Irrational Life

The Game of Life, invented by John Horton Conway over 30 years ago, is probably the best known cellular automaton. An infinite rectangular grid divides the plane into squares called cells. Each cell can be dead or alive, but at any given moment only finitely many cells are alive. Each cell has eight neighbors. The evolution in (discrete) time follows simple rules: if at the moment n a cell is alive and has two or three live neighbors, it stays alive at the next moment n + 1, otherwise it dies; if at the moment n it is dead and has exactly three live neighbors, then it becomes alive at the moment n+ 1, otherwise it stays dead. Both overcrowding and loneliness are intolerable. Cellular automata were introduced by von Neumann and Ulam in the late 40s. One of their goals was to study self-reproduction in these spatially discrete dynamical systems with spatially homogeneous, locally determined, discrete time dynamics. Cellular automata have since been studied from many other points of view, including computability and computational complexity, formal language theory, and statistical mechanics, and applied to model widely diverse phenomena in physics, chemistry, biology, and many other fields. Our primary tool will be substitution sequences. Given a finite alphabet and a suitable substitution rule, i.e., a finite word associated to each symbol in the alphabet, one obtains a sequence of symbols by starting with a symbol, replacing it with its associated word, similarly replacing each symbol in this word, and repeating this procedure. (Actually, one has to start with a letter for which the associated word begins with this letter, and this word has to have at least two letters.) The most