A Julia Set Mosaic

The Mandelbrot set and Julia sets share a close relationship, as emphasized in the latter applet: the Julia set with parameter \(c\) “looks like” the Mandelbrot set at \(z = c\). But how close is the resemblance? Something we might try is picking a large number of points in the complex plane and drawing a small Julia set at every one. We’d expect to get something vaguely resembling the Mandelbrot set — but it’s surprising just how accurate that picture is.