One tiny equation, absurd complexity
The Mandelbrot set lives in the complex plane. Choose a complex parameter c, start at z=0 and repeatedly apply:
If the orbit remains bounded forever, c is in the set. If |z| ever exceeds 2, that orbit is guaranteed to escape. Computers cannot test forever, so renderers stop after a chosen iteration limit.
Cardioids, bulbs, valleys and an impossible boundary
The largest heart-shaped region is the main cardioid. Circular bulbs attach at organized rational positions, while the boundary explodes into antennae, spirals, seahorse-like tendrils, elephant corridors and tiny copies of the full set.
On the real axis, the set occupies exactly the interval from −2 to 1/4. The boundary has Hausdorff dimension 2, capturing just how densely intricate it is.
The Mandelbrot set is a map of Julia behavior
Hold c fixed and vary the starting z to create a filled Julia set. For the quadratic family, parameters c inside the Mandelbrot set correspond to connected filled Julia sets; parameters outside correspond to disconnected ones. Every point in parameter space therefore describes another dynamical world.