Larger than Life: threshold-range scaling of Life’s coherent structures

The Game of Life has many coherent structures known as still lifes, oscillators, and spaceships. The most intriguing of these structures are the spaceships due to their ability to carry information across long spatial distances. Similar structures are supported by Larger than Life (LtL), which is a four-parameter family of two-dimensional cellular automata that generalizes the Game of Life to large neighborhoods and general birth and survival thresholds. Numerous examples of large range versions of Life’s spaceships are provided along with descriptions of the experimental methods used to find these objects. The empirical work illustrates that these structures are quite common, scale in a fairly coherent manner, and have a distinct geometry. A mix of rigorous results, questions, and conjectures are made about the existence of the generalized spaceships and other coherent structures for LtL rules with arbitrarily large neighborhoods as well as the convergence of such rules to “Euclidean automata”. Euclidean automata are deterministic rules that have Euclidean, rather than discrete universes.