2.4 The Game of Life
Now all of this may sound fine, philosophically speaking, but in practice, surely we, ourselves, can’t just be part of a mathematical structure? Equations don’t think! Well, yes, that’s true, if a little glib. By us being purely mathematical structures, or substructures, I don’t mean that we’re like programmed robots. What I mean is that, at the most fundamental level, there are no particles, only patterns, and by patterns, I mean the essence of the patterns, not the physical arrangement of things. In this scenario, the arrangement of particles and forces into chemical elements and stars and planets and people is no different from our current understanding of how things are put together, except that, ultimately, these particles and forces are purely mathematical structures. If our constituent parts are purely mathematical structures at the most fundamental level, then so are we!
An analogy may help: if you haven’t heard of it, let me introduce you to the Game of Life. I’m going to show you over the next four Steps that the Game of Life is purely a mathematical structure – it is not the simulation of the game that we see on our computer monitors – and yet substructures within the game can detect other substructures and react appropriately to them. Ultimately, we shall see that these purely mathematical structures can be “self-aware”!
The Game of Life (or simply “Life” to the afficionados) is not a competitive game; rather, it is played out on a two-dimensional grid of squares called cells, and displayed on a computer monitor screen. Once you have decided on the initial configuration of cells in the grid, you press the start button and see what happens to the cells you have chosen. A cell can be either alive or dead – typically, live cells are coloured black and dead ones white. The configuration of a cell – alive or dead – is determined by its eight neighbours. (Think of nine cells arranged in a square three cells wide by three cells high; then, with respect to the central cell, the remaining eight are its neighbours.)
The rules are simple:
- If a cell is dead and exactly three of its neighbours are alive, then that cell will come alive at the next time-step.
- If a cell is alive, it will die at the next time-step unless exactly two or three of its neighbours are also alive, in which case it will remain alive at the next time-step.
When these simple rules are applied simultaneously to every cell in the grid, effectively executing one time-step, then surprisingly complex patterns and processes can emerge. The Game of Life is an example of what is generally called a cellular automaton: a collection of cells on a grid arranged in a pattern that changes according to the rules of the game.
The Game of Life was developed in the late 1960s by the British mathematician, Professor John Conway, who left Cambridge in 1986 to settle in the USA, taking up the John von Neumann Chair of Mathematics in Princeton University. Conway always felt that the Game of Life rather overshadowed what he considered as his more important achievements in pure mathematics, but it is indeed the case that his is a household name among the millions of non-mathematicians who become engrossed with his invention.
Conway achieved overnight fame when the popular journal Scientific American featured an article on the Game of Life in 1970 – the columnist, Martin Gardner, received more readers’ letters about that piece than any other author whose work had been published by the journal up to that time. Among the many Life patterns that Gardner described was the glider, discovered by Richard Guy, a member of Conway’s group in Cambridge. Conway named it a “glider” for technical, mathematical reasons, but perhaps “duck” might have been more appropriate: the captivating feature of the glider is that it moves diagonally across the grid in a ducklike waddle, taking four time-steps to move one square horizontally and one square vertically.
John Horton Conway
A cycle in a glider’s diagonal progress across the board in the Game of Life. Live cells are coloured in black; dead cells are white. The number in each cell refers to the number of live cells neighbouring the cell. Starting at time-step h, these numbers, together with the two rules of the Game of Life, are sufficient to explain the configurations of the board in the subsequent time-steps.
The positions at five sequential time-steps are shown in the figure, where the time-steps are labelled h, h + 1, … h + 4. The number in each cell is the number of live neighbours for that cell. Using the above two rules, you should be able to see why a cell lives or dies from one time-step to the next. To avoid crowding the figure, I have limited the display to arrays of 7 × 7 cells, but the Game of Life is played on a much larger board, often nominally infinite in both directions, although, of course, computers will only display a finite subset of the total.
At each configuration of its four-stroke waddle, the glider retains its component five live cells, but Conway’s group found many other patterns that grew to enormous proportions, although these extended figures would commonly grind to a halt, with parts of the pattern vanishing, others remaining static and some local groups of cells caught in an endless loop of repeating shapes. Conway surmised that there would be no starting pattern from which the number of live cells would continue to grow indefinitely and Gardner even mentioned in his column that Conway had offered a $50 prize to the first person who could prove or disprove that conjecture before the end of the year.
Just three months later, Bill Gosper, an American mathematician and programmer at the Massachusetts Institute of Technology, discovered a pattern that oscillates over a period of 30 time-steps, emitting a glider at every repeat oscillation: the eponymous Gosper glider gun. Since the glider gun was forever adding five live cells to the game every 30 time-steps, Gosper was able to claim the $50 prize, his gun violating Conway’s conjecture that no pattern would grow without limit.
Now that gliders could be generated reliably and predictably with the Gosper gun, enthusiasts were quick to explore their potential. They soon found that two gliders whose paths intersected would destroy each other (if the aiming and timing were right), and they developed a range of tools to manipulate them including absorbing them (with an “eater”) and reflecting them directly back and at right angles without the tool destroying itself.
These facilities made it possible for gliders to store and transport information. Suppose you have a train of gliders moving diagonally across the board (they always move diagonally, of course), and suppose you aim at the train as it intersects the crosshairs of your Gosper gun: then you can pick off selected gliders in the train by firing your gun at the right moment, leaving holes in the train like gaping teeth.
The pattern made in a train of gliders is information that can be stored and transported, as in a stream of electronic pulses in a computer where a value “1” would indicate the presence of a glider and “0” would be the space where a glider should have been. In other words, the stream of gliders can represent a train of bits containing information. It is even possible to perform logical operations in the Game of Life, just as a digital computer does, by aiming the Gosper gun at these streams of gliders in particular configurations. With these logic operations, along with other tools like the reflectors which could, in essence, be made to capture the data streams and enclose them in the form of a memory, it became clear that it would be possible, in theory, to use the Game of Life to build and run an elementary computer called a Turing Machine!
