Both pictures come from repeating the same rule: $z_{n+1} = z_n^2 + c$. On the left, $c$ is the pixel itself and $z_0 = 0$ — that's the Mandelbrot set. On the right, $c$ is fixed (whatever you last clicked on the left) and $z_0$ is the pixel — that's the Julia set for that particular $c$. Click anywhere on the Mandelbrot to pick a new $c$ and watch its Julia set change on the right.

The Rule

Pick a complex number $c$. Start with $z_0 = 0$ and repeat $z_{n+1} = z_n^2 + c$. For some values of $c$, this sequence stays bounded forever; for others, it flies off to infinity. The Mandelbrot set is exactly the set of $c$ for which it stays bounded. Points are colored by how fast they escape (once $|z| > 2$, it's guaranteed to diverge) — that escape-time coloring is what produces the detailed, colorful boundary rather than a flat black-and-white shape.

Julia Sets Are the Same Rule, Reframed

Fix $c$, and instead ask: for which starting points $z_0$ does the sequence $z_{n+1} = z_n^2 + c$ stay bounded? That's the Julia set for $c$. Every point in the Mandelbrot set corresponds to a connected Julia set; every point outside it corresponds to a Julia set that's dust — totally disconnected. In this sense the Mandelbrot set is a map of Julia sets: it tells you, for every possible $c$, what shape of Julia set you'd get.

What to Try

• Click near the boundary of the Mandelbrot set (the black shape's edge) — that's where the most intricate Julia sets live.
• Click deep inside the main black cardioid — the corresponding Julia set becomes a simple, connected blob.
• Click just outside the set entirely — the Julia set falls apart into disconnected dust.
• Zoom into the Mandelbrot boundary and look for smaller copies of the whole set — they recur infinitely, at every scale.