Personal Projects

This is a list with projects that I work on in my free time. Through these projects I aim to sharpen my programming and abstract mathematical skills. Lastly, I hope that other people might find the code accompanying these projects useful and helpful.

Cellular Automata

For this project, I have the following two goals. First, I want to create an open-source package in Python that allows people to build custom cellular automata. Second, I want to use this package to play with cellular automata and study the mathematical theorems governing them. Specifically, I want to explore the following:

  • How important is the lattice geometry?
    • Can every two-dimensional automaton be described by an one-dimensional automaton?
    • Does a cellular automaton on a triangular lattice have the same behavior as one on a square lattice?
    • How similar is a cellular automaton on a lattice whose sites are randomly distributed to one whose sites are uniformly distributed?
  • What is the behavior of a cellular automaton which has an evolving governing function?
  • Probabilistic cellular automata.
    • How much do they differ from their deterministic counterparts?
  • Can a cellular automaton return to its initial state after a finite time?
  • Do all cellular automata map their initial states to the entire set of states? This questions tries to study if cellular automata are bijective functions. If not, how big is the subset of those that are bijective?
  • How important are the boundary conditions?

Most of the questions above have already been answered and I will try to provide the appropriate references as I tackle each point above.

Resources:

  • Github repository: https://github.com/stratisg/Cellular-automata
  • Wikipedia article on cellular automata: https://en.wikipedia.org/wiki/Cellular_automaton
  • Wikipedia article on self-replication: https://en.wikipedia.org/wiki/Self-replication
  • Wikipedia article on coupled map lattice: https://en.wikipedia.org/wiki/Coupled_map_lattice
  • Wikipedia article on probabilistic/stochastic cellular automata: https://en.wikipedia.org/wiki/Stochastic_cellular_automaton
  • Wikipedia article on Von Neumann universal constructor: https://en.wikipedia.org/wiki/Von_Neumann_universal_constructor
  • Wikipedia article on Greenberg–Hastings cellular automaton: https://en.wikipedia.org/wiki/Greenberg%E2%80%93Hastings_cellular_automaton
  • Wikipedia on Garden of Eden (cellular automaton): https://en.wikipedia.org/wiki/Garden_of_Eden_(cellular_automaton)
  • Wikipedia on digital physics: https://en.wikipedia.org/wiki/Digital_physics