The Mandelbrot set, arguably the most famous fractal ever discovered, has a group of very close cousins — more like children, in fact. Where the Mandelbrot set iterates the function \(f(z) = z^2 + c\) with \(c\) chosen as the starting \(z\)-value, Julia sets choose \(c\) beforehand and use it regardless of the starting \(z\)-value. This means that points close to \(c\) will look similar to the Mandelbrot set at that point, with the resemblance fading when moving farther away. To emphasize that resemblance, this applet uses the Mandelbrot set to choose \(c\). Hover over the Mandelbrot set to preview the corresponding Julia set, and click to generate a higher-resolution preview.
Enter values of \(a\) and \(b\), along with an image size in pixels, to generate an image of the Julia set for \(c = a + bi\). The values are filed in automatically when a high-resolution preview is created above, but custom values can also be used.