An infinite family of fractals indexed by the Mandelbrot set
The Mandelbrot set and Julia sets use the same formula, but differently! For Mandelbrot, c varies (the parameter); for Julia, c is fixed and z₀ varies. Each point in the Mandelbrot set corresponds to a connected Julia set—the Mandelbrot set is a "map of Julia sets"!
This is the key theorem: if c is inside the Mandelbrot set, the Julia set is connected (one piece). If c is outside, the Julia set is totally disconnected—"fractal dust" with infinitely many pieces. The Mandelbrot boundary is exactly where Julia sets transition between these states!
Julia discovered these sets in 1918 without computers! He won the Grand Prix of the French Academy but his work was forgotten for 60 years. Benoit Mandelbrot rediscovered it in the 1980s when computers finally allowed visualization of these mathematical objects.
From a single simple formula, an uncountably infinite collection of completely different fractals emerges. Move c slightly and the Julia set transforms dramatically—from smooth curves to wild dendrites to scattered dust. How can one equation encode such infinite variety?