Extended "Beehive" Rule Cellular Automaton Dynamics on Irregular Lattices: A Preliminary Study in the Form of a Visual Catalog

A 3-value cellular automaton (CA) is implemented on hexagonal lattices with irregular topologies. A lattice irregularity is defined as the substitution of a hexagon with an alternative polygon. For this study, polygons are constrained to the set [e|3 ≤ e ≤ 9], where e defines the number of edges of the substitute. Six irregular lattices are explored, each with a centrally located hexagonal cell replaced. The k-totalistic “beehive” rule is modified to account for higher-valued neighbor conditions where e > 6. The behavior of two starting configurations is demonstrated on a regular hex-lattice to provide a control. The same starting configurations are then implemented on the irregular lattices. The starting position of the configurations is varied to explore the influence of the irregularity on dynamics. For each irregular lattice, extended and novel dynamics are demonstrated in comparison to the control. Sensitive dependence on starting positions relative to the lattice irregularity is observed. The extension of spatio-temporal dynamics resulting from interactions with heterogeneous lattice features holds relevance for the representation of many dynamic systems. Irregular lattice topologies influence curvature in various physical systems, presenting opportunities for implementing embodied, situated, and performative CAs.