1. Elementary cellular automata
In Application
Roger Doles

2. - 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

3. Some Examples

6. 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?

7. 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?

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

9. Perimeter = 44
Perimeter = 38
Perimeter = 34
Perimeter = 32
Perimeter = 30
Perimeter = 30

10. Perimeter = 30
…for each of these boards

11. 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

12. 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

13. 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

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

15. 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

16. 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

17. Questions?