Elementary cellular automata In Application Roger Doles

- Define (game)board to be an m x n lattice of cells How it works To start, choose a seed - Define (game)board to be an m x n lattice of cells - Each cell can either be on or off - A cell with 2 or more neighbors on becomes on - Otherwise cell remains in same state

Some Examples

The problem Minimum number of trees in a forest to set the entire forest on fire? -Think about it as an m x n automaton - Assign each tree a square on the grid - How many need to start on fire?

So… How many do we need? We need another result first By definition there are only 4 ways, up to rotation, to turn a cell on How does the perimeter of the on cells change between steps?

Perimeter of on squares is 44 7 x 8 board 11 squares on to start Perimeter of on squares is 44 Seed pattern

Perimeter = 44 Perimeter = 38 Perimeter = 34 Perimeter = 32 Perimeter = 30 Perimeter = 30

Perimeter = 30 …for each of these boards

So… How many do we need? Since the perimeter of the on squares is non-increasing, the minimum number of squares is the ceiling of the average of m & n m n

So… How Do We position them? This pattern has 9, the ceiling of the avg. of m & n, turned on as a seed. This pattern will not fill the board The minimum number of cells is not guaranteed without specific orientation 8 9

So… How Do We position them? Consider this pattern as step t=1 It will produce step t from previous slide in one step Perimeter of full board= Per of seed = 34 = 2m + 2n For m = 8, n = 9 Number of seed cells, s = 9 8 9

So… How Do We position them? This orientation will always fill the board

For any natural numbers m,n So… the solution? Orientation of seed For any natural numbers m,n - If m + n is even, no more than (m + n) / 2 cells are required to fill the board - If m + n is odd, no more than (m + n + 1)/2 cells are required to fill the board

Tessellating cell Shapes Further work Board Shapes Tessellating cell Shapes We have already proven for square, rectangle and L –shape boards. What about a cross shape? Conjecture: Predominant L will fill using the same seed Remaining appendages of cross fill by choosing a similar pattern Show that behavior of growth is similar to squares and generalize to regular tessellating shapes Same rules must apply: half or greater of shared-edge neighbors must be on to turn on. Otherwise cell remains in same state Proposed for future work: triangles and hexagons

Questions?