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From 4-Colour Theorem you can also connect to the Mapping theme


Pure Painting by Theo van Doesburg (1920)
The architect Theo van Doesburg, a founder member of De Stijl in 1917, was the driving force behind the movement. De Stijl was a group of Dutch modernist artists whose work focussed on painting, furniture, architecture and interior design. The strictly non-representational geometrical designs of van Doesburg, Piet Mondrian and the other members of the De Stijl circle have been adopted as part of the universal language of modern design. What were once avant-garde works of art have inspired the design principles of the modern world. "The demand for pure means of expression, first formulated by De Stijl, has become a fact." Theo van Doesburg

Above: Map of the United States of America with the states coloured using just four colours, the minimum number that are necessary to colour all planar maps.

The four colour problem is an odd mathematical problem peripheral to mainstream mathematics. It is a classic problem, however, in that it is very simple to state, whilst being extremely difficult to prove. A precise statement of the problem is:

Can every map be coloured with at most four colours, in such a way that neighbouring countries are coloured differently?

The problem is notable in that over the years several eminent mathematicians supplied deficient proofs that were subsequently shown to be flawed. One such 'proof', by the London barrister and amateur mathematician Alfred Bray Kemp was believed correct for eleven years from 1879 until 1890, when Percy John Heawood demonstrated that it was invalid.

The oldest surviving reference to the 4-colour Theorem is in a letter written on the 23rd October 1852 from Augustus De Morgan, Professor of Mathematics at University College London, to the famous Irish mathematician Sir William Rowan Hamilton. The problem was introduced to De Morgan by his student Francis Guthrie, subsequently a physics Professor, who had learnt of it from his older brother Francis, later a Professor of Mathematics. About 120 years went by between the first mathematical investigations of the problem and its solution in the 1970s, when Kenneth Appel, Wolfgang Haken and John Koch finally proved that four colours are sufficient, but this theorem remains the most controversial theorem in mathematics. The controversy arises because Appel and Haken were able to demonstrate that all maps can be coloured with at most four colours or are equivalent to about two thousand 'irreducible' maps.

They then wrote a computer program to check through these 2000 cases. Nobody doubts that the computer demonstrated the truth of the four colour theorem, however, some mathematicians remain uncomfortable about a proof that is so cumbersome that it requires a computer to check numerous exceptional cases that a human mathematician could not reasonably be expected to check.

One curious feature of the problem is that, although it is so difficult to prove with regard to maps on a plane, it is quite easy to deduce the maximum number of colours that are needed to colour maps on compact surfaces other than the sphere. (The sphere is equivalent to the plane, as a map on a sphere can always be projected onto a plane and vice versa.)

For instance, any map on a torus, or tyre-shape, can be coloured with at most seven colours. The illustration above represents a map on a torus that requires seven colours. To form a torus opposite edges of the square must be glued together. The animation on the right shows the resulting torus.

For more information about the four colour theorem, see 'Four Colours Suffice' by Robin Wilson (Penguin 2002).

Text and computer-generated artwork by Nick Mee.