The Four Colour Theorem

 

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The famous Four Colour Theorem is concerned with mathematics as well as geography: it was first noted by August Ferdinand Möbius in 1840. In 1852 a student of De Morgan, then professor of mathematics at the University College London (who was also founder of the London Mathematical Society), conjectured with his brother, that any map or any figure divided in any way should be able to be coloured with four colours alone. This means that if you have a map of any number of states drawn on it, if you colour them with four colours, that would be enough to distinguish all the states and at no border on such a map will there be two states filled with the same colour.

Type of the problem can be studied through drawing of different diagrams like this one - see on the right-hand side of this page for a worksheet on this topic.

The problem persisted throughout the next twenty years or more and British as well as some American mathematicians tried to either prove or disprove this conjecture.

The colouring of geographical maps is a topological problem of a kind - it depends on the position of the countries, not on their shape, size, political systems or any other geographical, social or cultural features!

The first rule for colouring a map is that no two regions that are adjacent can be of the same colour (otherwise you wouldn't recognise the difference). It is ok for two regions to be of the same colour if they meet only at a point though. Until Francis Guthrie told his brother of the problem, and he in turn told his professor De Morgan, no one thought of proving that four colours suffice to colour any map!

Eventually the conjecture was proved by Wolfgang Haken and Kenneth Appel at the University of Illinois in 1976. They used a computer programme which run over 1200 hours and exhausted all the possible imaginary maps. This is one of the examples of the proof by exhaustion. There is another proof by exhaustion but I don't want to bore you to death by showing it to you.

 

   

See more about topology and the people who made it.

You can also visit a topological Atlas of the Internet here.

Download the worksheet on Four Colour Theorem by clicking on the number man.

Learn more about De Morgan and see the photograph of Francis Guthrie.

The Royal Geographical Society was the place where one of the important papers on the theorem were presented in the 19th century. It still exists - visit it here.

Here you can learn more about A Brief History of Map-making.

Or learn about the Mathematics of Cartography.

 

 

 

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