Fractals are among the most fascinating mathematical visuals capturing infinitely complex patterns that are self-similar and exhibit finer detail when magnified recursively. This project visualizes some of these fractals, particularly Julia Sets and the Mandelbrot Set. The latter popularly depicts the aesthetic beauty of mathematics. Fractals are witnessed in nature as well, in snowflakes, leaves and blood vessels, to name a few.

All images were generated using Python and the Python Imaging Library, the code for which can be found here.

Julia Sets

Images of Julia Sets obtained for various values of the complex constant c in f(z) = z2 + c

The Mandelbrot Set

The Mandelbrot Set is also computed using the function f(z) = z2 + c, except that c differs for each pixel. Thus the Mandelbrot Set can be considered as a map of all Julia Sets because it uses a different c at every location.

Julia Sets Animation

Click here to view an animation of all the Julia Sets obtained by varying the complex constant c = A + i B in f(z) = z2 + c. Here, A = -1 + 0.3*sin(θ) and B = 0.3*cos(θ), with θ varying from 0 to 2Π, in increments of 15°.