The automaton known as `Langton's ant' exhibits a dynamical transition from a disordered phase to an ordered phase where the particle dynamics (the ant) produces a regular periodic pattern. Despite the simplicity of its basic algorithm, Langton's ant has remained a puzzle in terms of analytical description. Insight can be gained from the one-dimensional and the triangular two-dimensional versions of this automaton for which analytical calculations can be performed, and interesting properties can be proved. We suggest that this class of simple automata may provide a prototype model for low-dimensional reactive processes.
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