The Four-Color Map problemDate: 11/11/97 at 23:25:27 From: Lisa Subject: The Four Color Map problem I just need help on what it is and where I can find information on it. I am having trouble finding information on it over the Internet. Thanks a lot. Date: 11/12/97 at 10:00:34 From: Doctor Anthony Subject: Re: The Four Color Map problem You can try Eric Weisstein's World of Mathematics, which will then refer you to a great deal of literature on the subject. http://mathworld.wolfram.com/Four-ColorTheorem.html I have not found a good summary as such on the Web, so will provide a brief note on the subject below. The subject was first raised in 1852 by a part-time mathematician, Francis Guthrie, who was colouring in a map of the counties of Britain. He was intrigued by the fact that four colours appeared to be sufficient regardless of the complexity of boundary shapes or how many regions had a commom border. He passed the problem on to University College London to the eminent De Morgan, who in turn passed it to the great William Hamilton. Hamilton was unable to invent a map which required five colours, but neither could he prove that no such map existed. Like Fermat's Last Theorem this apparently trivial problem generated a great deal of interest and activity. In 1879 a British mathematician, Alfred Kempe, published a 'proof' that was accepted by the mathematics establishment until in 1890 Percy Heawood of Durham University showed that the so-called proof was fundamentally flawed. The search continued, and like Fermat's theorem led to great advances in number theory, so the four-colour problem gave a stimulus to the new and increasingly important topic of topology. The first breakthrough in the four-colour problem came in 1922, when Philip Franklin ignored the general problem and settled for a proof which showed that any map containing 25 or fewer regions required only four colours. This was extended in 1926 to 27 regions, in 1940 to 35 regions, and in 1970 to 39 regions. Then in 1976 two mathematicians at the University of Illinois, Haken and Appel, came up with a new technique which would revolutionise the concept of mathematical proof. Haken and Appel used the ideas of Heinrich Heesch that the infinity of infinitely variable maps could be constructed from a finite number of finite maps. They reasoned that by studying these building-block maps it would be possible to attack the general problem. This proved very difficult in practice to achieve, because the number of building block configurations could not be reduced below 1482. To crank through all the permutations that might occur with this number of configurations would take a lifetime. Enter the age of the computer. In 1975, after five years of working on the problem, they turned to the new number-cruncher and in 1976 after 1200 hours of computer time they were able to announce that all 1482 maps had been analysed and none of them required more than four colours. The problem with this type of proof is that only another computer can carry out the customary check on its validity. Some mathematicians are most reluctant to accept it because no one, however patient, could work through the exposition line by line and verify that it is correct. It has been disparagingly referred to as a 'silicon proof'. The fact is, however, that mathematicians will increasingly have to rely on such methods. The age of the purist of pure logic as the only acceptable technique in mathematics has probably passed. -Doctor Anthony, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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