Exploring the Game of Life in Small Finite Universes with Various Boundary Conditions

INTRODUCTION

The Game of Life (initial rules and study done by John Conway)  implementation linked to below uses JavaScript and HTML to create the display for each generation of a given seed. To see what the Game of Life is all about read the associated exercise (even if you do not follow its directions) and check out the resources listed below.

Note Before Accessing The Fractal Generator Below

1. Your browser must be capable of interpreting JavaScript.
2. Your browser must have the saving of "cookies" enabled.

GAME OF LIFE EXAMPLES

A. Some Basic Forms

• Examples: Still life (block, beehive, boat), an period-2 oscillator (blinker), and period-4 glider (transforming into a pond still life after a collision) in a 21 X 21 torus model universe with all boundaries wrapping back on the opposite one (cyclic boundary conditions).

B. Some Interesting Seed Forms

• R-pentomino -- in a 25 X 25 torus model universe
• Open Cross (torus) -- 10 cells vertically and horizontally with an empty center cell in a 24 X 25 torus model universe in which all boundaries have cyclic boundary conditions that wrap the universe back on itself as in a torus
• Open Cross (box-0) -- 10 cells vertically and horizontally with an empty center cell in a 24 X 25 box model universe in which each edges does not contribute any count to the living cell.
• Open Cross (loop-1) -- 10 cells vertically and horizontally with an empty center cell in a 24 X 25 torus model universe in which the top and bottom edges do not contribute to the living cell count while the left and right have cyclic boundary conditions
• Two Open Squares (Möbius-1) -- two open squares  in a 14 X 10 Möbius strip model universe in with the same boundary conditions as above but with the left and right sides joined through a twist.

C. Hunting for Quilt Designs

• Quilt Design Search #1 -- using squares, instead of disks, in a 17 X 17 torus model universe with four open squares as seeds
• Quilt Design Search #2 -- using squares, instead of disks, in a 21 X 21 torus model universe with a cross, 17 cells high and 17 cells wide, as seed

D. Random Population of Living Cells with Various Boundary Conditions

User may set the size of lattice (default: 15 X 15). These use smaller lattice cells to facilitate larger universes.

• Random Seed Generator (torus) -- in a torus model universe ( larger-squares version )
• Random Seed Generator (box 0) -- in a box model universe where the borders contribute nothing to the living cell count ( larger-squares version )
• Random Seed Generator (box 1)-- in a box model universe where each border contributes one to the living cell count ( larger-squares version )
• Random Seed Generator (box 2)-- in a box model universe where each border contributes two to the living cell count ( larger-squares version )

E. Make Your Own Seed Arrangement

• Experiment on Your Own

F. Lab Exercise

• Guided Exercise

RESOURCES

Books And Articles

Berlekamp, E. R., Conway, J. H., and Guy, R. Winning Ways for Your Mathematical Plays (vo. 2).New York: Academic Press (1982).

Gardner, M. Mathematical games: the fantastic contributions of John Conway’s new solitaire game “life.” Sci. Am. (October 1970), 120-123.

Bowman, R. L. The “game of life” in small universes with various boundary conditions: an experimental approach. ACMSE 2006 Conference. [to be submitted]

Web Sites

Callahan, P. What is the game of life? Wonders of Math. http://www.math.com/students/wonders/life/life.html

Conway's game of life, Wikipedia. http://en.wikipedia.org/wiki/Conway's_Game_of_Life

Hensel, A. Conway's game of life (and Java applet). http://www.ibiblio.org/lifepatterns/

Martin, E. John Conway's game of life (and Java applet). http://www.bitstorm.org/gameoflife/

Weisstein, E. Eric Weisstein's treasure trove of the life cellular automaton. http://www.ericweisstein.com/encyclopedias/life/

COMMENTS?
Have any affirmations, questions, problems or suggestions? Contact:

Dr. Richard L. Bowman
Dept. of Physics
Bridgewater College
Bridgewater, VA 22812

phone: 540-828-5441

email: RBowman@bridgewater.edu

Created and maintained by: Richard L. Bowman (1999-2005; last updated: 04-Jan-06)