Poster (in Word format) - Bridgewater College

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Presented at the Virginia Academy of Science (Harrisonburg, VA), 21
-
May
-
10.

Exploring “The Game of Life” in Small Worlds

Richard L. Bowman,

Dept. of Physics, Bridgewater College, Bridgewater, VA 22812, USA
rbowman@bridgewater.edu


Abstract



While Conway’s “Game of Life,” an example of

2
-
D cellular automata,
has been shown to have parallels with the biological world, researchers debate the
role of boundary conditions in large universes on the patterns and behaviors
observed there. This paper examines the effects of various boundary cond
itions on
small worlds, 25 X 25 cells or smaller, and illustrates the dramatically different
behavior resulting from an identical starting arrangement of cells (seed), the R
-
pentamino. The boundary conditions are referred to by their geometric analogs:
tor
us, box, loop, and Möbius strip.



I. Introduction



During the past half century or so, many mathematicians, physicists and computer
scientists have noted that some very simple algorithms can begin to mimic the complexity found
in nature. Stephen Wolfram

recently summarized much of what is known [9]. One such
algorithm is the “Game of Life” (Life) developed by John Conway [2, 6]. When Martin Gardner
introduced Life to the world in his
Scientific American

column in 1970, he noted “its analogies
with the ri
se, fall and alternations of a society of living organisms” [6]. Since then many have
explored the behavior of Life. Most of these have used either cyclic or open boundary conditions
[1, 3, 5, 7]. Several large catalogs of stable forms are available via we
b sites [8].


The rules governing the behavior of this 2
-
D simulation are very simple and relate only to
one organism (or one cell) and its immediate eight neighbors. However, the aggregate behavior
of many cells may have its own resulting patterns just a
s individuals and their responses make up
the complicated patterns of an animal or human society.


The author has implemented the following rules of Life in JavaScript code on a web
page. [4] The results given here were obtained through the use of this we
b
-
based simulation.


A. Birth of a cell.
If a 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 a 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.


II. The R
-
pentamino Seed


One seed pattern, referred to as the R
-
pentomino (see Figure 1), consists of only five
living cells linked together. Yet, it only reaches its final state after 1103 iterations if it begins
near the center of a world that is at least 150 c
ell locations by 150 cell locations. The final state is
composed of 25 separate patterns of from 3
-
7 cells each.

2


Presented at the Virginia Academy of Science (Harrisonburg, VA), 21
-
May
-
10.


Figure 1. R
-
pentamino seed.



This paper explicitly explores the cell pattern development that occurs in small finite
worlds of Life, i.e.,
universes with lattices of 25 X 25 cells or smaller and with one of four
possible boundary conditions. The boundary conditions are referred to by their geometric
analogs: torus (living on a donut), box (a 2
-
D fish bowl), loop (a never
-
ending path with edge
s),
and Möbius strip (a never
-
ending path with

edges and

a twist). Figure 2 illustrates these
conditions.



Figure 2. Mapping a 3 X 2 Game of Life to each boundary condition:

(a) torus, (b)
box
, (c) loop, and (d) Möbius strip.


A. Torus Boundary Conditi
on


For the R
-
pentamino seed in the 3 X 3 up through the 8 X 8 square worlds, the death of
the whole community occurs quickly (in 2, 5, 16, 12, 20, and 35 generations, respectively). Table
1 documents the ending scenarios for a variety of other
-
sized squa
re worlds.


The ending scenarios for variously sized rectangular worlds also show a similar non
-
definable progression.


B. Box Boundary Condition


When the box boundary condition is applied then the question emerges of what effect
will the wall or edge o
f the box have on how the cells living next to it will sense the edge. Three
possibilities have been explored: (1)
the edge

does not contribute
to the nearest neighbor count
(sometimes referred to as the open boundary condition), (2)
the edge

adds one
to t
he nearest
neighbor count, or (3)
the edge

adds two
to the nearest neighbor count.


3


Presented at the Virginia Academy of Science (Harrisonburg, VA), 21
-
May
-
10.

Table 1. Fates of living cell

groups in small universes with

an R
-
pentomino seed (torus model).




The results of several experiments with variously sized worlds are recor
ded in Table 2 for
two of these box boundaries. The having the wall contribute two to the nearest neighbor counts
gave the same increase of stability as the additional count of one did, with all communities
beginning with the R
-
pentamino seed remaining ali
ve after more than 1000 iterations

in a 13 X
13 world.

Table 2. Fates of living c
ell groups in small worlds with

an R
-
pentomino seed (box model).


4


Presented at the Virginia Academy of Science (Harrisonburg, VA), 21
-
May
-
10.

C. Loop and Möbius Strip Boundary Conditions



The results for the loop and Möbius strip boundary condition
s applied to the R
-
pentamino seed showed similarly different results from that found for a large universe. For
example, Figure 3 shows the ending results for a 13 X 13 world with the R
-
pentamino seed
beginning in the center. The torus (open) boundary condi
tions were on the left and rgith and the
box conditions on the top and bottom. The final state for the seed in the world with an edge not
contributing to the nearest neighbor count was

reached in the 55
th

generation. The one where the
edge contributed one
to the nearest neighbor count ended in the 398
th

generation after going
through several states where various still life forms emerged and existed for up to 40 or more
generations only to disappear as Life moved on.



Figure 3. The ending states for an R
-
p
entamino seed in a loop world where the top and
bottom edge did not contribute to the nearest neighbors (left) and where the edges
contributed one to the nearest neighbors (right).


The pairs of cells that exist at the top and bottom edge in the world on t
he right are a new
forms of still
-
life that come from the edge boundary condition. Other more complex forms of
still life and oscillators with very long periods (as high as 95 generations) have also appeared
when these edge conditions were applied to squar
e worlds.


III. Further Work


Experiments with various other seeds have also been conducted and continue and will be
reported later.


5


Presented at the Virginia Academy of Science (Harrisonburg, VA), 21
-
May
-
10.


References

[1]
Alstr

m
, P. and Leão, J. Self
-
organized criticality in the “game of Life.”
Phys. Rev. E 49

(1994), R25
07
-
R2508.

[2] Berlekamp, E. R., Conway, J. H., and Guy, R.
Winning Ways for Your Mathematical Plays

(vol. 2).

New York: Academic Press (1982).

[3] Blok, H. J. and Bergersen, B. Effects of boundary conditions in scaling on the “game of
Life.”
Phys. Rev. E 5
5

(1997), 6249
-
6252.

[4] Bowman, R. L.
Interactive Science Activities on the Web

(ISAW).
http://
www.bridgewater.edu/~rbowman/ISAW/


[5] de la To
rre, A.C. and Martin, H. O. A survey of cellular automata like the “game of life.”
Physica A, 240

(1997), 560
-
570.

[6] Gardner, M. Mathematical games: the fantastic contributions of John Conway’s new solitaire
game “life.”
Sci. Am.

(October 1970), 120
-
123.

[7] Ninagawa S. et al. 1/f fluctuations in the “Game of Life.”
Physica D 118

(1998), 49
-
52.

[8] Weisstein, E. Eric Weisstein's treasure trove of the life cellular automaton.
http://www.ericweis
stein.com/encyclopedias/life
/


[9] Wolfram, S.
A New Kind of Science
. Wolfram Media, Champaign, IL. 2002.



---

Presented at the

meeting of the

Virginia
Academy of Science, James Madison U., V
A
, 21
-
May
-
10.