SKM 2023 – scientific programme

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DY: Fachverband Dynamik und Statistische Physik

DY 4: Pattern Formation, Delay and Nonlinear Stochastic Systems

DY 4.11: Talk

Monday, March 27, 2023, 12:15–12:30, ZEU 250

Sampling from the rule 150 fractal though an iterated stochastic process — •Jens Christian Claussen — University of Birmingham, UK

A widely known, but surprising way of sampling points from the Sierpinski fractal is through an iterated stochastic process where in each time step one of three operators is applied, which can be interpreted from their number representation, or as a geometric operation. While the Sierpinski fractal can also be generated by the rule 90 elementary cellular automaton (ECA), the ECA rule 150 generates a fractal pattern with a 2-step self-simularity resembling a generalization of a Fibonacci iteration [1]. Here we show that the rule 150 fractal can be generated without a 2-step iteration. We introduce a set of 6 operators, which allow to generate the rule 150 fractal from a stochastic process. We show that these 6 operators can be reduced to 4 operators, by adding one operator to the 3 operators from the rule 90 case. The operators for the rule 150 can be interpreted both from their number representation and geometrically. Further each point of the rule 150 fractal can be represented by a any base-6 number, or by a 4-letter symbolic sequence with a grammar restriction.

[1] Jens Christian Claussen, J. Math. Phys 49, 062701 (2008)