GAME OF LIFE EXERCISE

Richard L. Bowman, PhD
937 College Ave., Harrisonburg, VA 22802, USA
richard.bowman@edtechbybowman.net

Introduction

The Game of Life (Life) was first created and studied by John H. Conway and was introduced to the world in 1970 by Martin Gardner in his column on Mathematical Games in Scientific American [1]. Later Conley related more of his studies in a book edited by Elwyn R. Berlekamp, Conway and Richard Guy [2].

The rules are simple, but the outcomes appear very complex, and even though they are on a two-dimensional surface, they have some similarities to human societies. Thus this is called the Game of Life. Here is the procedure for determining the generations of a Life seed pattern.

1. Choose a beginning seed pattern of “living cells” on a 2-D grid of blank cells.
2. Determine what locations will have living cells in the next generation (iteration) of the group of cells by the following rules.
   A. Birth of a cell. If any blank cell has exactly three living neighbor cells, then the empty cell will give birth to a living cell in the next generation.
   B. Survival of a cell. If any living cell has two or three living neighbor cells, then it will remain alive in the next generation.
   C. Death of a cell. If a living cell has more than three living neighbors than it dies from “overcrowding.” If a cell has less than two living neighbors, than it dies from “loneliness” or “exposure.”
3. Replace the original seed group with the new generation and repeat from step 2 until a stable pattern results or for as long as the user wishes. Numerous patterns have been used as seeds and a large array of stable patterns have been observed and named [1-6].

Studies cared out by myself have examined small universes in Life under various boundary conditions. This research have shown that these two factors can dramatically effect the final fate, and the journey to that final state, of a particular beginning seed pattern. Boundary conditions can be described by the geometrical shapes to which they can be associated [3].

• Torus model -- a model in which each edge of the finite universe wraps back on the opposite edge. This is often referred to as cyclic boundary conditions.
• Box model -- the finite universe in this model stops at the edges. Each edge can contribute or not contribute to the living neighbor cell count. If the edges contribute nothing, this is often referred to as open boundary conditions. I have explored Life where the edges contribute one or two to the living neighbor cell count. In small universes these tend to increase the longevity and complexity of the patterns that result in each generation.
• Loop model -- a model in which the two edges at the ends of the horizontal direction act as they do in the torus model, and the two edges at the end of the vertical direction behave as they do in the box model (again contributing 0, 1, or 2 to the living neighbor cell count.
• Möbius strip -- the edges in this model behave as they do in the loop model but with the two edges along the horizontal direction being "pasted" together after one of them has been given a twist.

Explorations

1. Print out a copy of these directions for each member of the lab group, unless your instructor has already made copies for your use. Read the introductory material of this exercise before continuing!!

2. Have the Life web page of the ISAW web site ( http://www.edtechbybowman.net/PhysAstroSims/life/ ) open and read to use. All of the assigned activities will reference results from executing the different Life seeds and universes on that page.

3. First learn to recognize a few of the basic stable forms that may result from a seed moving through a number of Life generations. On the web page select the "Examples" link and print out the page when it fully displays. Circle the various objects living in that universe and write the type of object each is (still life, oscillator, or gliders) and its name next to the object.

Note: To find the names of patterns in Life, use the hand-out given to you by your instructor or visit one of the web sites listed as a reference at the end of this exercise page. Martin Gardner's article in Scientific American is a good place to begin, if you can locate a copy.

4. Run these "Example" patterns through several generations and observe what happens. Keep going until the glider collides with one of the other patterns and generates a new pattern. Print this page out and circle the new object giving its pattern type and name.

Note for the steps below: To save you time in carrying out this lab, you do not have to run any of these seed patterns through more than 100 generations. You are encouraged to come back later and try for more generations.

5. In turn run each of the linked web pages under the section, "Some Interesting Seed Forms." Print out the first generation page and then the page showing the final fate of the society generated by each different seed. Also, print out any interesting pages in the middle of the journey, especially if you observe any stable patterns that exist for at least three generations.

6. Next run the web pages that allow the user to look for patterns that might be used to make quilts. Print out the starting seed pattern and any pages that you think would work for this purpose (at least four of them). If these societies end in some type of stable state, print out the page that shows which generation this fate first emerges.

7. Finally, try the "Random Seed Generator" and watch what happens in each generation. Print out the beginning seed page and any final fate page. If any interesting pages show up in the journey, print them out, also.

(Optional) 8. If your instructor has set it up so that you can modify the seed pattern and universe size for a Life page, try you hand at proposing and observing a new pattern and universe of your choosing.

Conclusion

Collect all of your lab sheets, including all printouts that you used; staple them together; place your name and your lab partners' names on the top of the first sheet, and hand them into your instructor. You have only begun to explore the many patterns and journeys of various seed patterns in selected universes with differing boundary conditions that can come from the Game of Life. Keep exploring on you own using the activities on the ISAW web site or check out the references below or others from your library to learn more. In some ways the Game of Life does mimic the complexity of nature with only simple rules. Have fun exploring!

Bibliography

1. Gardner, M. Mathematical games: the fantastic contributions of John Conway’s new solitaire game “life.” Sci. Am. (October 1970), 120-123.
2. Berlekamp, E. R., Conway, J. H., and Guy, R. Winning Ways for Your Mathematical Plays (vo. 2).New York: Academic Press (1982).
3. Bowman, R. L. The “game of life” in small universes with various boundary conditions: an experimental approach. ACMSE 2006 Conference. [to be submitted]
4. Callahan, P. What is the game of life? Wonders of Math. http://www.math.com/students/wonders/life/life.html
5. Conway's game of life, Wikipedia. http://en.wikipedia.org/wiki/Conway's_Game_of_Life
6. Weisstein, E. Eric Weisstein's treasure trove of the life cellular automaton. http://www.ericweisstein.com/encyclopedias/life/
7. Hensel, A. Conway's game of life (and Java applet). http://www.ibiblio.org/lifepatterns/
8. Martin, E. John Conway's game of life (and Java applet). http://www.bitstorm.org/gameoflife/

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Created and maintained by: Richard L. Bowman (2005-2011; last updated: 14-Sep-11)