Cellular Automata Ordered by Entropy

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Entropy can be used to study the amount of information in the evolution of a cellular automaton.


The entropy of a list is defined by , summing over the elements of of , where represents the probability of a white cell and represents the probability of a black cell. The initial condition is a finite set of random binary numbers.

The 256 elementary cellular automata (CA) can be ordered according to their entropy. In the graphic you can see that the most complex (noisy) patterns are on the left, while the ones with more information and the simplest (repetitive) patterns with less information are on the right. The entropy value is lower in ordered systems and higher in chaotic (disordered) systems. Before calculating the entropy, redundant pattern data is removed, and the minimum entropy is selected from the vertical, horizontal, and diagonal directions.


Contributed by: Daniel de Souza Carvalho (January 2009)
Open content licensed under CC BY-NC-SA



This Demonstration extends Compressing Elementary Cellular Automaton Rules.

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