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Paul Reiners

Rochester, Minnesota

Registered: Feb 2004
Posts: 34

Q about Game of Life

I have a few questions about Conway's Game of Life:

Is it known whether Life universes that start from a random initial state tend to 'settle down' to a certain density of live cells? If so, is this density a constant or does it depend on the density of the initial state? If the latter, does anyone have any results on the relationship between initial state densities and the 'final' density after the universe has settled down?

By the way, these questions have to do with a video game I'm writing in Python that uses Life. If you're curious, you can download a beta version of the program from here:


If you do try the game, please let me know of any problems you run into or any suggestions you have.

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Old Post 09-04-2007 01:51 PM
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Jason Cawley
Wolfram Science Group
Phoenix, AZ USA

Registered: Aug 2003
Posts: 712

There is a typical background density in life, but subject to considerable random variations from run to run. It is a statistical thing (property of an ensemble of initials) rather than a steady state always hit, in other words (each run, the actual final concentration will vary around it), consisting of a mix of stable and periodic configurations each produced with some probability. The ensemble average is quite low, below 3% of sites occupied - I've see a figure of 0.0287 given from some experiments. (See link below).

It isn't terribly sensitive to the initial density, but there are endpoint extremes that will change it. For instance, if the initial density is very low, everything will die out immediately with very high probability. You can also get the whole pattern to die instantly from a totally filled pattern.

For patterns "overfilled", the concentration drops rapidly, with some whole regions dying out and others moving to about the background density. But there are also significant run to run variations - a small portion of initials will make larger than usual persistent structures or leave more gliders etc.

A paper on the subject from 1993 is Nonlinear Dynamics of the Cellular Automaton "Game of Life", Physical Review E, pp. 3345-3351 by Garcia et al (a group in Brazil).

A website that catalogues the most common persistent structures seen in large random life initials (though he fixed the initial density at 3/8 to get about as many as possible, really) can be found here -


Note that the small simple patterns are by far the most common, but there are large numbers of somewhat more complicated ones, with low frequencies dropping in roughly power law fashion.

Note also that special initials are possible that "tile" with densities up to 0.5, but there is essentially no chance of hitting one from random initials.

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Old Post 09-04-2007 06:41 PM
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