A Tour in the Wonderland of Math with Python

A collection of python scripts for drawing beautiful figures and animating interesting algorithms in mathematics.

UPDATE: This repo is currently under a huge reconstruction, more fancy and advanced features will be added, some dirty code will be removed. Also I will make more shadertoy live demos to help people understand the math. So you can look forward a more fantastic version :smile:.​

Known issues:

1. The code for drawing E8 root system is not mathematically correct (though the result image looks right), I didn't take the bilinear form into account. Same reason for drawing polytope on its Coxeter plane in polytopes project.

About this repo

The purpose of this project is to show the beauty of math with python. It consists of several independent subprojects with each one illustrates a special object/algorithm in math. The current list is:

• Aperiodic tilings like Penrose tiling, Ammann-Beenker tiling, etc.
• Triology on perfectly random sampling algorithms. See
  1. Domino shuffling algorithm on Aztec diamonds
  2. Wilson's uniform spanning tree algorithm on 2d grids
  3. Coupling from the past algorithm on lozenge tilings.
• Hopf fibration
• 3D and 4D Uniform polytopes.
• 2D and 3d Uniform tilings in Euclidean, spherical and hyperbolic spaces.
• Make gif animations of various algorithms.
• Reaction diffusion simulation with pyglet and glsl.
• Raymarching fractals with pyglet and glsl.
• Raymarching Möbius transformation animations with pyglet and glsl.
• Limit set of rank 4 hyperbolic Coxeter groups
• Miscellaneous scripts like E8 root system, Mandelbrot set, Newton's fractal, Lorenz attractor, etc.

These topics are chosen largely due to my personal taste:

1. They must produce appealing results.
2. There must be some non-trivial math behind them.
3. The code should be as simple as possible.

I'll use only popular python libs and build all math stuff by hand (tools like sage, sympy, mathemetica will not be used here). Also I will only maintain the code for python >= 3.6.

Note: Python3.5 is deprecated now because it's a bit tricky to install the latest numba on Ubuntu16.04 for python3.5 (if you are using anaconda for package management then you need not worry about this because anaconda will fix it for you). Note numba is only used in a few fractal scripts in the misc directory and all other projects should also work for python>=2.7.

How to use

All projects here are implemented in a ready-to-use manner for new comers. You can simply run the examples without tweaking any parameters once you have the dependencies installed correctly. Each subdirectory in src/ is a single program (except that glslhelpers is a helper module for running glsl programs and misc is a collection of independent scripts), any file named main.py, run_*.py, example_*.py is an executable script that gives some output.

List of algorithms

Here is a list of some algorithms implemented in this project:

• Domino shuffling algorithm
• Hopcroft's DFA minimization algorithm
• Lempel–Ziv–Welch compression algorithm
• Propp-Wilson's coupling from the past algorithm
• Todd-Coxeter coset enumeration algorithm
• Wilson's uniform spanning tree algorithm
• Casselman's minimal roots algorithm
• Encoding and decoding algorithms for Gray code

Dependencies

The recommended way to install all dependencies is simply running the bash script install_dependencies.sh.

sudo bash install_dependencies.sh

Or you can install the python libs by pip:

pip install -r requirements.txt

Open source softwares required:

• python3-tk (for file dialog)
• ImageMagick (for making gif animations)
• FFmpeg (for saving animations to video files)
• POV-Ray (for generating high quality raytracing results)
• graphviz (for drawing automata of Coxeter groups)
• Inkscape (optional, for convering large svg files to png)

They can all be installed via command-line:

sudo apt-get install python3-tk imagemagick ffmpeg povray graphviz inkscape

Note pygraphviz also requires libgraphviz-dev:

sudo apt-get install libgraphviz-dev

In the scripts these softwares are called in command line as povray, ffmpeg, convert (from ImageMagick), etc. For Windows users you should add the directories contain these .exe files to the system Path environment variables to let the system know what executables these commands refer to. For example on Windows the default location of POV-Ray's exe file is C:\Program Files\POV-Ray\v3.7\bin\pvengine64.exe, so you should add C:\Program Files\POV-Ray\v3.7\bin to system Path and rename pvengine64.exe to povray.exe, then you can run the scripts without any changes and everything works fine.

TODO

I have a long list of projects to do in mind and they may take a few years to accomplish:

• Knots, inspired by knotilus but I hope I could do better than that.

• Minimal surfaces. Implement two ways to construct minimal surfaces: either by solving Plateau's problem or use a pair of analytic functions. Render the result in POV-Ray.

• Uniform tilings: Replace the hyperbolic module currently used in uniform-tilings project by a custom one (since I want to include LaTeX expressions in the images), with upper half space model added. Also find an efficient way to render the upper half space boundary images in python. Inspired by Roice's artwork.

• Shader animations of polyhedral, euclidean, hyperbolic tilings with gears.

• Escher circle limits in svg format.

• Knots and dynamic systems.

Thanks

I have learned a lot from the following people:

• Bill Casselman
• Roice Nelson
• Possibly Wrong
• Jos Leys
• Greg Egan

License

see the LICENSE file.