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Is there any function defined for say the plane, that has interesting nontrivial behaviour similar to Conway's Game Of Life, but where every point's on/off status is decided by something like the density of on points in some finite neighbourhood of the point in the plane. Alternatively the status is decided by the average status weighted by 1/distance. Preferably every point should be part of a continuous curve for easier visualization.

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    @Rahul: Thanks.. I've been having trouble with special characters in my links.2012-01-18

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Yes, it exists: M. Pivato, RealLife : the continuum limit of Larger Than Life cellular automata.

This is exactly what you're thinking about (update according to local density) and it can be interpreted as the limit of classical CA with larger and larger neighborhood.

The nice thing apart from the rigorous mathematical setting introduced, is that the limit shapes of gliders/blinkers/etc of CA with larger and larger neighborhood are gliders/blinkers/etc of the continuum limit CA.