Julia sets are named after French mathematician Gaston Julia, who — alongside Pierre Fatou — studied the iteration of complex functions during the First World War, decades before computers existed to visualize the results. Where the Mandelbrot set fixes the starting point at zero and varies c across the whole image, a Julia set does the opposite: c is fixed for the entire fractal, and every pixel becomes its own starting value z(0). Iterating z = z² + c from each point either keeps it bounded (colored black, part of the "filled Julia set") or sends it to infinity at some rate (colored by escape speed).
The shape of a Julia set changes dramatically with c. Values of c taken from inside the Mandelbrot set tend to produce a single connected fractal — swirling spirals, feathery dendrites, or the famous rabbit-eared "Douady Rabbit". Values of c from outside the Mandelbrot set shatter the Julia set into an infinite dust of disconnected points, a "Cantor set" scattered across the plane. This deep relationship is exactly why the Mandelbrot set exists: it is a map of every possible Julia set, colored by whether that set is connected.
Drag the two c sliders (or tap a preset) to morph continuously between these forms in real time, then scroll to zoom and drag to pan into the fractal detail at your chosen c value. The Spiral preset uses the same c as the algorithm's classic textbook example; try the Dendrite (c = i) for a branching tree with no interior region at all, or Airplane for a fractal that resembles exactly what its name suggests.