Mandelbrot.
The Mandelbrot set of the complex quadratic map z ↦ z² + c, rendered with smooth escape-time colouring. Drag to pan, scroll or use the buttons to zoom — there's structure all the way down. Hover any pixel and the inset shows the Julia set for that c.
What you're looking at.
A complex number c belongs to the Mandelbrot set iff the orbit of 0 under z ↦ z² + c stays bounded forever. Everywhere outside the set, the colour encodes how fast that orbit escapes to infinity — the smoother the gradient, the slower the escape near that direction. The n + 1 − log₂ log|z| smoothing turns the integer escape count into a continuous value, which is why the bands look continuous rather than stair-stepped.
The Mandelbrot and Julia sets are joined at the hip: the Mandelbrot set is precisely the locus of c for which the corresponding Julia set is connected. Pixels inside the main cardioid produce simple filled-disc Julias; pixels on the boundary produce delicate dendritic Julias that look like the Mandelbrot itself locally; pixels far outside fragment into a Cantor-dust of disconnected points — totally disconnected. Hover the cursor across the boundary in the inset and watch the topology change.
The set's boundary has Hausdorff dimension 2 — wrinkled enough that, in a measure-theoretic sense, it fills the plane while still enclosing zero area. Pick any point on the edge and zoom; you will never run out of structure (until double-precision floats start quantising the plane around 10¹³ zoom). The deep eddies, mini-brots and seahorse valleys are all the same equation, just iterated on smaller and smaller windows.