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DY: Fachverband Dynamik und Statistische Physik
DY 10: Nonlinear Dynamics, Synchronization and Chaos
DY 10.13: Talk
Monday, March 18, 2024, 18:15–18:30, BH-N 128
Self-similar growth patterns in 2-dimensional von Neumann elementary cellular automata: a complete exploration of all outer-totalistic rules — •Kian Siadat and Jens Christian Claussen — School of Computer Science, University of Birmingham, UK
Growth patterns of cellular automata emerging from a localized single seed initial condition have intrigued the nonlinear dynamics community through the identification of two universality (sub)classes within the Wolfram class IV cellular automata [1], where rule 90 (Sierpinski) and rule 150 are representatives of these classes. In this work, we perform an exhaustive exploration of all 2-dimensional cellular automata rules which in addition obey the criterion of outer-totalistic, i.e., the output depends on the number of active neighbour cells, but not about their spatial pattern. While being a subset of all possible 2-dim CA rules, this is the most central class of CA rules as they retain a maximum of symmetries. We identify several distinct time series related to the growth patterns, and beside the 1-dim and 2-dim versions of rule 90, we also identify a rule providing a triple replication, and generating a 2-dim spatial Sierpinski pattern.
Our findings indicate that despite the large number of possible cellular automata rules, we observe only a few self-similar growth patterns and associated time series.
[1] Nagler, J. Claussen, J. (2005) 1/fα spectra in elementary cellular automata and fractal signals, Phys. Rev. E 71, 067103 (2005)
Keywords: Elementary cellular automata; Self-similar growth patterns; Outer-totalistic cellular automata; 1/f scaling; universality classes