The Ising model is an example of very simply defined model that makes complex behavior emerge.
Originally introduced by Wilhelm Lenz and his graduate student Ernst Ising, its purpose was to understand why magnets lose their attractive power when heated past a certain temperature. The model was first tried in 1D, where it fails to show that a magnet stays magnetized, and therefore abandoned.
More than 20 years later, Lars Onsager tried to solve it again for the 2D case. The solution was published in 1944 but still did not attract a lot of interest.
In the 50s, when experimental measurements showed that the Ising model accurately predicted “critical exponents” of various gases, physicists eventually got interested.
At the critical temperature, islands of all sizes coexist, from dots to continents. Here, one arrow can flip another, distant arrow, despite their not being neighbors — an indication that the system’s macroscopic properties have detached from its microscopic details. This detachment is the magic of universality. All systems with the same number of dimensions and the same symmetries go through identical phase transitions, regardless of whether their microscopic parts are iron atoms, water molecules or little arrows.
A striking property of the model is it describes something that looks like a deep universal property of complex systems (composed of many components interacting locally with each other). This has been applied to Physics, Biology, and other fields. It also seems to indicate that complex systems are a relevant framework to analyze natural phenomena and try and reproduce some of its emergent properties.