Julia sets

On the left is the Mandelbrot set; on the right, the Julia set of z ↦ z² + c. Drag c around the left half. Inside the main cardioid the Julia set is a single crumpled loop; move into one of the bulbs on its rim and the loop pinches together at many points, with as many lobes meeting at each pinch as the bulb's period. Take c outside the Mandelbrot set and the Julia set falls apart into dust.