Gliders on the Stranded Cellular Automata Model
Alexa Renner
TL;DR
The paper develops a rigorous algebraic framework for Stranded Cellular Automata ($SCA$), defining grid patterns, generations, and gliders, and then proves a complete classification for $1$- and $2$-stranded gliders, including their speeds and turning/crossing rules. The authors introduce a comprehensive construction and continuity theory for grid patterns, enabling precise, repeatable descriptions of gliders and their subpatterns, shifts, and nesting. Key contributions include explicit catalogs of all $1$-stranded gliders and a thorough enumeration of $31$ two-stranded repeating patterns (with unique speed-$1$ and speed-$(-1)$ gliders), along with a formal treatment of pure gliders and nestedness. These results provide a rigorous foundation for systematic glider synthesis and analysis in the SCA framework, with potential implications for weaving-inspired cellular automata and algebraic pattern theory.
Abstract
The Stranded Cellular Automata (SCA) model consists of a grid of cells which can each contain between zero and two strands apiece and two turning rules that control when strands turn and when they cross. While patterns on this model have been studied previously, such research has not needed an algebraic description of the model. We provide a formal algebraic definition of patterns on the model, define gliders on the model in a way which is semi-compatible with definitions of gliders in other cellular automata models, and classify all 1- and 2-stranded gliders on this model. In addition, we prove an equivalence of two classes of gliders and design an algorithm to generate all such elements of that class.