Are periodicity and symmetry the properties of a discrete space? (On one paradox of cellular automata)

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• 作者：Vladimir Ya Shevchenko ; Sergey V. Krivovichev
• 关键词：Crystal structure ; Periodicity ; Aperiodicity ; Cellular automata ; Giant unit cells ; Complexity
• 刊名：Structural Chemistry
• 出版年：2017
• 出版时间：February 2017
• 年：2017
• 卷：28
• 期：1
• 页码：45-50
• 全文大小：
• 刊物类别：Chemistry and Materials Science
• 刊物主题：Computer Applications in Chemistry; Physical Chemistry; Theoretical and Computational Chemistry;
• 出版者：Springer US
• ISSN：1572-9001
• 卷排序：28

文摘

We consider a three-color cellular automaton (CA) that acts on a square lattice. In ternary numeral system, the CA has the number 111010011111110111011102010, which corresponds to 3704707887996 in the decimal system. With the finite size of the input row, CA-3704707887996 generates aperiodic structures with repeating self-similar patterns. In order to investigate the formation of periodic patterns by the CA-3704707887996, the input row was used that represents a periodic structure containing variable number of cells. The behavior of the CA-3704707887996 work for the input rows with periods up to 21 cells was studied. All generated patterns were analyzed from the viewpoint of their periodicity along the vertical direction. It was found out that the behavior of the system depending upon the length (l_input) and structure of the periodic input row is strongly nonlinear. The complexity C of a pattern considered as a size of a unit cell increases exponentially with the increasing l_input value and can be described by the function C = 1.8986 exp [0.3334 l_input], with the correlation coefficient R2 = 0.8967. As a rule, initially the CA generates a metastable aperiodic structure and then adopts a stable regime of generation of a stable periodic pattern. The abstract properties of the CA-3704707887996 have several important consequences for structural chemistry: (1) relatively simple schemes of local interactions of particles may result in the formation of very complex structures depending upon the initial conditions and abstract properties of the interactions; (2) symbolic complexity of the generated patterns increases exponentially depending upon the structure of initial conditions; (3) under certain initial conditions, explosive fluctuations of complexity are possible that lead to the formation of giant superstructures with extremely large periods; and (4) under certain conditions, the border between aperiodic and periodic structures virtually disappears.