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 (PIL).
More details can be found at the project page - Fractals.
The equation for Julia Sets is z = z2 + c for a complex number z and constant c.
Modify the following parameters in julia.py to generate your own animations:
for angle in np.arange(start, stop, step):
c = complex(k1 + k2*math.sin(angle), k3*math.cos(angle))
Here, start and stop denote the angles to begin and stop the animation at, incremented by step for each Julia Set. The constant c can also be a complex number, which has constants k1, k2 and k3 in this case and is a parameter that can be tweaked arbitrarily.
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.
- Visualizing Julia Sets for varying values of the complex constant c = A + i B in f(z) = z2 + c. Here, A = -1 + 0.3sin(θ) and B = 0.3cos(θ), with θ varying from 0 to 2Π, in increments of 15°.
