Cellular Automata

Mathematical Principles & Theorems

Cellular automata are discrete dynamical systems. Conway's Game of Life operates on a 2D grid \(\mathbb{Z}^2\) under transition rules: (1) Live cell with 2–3 neighbors survives, (2) Dead cell with exactly 3 neighbors becomes alive, (3) All other cells die of under/overpopulation. Wolfram 1D Elementary Automata define 256 discrete rules mapping 3-cell neighborhoods \(\{0, 1\}^3 \to \{0, 1\}\), exhibiting Class 1 (homogeneity), Class 2 (periodicity), Class 3 (chaos, Rule 30), and Class 4 (Turing completeness, Rule 110). Langton's Ant exhibits emergent order around step 10,000.

Operating Instructions

• Select mode: Conway's Game of Life, 1D Elementary Automata, or Langton's Ant.
• For Game of Life: Draw cells on canvas or choose a preset (Glider, Gosper Gun, Pulsar); adjust speed via numeric input.
• For 1D Automata: Enter rule number (0–255) and generation count, then click Draw to view spacetime fractal triangles.
• For Langton's Ant: Configure steps per frame and click Play to observe emergent highway construction.