Elementary Cellular Automaton Simulator - All 256 Rules
The simplest type of cellular automaton: each cell has two states (0/1), and the next state is determined by the cell and its two neighbors (3 cells total). With 8 possible combinations, there are exactly 256 rules. Notable examples include Rule 110, which has been proven Turing complete, and Rule 30, known for its chaotic patterns.
| Pattern | 111 | 110 | 101 | 100 | 011 | 010 | 001 | 000 |
|---|---|---|---|---|---|---|---|---|
| New state | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 0 |
Rules
- In an elementary cellular automaton, each cell in a one-dimensional row has a state of 0 (dead) or 1 (alive). The next state of each cell is determined by itself and its two neighbors (3 cells total).
- The rules are defined as follows:
- 1. There are 8 possible combinations of 3 cells: 111, 110, 101, 100, 011, 010, 001, 000
- 2. A rule specifies the next state (0 or 1) for each combination. With 8 bits, there are 2^8 = 256 possible rules
- 3. The rule number is the decimal value of the 8-bit output. For example, Rule 110 means 01101110₂ = 110₁₀
What is an Elementary Cellular Automaton?
Elementary Cellular Automata (ECA) were systematically studied by Stephen Wolfram in the 1980s and represent the simplest form of cellular automaton. Wolfram classified ECA behaviors into four classes: Class 1 (uniform), Class 2 (periodic), Class 3 (chaotic), and Class 4 (complex).
Rule 110, a Class 4 automaton, was proven Turing complete by Matthew Cook in 2004. Its name comes from reading the output pattern as binary "01101110", which equals 110 in decimal. Rule 30, a Class 3 automaton, produces such chaotic patterns that Mathematica uses it for random number generation.
Learn more about the history and applications of cellular automata →