c = −0.7 + 0.27i
why this map matters
Mandelbrot (1980) noticed that the simple iterated map z² + c produces a boundary of infinite complexity. Its boundary has Hausdorff dimension 2 — it fills space in a measure-theoretic sense — yet has zero area. Every detail at every scale looks like the whole thing distorted.
Each point c in the Mandelbrot generates a corresponding Julia set. Julia sets for c inside the Mandelbrot are connected; for c outside, they're Cantor dust. The fact that picking different "control parameters" in a simple equation gives you topologically distinct universes is the fingerprint of complex dynamics, a field that came out of nothing in the late 1970s and now connects to: holomorphic dynamics, renormalization theory, and even random matrices and the Riemann hypothesis.