Have you ever wondered how complexity can arise from the simplest rules?

That’s the strange beauty of Cellular Automata — pretty grids of colourful little squares that update step by step, following rules so simple you could write them on a napkin, yet producing patterns of astonishing variety, beauty, and complexity.

Researchers study them in two main ways. From a dynamical perspective, we ask: what kinds of patterns emerge after many steps? From a computational perspective, the question is: how powerful are they? Could they, for instance, rival your abilities — or even those of your computer?

This summer, I had the chance to bring these two perspectives together. Working at the Chair of Statistical Field Theory at EPFL, I first developed a framework to classify Cellular Automata by their dynamical properties and then asked: which of these properties actually matter for computational power? More precisely, we focused on the computational ability of one Cellular Automaton to simulate others — that is, to reproduce the behaviour of another automaton under certain “translation” conditions.

The surprising — though perhaps not the hoped-for — result is that it doesn’t seem to take any very specific kind of dynamics! Automata with very different dynamical “personalities” — surjective, chaotic, mixing, transitive — might all still be able to pull off some pretty impressive computations. In other words, these squares aren’t just pretty to look at, they can also be clever in many different ways!

The broader message is that dynamics and computation are not isolated viewpoints, but neither are they equivalent. They are complementary. The richness of Cellular Automata lies precisely in the diversity of ways these aspects interact — a reminder that complexity often emerges where we least expect it… a bit like flowers in the street, or bad grades on your transcript.