1. Introduction
A cellular automaton is a discrete system composed of a grid of cells, each in a finite state, that evolves according to a local rule applied simultaneously to all cells. Despite their simplicity, these systems can produce an extraordinary variety of behaviors, from stable structures to chaotic patterns including rotating spirals.
2. Definition and Rules
A 1D cellular automaton is defined by: an alphabet of states (often {0,1}), a neighborhood (the k adjacent cells), and a transition rule that determines the next state of a cell based on its current state and that of its neighbors. For a binary automaton with 3-cell neighborhood, there are 2^(2^3) = 256 possible rules, numbered 0 to 255 by Wolfram.
3. Conway's Game of Life
The Game of Life (Conway, 1970) is a 2D two-state automaton with Moore neighborhood (8 cells). Its rules: a living cell survives with 2 or 3 living neighbors, dies otherwise; a dead cell is born with exactly 3 living neighbors. These simple rules produce oscillators, gliders, glider guns and universal computation structures.
4. Wolfram's Classification
- Class I: convergence to a homogeneous state
- Class II: stable or oscillating periodic structures
- Class III: chaotic behavior, apparently random
- Class IV: complex localized structures, edge of chaos (e.g. Rule 110)
5. Spirals in Cellular Automata
Spirals appear in multi-state automata, notably in the Greenberg-Hastings model (1978), which simulates excitation waves in excitable media. This model produces stable rotating spirals resembling the spiral waves observed in Belousov-Zhabotinsky reactions and in fibrillating cardiac muscle.
6. Conclusion
Cellular automata demonstrate that extremely simple local rules can generate remarkable global complexity. Spirals only appear in certain classes of automata — those that combine excitation, refractoriness and diffusion — making them indicators of specific dynamics rather than generic properties of complexity.