# Visions of Infinity: The Great Mathematical Problems by Ian Stewart. PART 2 of 2.

Author (Photo: n mathematics, the four color theorem, or the four color map theorem, states that, given any separation of a plane into contiguous regions, producing a figure called a map, no more than four colors are required to color the regions of the map so that no two adjacent regions have the same color. Two regions are called adjacent if they share a common boundary that is not a corner, where corners are the points shared by three or more regions.[1]

Despite the motivation from coloring political maps of countries, the theorem is not of particular interest to mapmakers. According to an article by the math historian Kenneth May (Wilson 2014, 2), “Maps utilizing only four colors are rare, and those that do usually require only three. Books on cartography and the history of mapmaking do not mention the four-color property.”

Three colors are adequate for simpler maps, but an additional fourth color is required for some maps, such as a map in which one region is surrounded by an odd number of other regions that touch each other in a cycle. The five color theorem, which has a short elementary proof, states that five colors suffice to color a map and was proved in the late 19th century (Heawood 1890); however, proving that four colors suffice turned out to be significantly harder. A number of false proofs and false counterexamples have appeared since the first statement of the four color theorem in 1852.... https://en.wikipedia.org/wiki/Four*color*theorem)
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Visions of Infinity: The Great Mathematical Problems by Ian Stewart. PART 2 of 2.

From Booklist
*Starred Review* Few of us share Stewart’s mathematical skills. But we relish the intellectual stimulation of joining him in exploring mathematical problems that have pushed even genius to the limit. We thrill, for instance, to the ingenuity of a great Chinese mathematician coming tantalizingly close to proving the centuries-old Goldbach Conjecture. And we feel the human meaning of mathematical achievement when a triumphant British analyst weeps before television cameras after finally proving a seventeenth-century algebraic theorem. We feel that meaning again when a brilliant Russian mathematician retreats into reclusive isolation, distressed because of initial skepticism toward his groundbreaking work on a nineteenth-century riddle. But high-level mathematics stirs deep emotions largely because it taps into the mind’s deepest impulses. Stewart repeatedly shows how a trivial mathematical curiosity can open up vital new conceptual insights. Readers learn, for example, that the apparently inconsequential four-color problem has led investigators deep into theoretical physics and has compelled fundamental rethinking of what constitutes a mathematical proof in a computerized age. Proofs incorporating computer-generated calculations may strike old-school mathematicians as unsatisfying, but Stewart assures readers that mathematics still depends on human investigators and that such investigators will not soon run out of daunting mathematical problems. A bracing mental workout for amateur mathematicians. --Bryce Christensen

Review Science News “As a guide to the inner workings of the mathematical jungle, Stewart provides an engaging and informative experience. If you wish to intelligently discuss the Riemann hypothesis, P/NP problems or the Hodge conjecture, you ought to read this book first.”

Choice “A designated math popularizer, Stewart writes books that are always enlightening and enjoyable.... Again, Stewart provides another interesting read for anyone intrigued by mathematics.”

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