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.
Highlights
IntroductionLeopards living in closed habitats, such as forests, have a frequency of stripes that is much higher, more irregular, and more complex than those living in open habitats [1]
Animals have many kinds of pattern formation
The results show that under all initial conditions, the numbers of symmetries of the cellular automata (CA) patterns correspond to three qualitative classes of CAs, identified as Classes II, III, and IV, according to Wolfram’s original classification [7,8]
Summary
Leopards living in closed habitats, such as forests, have a frequency of stripes that is much higher, more irregular, and more complex than those living in open habitats [1] These body patterns reflect adaptations to the surrounding environment and, pattern formation is substantially related to the conservation of species [1]. Studies have revealed the theoretical process of pattern formation, which reduces entropy and increases specific symmetry (self-organization) [4], and the real biological process of pattern formation, which reduces entropy and increases specific symmetry. Pattern formation of a biological or animal surface does not necessarily start from such a random state, but rather from a regular state. Clarifying the theoretical process of regular pattern formation has the possibility of revealing general features and the real biological process of pattern formation
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