Julia Set Generator

This code creates sequences of images of Julia sets by iterating complex numbers through the function f(z) = z^p + c for complex constants c and (usually) integer exponents p. Images are RGB .png files and can be combined to form animations.

Requirements

These scripts are run in Python 2.7.9. The following third-party packages are used:

• numpy - efficient array operations with complex numbers
• numexpr - optimizes numpy array operations
• PIL.Image - creates images from numpy arrays and saves them to disk
• pathos - allows for multiprocessing to be used in Julia.py with JuliaTools.py imported as a module

Running the scripts

First, run JuliaInputs.py. This creates the file input.json which contains all of the constants that are read into Julia.py. You can now run Julia.py, which imports JuliaTools.py as a module. The sequence of Julia fractals will be written to images/.

Output

There are three types of image sequences that Julia.py can create.

• Linear - Every frame shows a fixed region of the complex plane centered at the origin. Each successive frame is made by moving c along a straight line in the complex plane. (Example)
• Radial - Every frame shows a fixed region of the complex plane centered at the origin. Each successive frame is made by moving c along a circular arc centered at the origin. (Example)
• Zoom - Every frame shows the same Julia fractal, i.e. c is not changed. The center of each image is also constant, but it may be centered anywhere in the plane. Each successive frame shows a slightly smaller or larger region of the complex plane to give a zooming effect. (Example)
• Power - Every frame shows a fixed region of the complex plane centered at the origin, and c is fixed. The exponent p in f(z) = z^p + c is changed in each frame, usually taking on fractional values.

Twitter Bot

This code is being used by the twitter bot @JuliaFractalBot to generate and tweet videos of Julia fractals. A sequence of images produced by Julia.py is converted to a .mp4 video compatible with Twitter using ffmpeg:

ffmpeg -r 30 -f concat -i framelist.txt -vcodec libx264 -crf 12 -pix_fmt yuv420p -s 1024:1024 -aspect 1:1 JuliaVid.mp4

The video file is then asynchronously uploaded to Twitter using large-video-upload-python. One can also create an animated .gif from the images:

ffmpeg -r 10 -f concat -i framelist.txt JuliaGif.gif

This bot uses .mp4s over .gifs because Twitter's file size limit for videos is 512MB, while it is only 15MB for animated .gifs.