Complex Systems

Symmetry and Entropy of One-Dimensional Legal
Cellular Automata Download PDF

Kazuhito Yamasaki
Department of Earth and Planetary Sciences, Kobe University
Nada, Kobe, 657-8501, Japan

Kazuyoshi Z. Nanjo
Earthquake Research Institute, University of Tokyo, 1-1-1
Yayoi, Bunkyo-ku, 113-0032, Tokyo, Japan

Satoshi Chiba
Department of Ecology and Evolutionary Biology, Tohoku University
Aobayama, Sendai, 980-8578, Japan


The one-dimensional legal cellular automata (CAs) used in Wolfram's original classification from a viewpoint of symmetropy (an object related to symmetry and entropy) are quantified. For this quantification, the discrete Walsh analysis that expresses the two-dimensional discrete pattern in terms of the four types of symmetry is used. The following was found. (1) The relationship between symmetry and entropy of the CA patterns corresponds to the three qualitative classes of CAs: II, III, and IV. (2) The change in symmetropy shows that class IV (complex) exists between class II (periodic) and class III (chaotic). As an application of these findings, the scale dependence of the symmetropy of the CA patterns is considered, and it is shown that class IV is useful for drawing complicated patterns when the system must keep the number of cells low.