1. Conway’s Game of Life Jess Barak Game Theory

2. History Invented by John Conway in 1970 Wanted to simplify problem from 1940s presented by John von Neumann ◦ Machine that could build copies of itself ◦ Very complicated rules on rectangular grid

3. History continued… Opened new field in mathematical research ◦ Cellular automata  Collection of cells on a grid of specified shape that evolves through a number of discrete time steps according to a set of rules based on the state of neighboring cells

4. Why Life? Simple example of what is called emergent complexity or self-organizing systems Study of how elaborate patterns emerge from simple rules ◦ Example: Understand stripes on zebra that arise from tissue of living cells growing together

5. Rules Played on grid of square cells Cell can be live or dead ◦ Live = cell colored ◦ Dead = cell empty Each cell has neighborhood consisting of 8 cells in every direction (even diagonals)

6. Rules continued… Dead cell with 3 live neighbors becomes live (birth) Live cell with 2 or 3 live neighbors stays live (survival) All other cases, cell dies or remains dead (overcrowding or loneliness)

7. Patterns Still Lifes ◦ Block ◦ Behive ◦ Boat ◦ Loaf

8. Patterns Oscillators ◦ Blinker ◦ Toad Gliders

9. Do all Patterns Stabilize? Stabilizes when a pattern’s population stops growing ◦ Never collide with other patterns Reward to see if there were any patterns that grow forever ◦ Soon after, there were 2 patterns found that grow forever

10. Example

11. What is Life Good For? Studying the patterns of Life can result in discoveries in other areas of math and science Behavior of cells or animals can be understood using simple rules Traffic problems can be solved by analyzing them these simulations Fixing computer viruses, curing human diseases, and exploring galaxies

12. References http://www.math.com/students/wonders/li fe/life.html http://www.math.com/students/wonders/li fe/life.html http://mathworld.wolfram.com/CellularAu tomaton.html http://mathworld.wolfram.com/CellularAu tomaton.html http://www.bitstorm.org/gameoflife/