RealLife: the continuum limit of Larger Than Life cellular automata

Abstract

Let A:={0,1}. A `cellular automaton' (CA) is a shift-commuting transformation of A^{Z^D} determined by a local rule. Likewise, a `Euclidean automaton' is a shift-commuting transformation of A^{R^D} determined by a local rule. `Larger than Life' (LtL) CA are long-range generalizations of J.H. Conway's Game of Life CA, proposed by K.M. Evans. We prove a conjecture of Evans: as their radius grows to infinity, LtL CA converge to a `continuum limit' Euclidean automaton, which we call `RealLife'. We also show that the `life forms' (fixed points, periodic orbits, and propagating structures) of LtL CA converge to life forms of RealLife. Finally we prove a number of existence results for fixed points of RealLife.