Activity 2-1: The Game of Life

www.carom-maths.co.uk

The Game of Life is a fascinating simulation of how a population (of maybe bacteria?)

might grow if subject to a few simple rules.

It was invented by John Conway in 1970 and was gradually refined

by him and his team until it reached the form here.

The game is ideally suited to the computer, which can calculate successive generations

of the population for us with ease.

So what are these simple rules?

Each square is either black (alive) or white (dead.)

The action takes place on a square grid that shows the current generation of the population.

If a square is dead to start with, then it can become alive in the next generation if it has exactly three live neighbours.(This is called the ‘birth’ rule.) Otherwise , it remains dead.

Note that each square

has eight neighbours.

If, on the other hand, a square is live to start with, thenif the square has 0 or 1 live neighbours, it dies (of ‘loneliness’?)

If the live square has 2 or 3 live neighbours, it remains alive in the next generation.

If the live square has 4 or more live neighbours, it dies (through ‘overcrowding.’)

Before turning to the computer,it is a good idea to try these rules out for yourself

to see how they work.

Task: take some squared paper, and take the following as your starting population.

Use the rules to find the next four generations.

You should get this:

Configuration 5 just stays as it is.

Conway and his team found themselves asking several questions;

Are there starting populations that get bigger and bigger?

Can we find a starting shape that moves across the page?

(Conway called such a shape a ‘glider’.)

Can we find a starting population that generates gliders?(Such a configuration would be a ‘glider-gun’.)

Task: visit the link below for a computer program that will do the hard work for us. Invent some shapes of your

own, and then try the suggested shapes.

http://www.ies-math.com/math/java/misc/life_game_L/life_game_L.html

So shape A oscillates (period 2) while shapes B and C head towards a stable shape (the 2x2 square.)

Shape D arrives at an oscillating pattern, while Shape E is a glider.Shape F is periodic, period 3, and is called the pulsar.

Game of Life link

Some recurring patterns have been given names.

The patterns on the leftare famous enoughto have been named

appropriately.

Other types ofconfiguration

have been called breeders, puffers,

space fillers, and so on.

So is the glider gun possible?

Try this! This proves that some configurations can produce arbitrarily large numbers of live cells,

something that Conway initially doubted.

Try out some shapes of your own!

Stable reflector http://radicaleye.com/lifepage/patterns/p1/stillref.html

Puffer 1 http://www.youtube.com/watch?v=oHgLOOJ0mnY

Period-88 oscillator http://radicaleye.com/lifepage/patterns/p1/osc88.html

A huge range of possible behaviours are possible:you could try the ones below.

With thanks to:John Conway and his team.

Wikipedia for another very helpful article.To IES for their excellent collection of Java applets.

Paul Callahan for his terrific collection of Life material.Bill Gosper, Dean Hickerson and David Buckingham.

Carom is written by Jonny Griffiths, hello@jonny-griffiths.net