A universe with four rules
Tonight's idea
Take a huge sheet of squared paper. Each square is a cell, either alive (filled in) or dead (empty). Every cell has eight neighbours: the squares that touch it, corners included. Time moves in steps. At each step, count the live neighbours of every cell and apply four rules.
A live cell with fewer than 2 live neighbours dies.
A live cell with 2 or 3 lives on.
A live cell with more than 3 dies.
A dead cell with exactly 3 comes alive.
All cells change at the same moment. Do all the counting on the old grid, then draw the new one. That is the whole of the Game of Life, invented by the mathematician John Conway. It has no players. You choose the starting pattern and the rules do the rest.
Some patterns never change. They are called still lifes. Some repeat. And one five-cell pattern, the glider, rebuilds itself every four steps, one square further along a diagonal. Nothing in the rules mentions movement.
Watch one: the blinker
Start with three live cells in a row.
The two end cells have 1 live neighbour each, so they die. The middle cell has 2 and lives on. The dead cells just above and just below the middle touch all three live cells, so they come alive. Every other dead cell counts 2 or fewer and stays dead.
The row has turned upright. One more step turns it back, so the blinker repeats every 2 steps.
Draw a pattern and press Step or Run: mathdetours.com/demos/life.html.
Your turn
- What is left after one step if you start with (a) a single live cell, (b) two live cells side by side, (c) a block: four live cells in a 2 × 2 square?
- Start with three live cells in an L shape: a block with one corner missing. Draw the next step, and the step after that.
- Start with four live cells in a row. Draw the next two steps. What happens after that?
- Look at the glider. (a) Where is it after 100 steps? (b) A dead cell with no live neighbours stays dead. Use this to explain why no pattern can spread faster than one square per step. How does the glider's speed compare with that limit?
Stretch: no way back
Find two different patterns that both turn into a block after one step. Then find two different patterns that both turn into an empty grid. What does this tell you about working out the previous step from the present one?
Answers
- (a) Nothing. The cell has 0 live neighbours and dies, and no dead cell has 3. (b) Nothing. Each has 1 live neighbour and dies. (c) The same block. Each live cell has 3 live neighbours and lives on, and no dead cell touches more than 2 of them. The block is a still life.
- Each of the three live cells has 2 live neighbours and lives on. The missing corner touches all three, so it comes alive. No other dead cell touches more than 2. The next step is a block, and it stays a block for ever.
- Step 1: the two end cells die (1 neighbour each), the middle two live on (2 each), and the four cells above and below the middle two come alive (3 each). That makes a solid rectangle, 2 wide and 3 high. Step 2: the two cells in its middle row have 5 neighbours and die, the four corners have 3 and live on, and the dead cells just left and right of the middle row touch 3 and come alive. The result is a ring of six cells round two empty ones, called a beehive. Every live cell in it has 2 neighbours and no dead cell has exactly 3, so it is a still life.
- (a) It moves one square right and one square down every 4 steps, so after 100 steps it is 25 squares right and 25 down, in its starting shape. (b) A cell can only come alive if it already touches a live cell. So in one step the live region can grow by at most one square in any direction. The glider covers one diagonal square in 4 steps: a quarter of the limit.
Stretch. A block and the L shape from question 2 both become a block. A single cell, a pair of cells and an empty grid all become an empty grid. So the present does not tell you the past: Life can be run forwards but not backwards. The rules decide the future completely, yet they throw information away at every step.