Cellular automata, of which Conway’s thing is an instance, capture some essential feature of computation in an interconnected environment. Even with a 1D automaton, some very simple update rules are unexpectedly able to produce “complexity”… chaos, randomness ; and some are provably turing-complete. This has been studied by Stephen Wolfram (author of Mathematica, Wolfram alpha…) and explained in his “A new kind of science” book.
(I own a hard cover at home, this is cool stuff…)