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The Game of Life & Cellular Automata

Reid Savage

The game of life , coined by mathematician John Conway in the 1960s, is a zero-player game

based on cellular automata (hence abbr. CAM). In its most elementary form, it is played by

using a grid system of some sort and a method of flipping the states of the cells from two

states (on/off, dead/alive, 0/1, etc); this was originally accomplished with a Go board. In this

game, a certain amount of cells states are flipped to trigger the beginning of the game. With

each iteration of time, a calculation is performed on each cell. John Conways variation

included these rules:

If the cell is alive, then it stays alive if it has either 2 or 3 live neighbors.

If the cell is dead, then it springs to life only in the case that it has 3 live neighbors. 1

These rules are arbitrary in that they hold no special significance other than allowing for a

CAM game that is both interesting and hard to predict.

If we consider the state of 3 adjacent cells, how many possible automata (evolutions) are

there? If each cell has two states, and there are three cells, and order matters, this would be

23 combinations. However, since the state of the middle cell can be picked independently,

there are actually 2 8 combinations. This can be generalized as k^(k^(s)), where k is the

number of states a cell can occupy, and s is the number of neighboring cells. As you can tell,

the number of possible automata grows really fast. The function k^(k^(s)) is quite scary

once you consider the number of possibilities on a theoretically infinite grid and more than two

states. 2

1 http://www.math.cornell.edu/~lipa/mec/lesson6.html2 And/or the third/fourth dimension, man.

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Conways game of life is far from the only set of rules that have been created based on the

cellular automaton. The inventor of the key principles of the computer, John von Neumann,

devised his own set of rules to explore machine self-replication. von Neumanns cellular

automaton is based on a Cartesian grid with each cell having 29 possible states categorized3

into the ground state, transition/sensitized states, confluent states, ordinary transmission

states, and special transmission states. As you can see, considering the above equation, the

number of possible automata can be practically infinite, and can be used to model many

different systems. von Neumanns original use for CAA was created for the modeling of fluid

dynamics; with todays extremely powerful computers, there are far more complex systems

that could be modeled to higher resolution.

CAM have applications in traffic modeling, urban growth modeling, pattern recognition,

cryptography, machine learning, particle models, fluid dynamics, thermo/hydrodynamics,

seismic elastodynamics, gas behavior, and just about any system of which each cells state

can be determined by the previous state of its neighbors . 4

As you can see, the complexity of CAM can vary wildly, and is highly applicable to many

different problems. It is likely that they will be popular modeling tools for quite some time. 5

3 Unit square based.4 Which turns out to be a lot.5 At least until we figure out that one system that explains all of the other systems.