About this experiment
A cellular automaton is a grid of cells that are either on or off. At every tick, each cell looks at its neighbours and follows the same simple rule to decide its next state. Nothing else is programmed in, yet gliders crawl across the grid, guns fire them out one after another, and some one-dimensional rules produce patterns so irregular they are used as random number generators. Run John Conway's Game of Life, draw your own starting patterns, and explore all 256 elementary rules.
Learning goals
- Apply the Game of Life rules by hand to work out the next generation
- Recognise still lifes, oscillators and spaceships
- Explain how a rule number encodes an elementary one-dimensional automaton
- Describe how complex behaviour can emerge from simple local rules
- Compare rules that give repeating, nested, chaotic and complex patterns
How to use it
- 1Load the Glider and press Step four times. How has it changed after four generations?
- 2Load the Pulsar. How many generations does it take to repeat?
- 3Run the Glider gun. How often does a new glider appear? Watch the graph of live cells.
- 4Clear the grid and draw a row of three cells. Predict the next generation on paper, then press Step.
- 5Load the R-pentomino and run it. Only 5 cells grow into what? Switch off edge wrapping and compare.
- 6Switch to One-dimensional rules. Compare rules 30, 90 and 110. Which looks random, which looks nested, which has structures that move?
- 7Flip one result in the rule table of rule 90. What rule number do you get, and how does the picture change?
The model
Assumptions
- The Life grid is 72 by 44 cells. With wrapping on, the edges join so the grid is a torus; with it off, cells beyond the edge always count as dead.
- All cells update at the same time from the previous generation.
- One-dimensional rows are 121 cells wide and wrap around at the ends. The random start is seeded, so it is the same every time.
Try this in class
- Q1A dead cell has exactly three live neighbours. What happens to it in the next generation?
- Q2Why does a 2 by 2 block of live cells never change?
- Q3Write rule 30 in binary and use it to find the new state of a cell whose neighbourhood is 100.
- Q4The glider moves one cell diagonally every four generations. How long does it take to cross the 72-cell grid?
- Q5Rule 110 can carry out any computation a computer can. What does that suggest about simple rules and complex behaviour?
Frequently asked questions
Who invented the Game of Life?
The British mathematician John Horton Conway in 1970. It became famous through Martin Gardner's column in Scientific American, and people have been discovering new patterns ever since.
Is the Game of Life really a game?
It is a zero-player game: once you have drawn the starting pattern, the rules decide everything. The fun is in designing patterns and finding out what they do.
Where are cellular automata used?
In models of forest fires, traffic, crystal growth, the spread of disease and the patterns on sea shells. Rule 30 was once used as a random number generator, and grid-based simulations in physics and biology often work the same way.