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RETURN TO:
SYSTEMS THEORY & cHAOS
 
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Conway's Game of Life
Author: Michael Colebrook
 
 
 
 
 
 
 
 
 
 
 
 
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One of the pioneers of systems theory, John von Neumann, developed the concept of cellular automata in an attempt to produce an abstract mathematical 'machine' that would reproduce itself. Picture a huge chess board, each of whose squares exhibits a quality such as colour. Each move of the automaton is based on a set of rules by which the colour of each square changes depending on the colours of its neighbours. Von Neumann was successful in devising rules and an initial configuration such that after a number of moves the original state reproduced itself, although he needed complex rules and many colours.
Subsequently John Horton, Conway developed a cellular automaton that uses only two colours and a very simple set of rules that is amazingly versatile. He called his system the Game of Life.
 In Creek mythology, the machinery of the universe was the gods themselves ... In more recent conceptions, the universe is created complete with its operating mechanism: once set in motion, it runs by itself. God sits outside of it and can take delight in watching it.
Cellular automata are stylized, synthetic universes ... They have their own kind of matter which whirls around in a space and a time of their own. One can think of an astounding variety of them. One can actually construct them, and watch them evolve. As inexperienced creators, we are not likely to get a very interesting universe on our first try; as individuals we may have different ideas of what makes a universe interesting, or of what we might want to do with it. In any case, once we've been shown a cellular-automaton universe we'll want to make one ourselves; once we've made one, we will want to try another one. After having made a few, we'll be able to custom-tailor one for a particular purpose with a certain confidence.
A cellular automata machine is a universe synthesizer. Like an organ, it has keys and stops by which the resources of the instrument can be called into action, combined, and reconfigured. Its color screen is a window through which one can watch the universe that is being "played."
 Toffoli, T. and N. Margolus. Cellular Automota Machines: A New Environment for Modelling (MIT Press, 1987), p.1.
 Quoted in Evelyn Fox Keller. Making Sense of Life (Harvard University Press, 2002)

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This configuration contains a mix of static patterns and other patterns, which while changing shape, oscillate to-and-fro in endless loops of moves. The two complex oscillators on the left of the grid produce a stream of gliders which travel south-east across the grid. Each of these interacts with the two oscillators across the bottom of the grid to produce a secondary stream of more complicated gliders. The whole process occupies a loop of 47 moves. It is important to stress that every coloured square is subject to the same set of rules all the time.
Conway was able to show that, even with his simple rules, there are initial configurations for which the outcome is unpredictable.

This simple initial pattern runs for over 17,000 moves creating chaotic patterns of coloured squares before effectively 'dying' as a set of non-interacting patterns. The population of coloured squares reaches a maximum of over 2000. Moving any one of the squares in the initial pattern produces very different results, most of which 'die' after only a few moves.
There is at least one initial state that gets close to von Neumann's objective of producing a self-replicating system. It consists of a particular pattern of gliders.

The gliders interact to produce the four static elements shown in the central diagram. The two remaining gliders then interact with these to produce the two oscillating patterns shown in the right hand diagram. These interact to produce a stream of gliders. The system can be considered to be analogous to the reproductive system of slime moulds in which separate individuals gather together to form a fruiting body which then produces numerous offspring.

There is one elaboration of Conway’s rules that does replicate almost any initial pattern. For a red square, if 1, 3, 5 or 7 adjacent squares are red the square stays red. For a white square if 1, 3, 5 or 7 adjacent squares are red the square turns red. For all other states the square turns or stays white. The diagram, running from left to right, shows an initial pattern and the patterns after 4, 8, 12 and 16 moves.

The significance of the Game of Life is that given simple rules, various initial states can result in almost infinitely varied outcomes involving a very full spectrum of states ranging from completely chaotic, to chaotic involving some order, to a variety of ordered states from complex oscillating and moving patterns, to simple loops to static patterns.

Since the outcomes are unpredictable the patterns produced by any initial state emerge through what have to be regarded as creative processes (see Emergence).

The following web site contains a  downloadable programme to enable you to play for yourselves. http://psoup.math.wisc.edu/Life32.html

Another page with lots of information and games that can be downloaded and played with Life32 is at http://entropymine.com/jason/life/