# Surreal numbers and monster groups

## The life of John H. Conway

John H. Conway, who died on April 11th, was one of the most original and innovative mathematicians of our time. He made several important discoveries in topology and group theory, finding some very large exceptional finite groups, , and developed a novel numerical system, based on a close correspondence between numbers and games, which he termed surreal numbers. He considered this his finest achievement, but he was best known among the wider public for inventing the world’s first cult computer game, which exploded onto computer screens around the world in 1973.  While much of Conway’s work, including his surreal numbers, were based on the theory of two-player games, this game requires only one player, whose only move is to set up an initial pattern of squares on a square grid, like an infinite chessboard, and then sit back and watch as the chosen pattern is repeatedly transformed by a very simple rule, which selects, each move, those squares which have exactly 3 adjacent squares selected the previous move, and deselects those which have fewer than 2 or more than 3 . The entire subsequent development of the game is then determined, but it is, in general, very difficult to predict what will happen without actually playing the game: even simple initial conditions lead to complex shifting patterns spreading across the screen in an apparently autonomous manner, which gave the game its name, the game of Life. Conway initially conjectured that every game would eventually fizzle out to a fixed state, but it was soon discovered that there are patterns which perpetuate themselves while moving through space, meaning that some games continue to expand indefinitely. Such stable yet shifting shapes are called gliders, and it was later discovered that there are other shapes which emit a continuous stream of gliders: these were called glider guns. It was then realised that streams of gliders could be seen as carrying bits of information, in the same way that charge or voltage does in a computer, and that judiciously placed glider guns could act as logic gates on these information streams, and in this way all the essential components of a computer could be constructed within a game of Conway’s Life, showing the game to be Turing complete, capable of performing any computation whatsoever. In particular, if consciousness is computational, then it can be embodied in a game of Conway’s Life.

Conway’s Life, in which each square in a grid reacts according to the state of its neighbours, is the best known example of a cellular automaton. Cellular automata were invented by John von Neumann to solve the deep philosophical problem of reproduction: how can one become two, or two become three? Since antiquity it was recognised that, in humans at least, sperm has something to do with it, which lead some to believe that each sperm must contain a tiny homunculus. Von Neumann devised a complex cellular automata as a theoretical model to demonstrate that there was no logical or philosophical contradiction involved in a shape reproducing itself. His model relied on structures encoding information about themselves, a few years before the discovery of DNA revealed a similar process unfolding in every living cell. Conway’s life also contains self-replicating shapes. Given a big enough grid, it’s been suggested, these self-replicators would evolve into organisms of ever greater complexity.

Von Neumann realised that, while the laws of physics are far too complicated to model accurately, we can gain important insights by studying fictitious realities. The game of Life does not model the laws of nature as we know them, but it shows the kind of behaviour that might evolve in any possible universe. The shifting patterns of gliders have notable similarities to particles in our universe, reacting to other gliders in complex ways that allow the evolution of self-replicating shapes, logic gates and even universal computers within the cellular universe.

Conway’s life is more than just a game: it’s a research tool, a philosophical thought experiment, and one of the earliest examples of generative algorithmic art. It has given birth to the field of ALife, which explores computationally the space of possible lifeforms. It is a perfect demonstration of the phenomenon of computational emergence, where simple rules produce a complex behaviour, out of which arises a simplicity of form which is in no way apparent from the rules that produce it. Steven Wolfram has shown that this emergence is ubiquitous even in 1-dimensional cellular automata, and it’s been shown that even these can be Turing complete. Variations on Conway’s Life transpose the action to continuous or multi-dimensional realms (you can try some here) but Conway’s game remains the classic archetypal example of the genre.