Patterns and algorithms

Recent work at Ohio State University on algorithmic pattern generation is reviewed. A display manifold (quadratic lattice) is filled with symbols satisfying axioms of some algebraic structure by using the production rules of the structure and some initial or boundary condition to start the filling process. The algorithms operate "locally" to generate "global" patterns, often of great beauty. The tremendous variety in pattern obtained with a fixed algorithm can be interpreted as "adaptations" of the algorithms to different conditions, imposed externally. Computer generated examples will be shown of both periodic and non-periodic patterns using a variety of systems, and some of the general theorems discussed, Examples from nature strongly suggest that inorganic and biological structures of many kinds may well be constructed in accordance with similar principles. Connections with other branches of mathematics and computer science (universal algebras, number theory, parallel computation, cellular automata) are pointed out.