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Symmetry 2015, 7(4), 1768-1779; https://doi.org/10.3390/sym7041768

Effects of Initial Symmetry on the Global Symmetry of One-Dimensional Legal Cellular Automata

Department of Earth and Planetary Sciences, Kobe University, Nada, 657-8501 Kobe, Japan
Received: 10 July 2015 / Revised: 22 August 2015 / Accepted: 25 September 2015 / Published: 29 September 2015
(This article belongs to the Special Issue Symmetry and Asymmetry in Biology)
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Abstract

To examine the development of pattern formation from the viewpoint of symmetry, we applied a two-dimensional discrete Walsh analysis to a one-dimensional cellular automata model under two types of regular initial conditions. The amount of symmetropy of cellular automata (CA) models under regular and random initial conditions corresponds to three Wolfram’s classes of CAs, identified as Classes II, III, and IV. Regular initial conditions occur in two groups. One group that makes a broken, regular pattern formation has four types of symmetry, whereas the other group that makes a higher hierarchy pattern formation has only two types. Additionally, both final pattern formations show an increased amount of symmetropy as time passes. Moreover, the final pattern formations are affected by iterations of base rules of CA models of chaos dynamical systems. The growth design formations limit possibilities: the ratio of developing final pattern formations under a regular initial condition decreases in the order of Classes III, II, and IV. This might be related to the difference in degree in reference to surrounding conditions. These findings suggest that calculations of symmetries of the structures of one-dimensional cellular automata models are useful for revealing rules of pattern generation for animal bodies. View Full-Text
Keywords: cellular automata; pattern formation; symmetry; discrete Walsh analysis; complex system cellular automata; pattern formation; symmetry; discrete Walsh analysis; complex system
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This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited (CC BY 4.0).
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Tanaka, I. Effects of Initial Symmetry on the Global Symmetry of One-Dimensional Legal Cellular Automata. Symmetry 2015, 7, 1768-1779.

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