Larger than Life’s Extremes: Rigorous Results for Simplified Rules and Speculation on the Phase Boundaries

Larger than Life

(LtL), is a four-parameter family of two-dimensional cellular automata that generalizes John Conway’s

Game of Life

(Life) to large neighborhoods and general birth and survival thresholds. LtL was proposed by David Griffeath in the early 1990s to explore whether Life might be a clue to a critical phase point in the threshold-range scaling limit. The LtL family of rules includes Life as well as a rich set of two-dimensional rules, some of which exhibit dynamics vastly different from Life. In this chapter we present rigorous results and conjectures about the ergodic classifications of several sets of “simplified” LtL rules, each of which has a property that makes the rule easier to analyze. For example, these include symmetric rules such as the threshold voter automaton and the anti-voter automaton, monotone rules such as the threshold growth models, and others. We also provide qualitative results and speculation about LtL rules on various phase boundaries and summarize results and open questions about our favorite “Life-like” LtL rules.