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Fun with math

October 1st, 2010

BY JOEL HESS

I just play all the time and am fortunate enough to get paid for it.
— Martin Gardner, 1998

Although this column is primarily concerned with language and linguistics, I would be remiss not to acknowledge the passing in May of one of the heroes of my youth, the inestimable Martin Gardner.

Gardner, born in Tulsa in 1914, never took a math course beyond high school, where he struggled with calculus. Even though at first he considered himself poor at mathematical puzzles, and majored not in mathematics but in philosophy at the University of Chicago, he ended up almost single-handedly reviving interest in recreational mathematics in the United States, first through his editorship of the children’s magazine Humpty Dumpty, where his innovative stories, puzzles and games in the 1950s inspired multitudes of wide-eyed kids, and later through his column entitled “Mathematical Games” in Scientific American from 1956 to 1981, and in a series of books based on those columns, in which he entertainingly delved into such mathematical curiosities as flexagons, game theory, tangrams, Penrose tiling, polyominoes, fractals, the board games Nim, Hex, and Mill, the artwork of M.C. Escher, Turing machines, hypercubes, Möbius strips, and much more. He even touched on recreational linguistics through his explorations of codes and ciphers.


Gardner was a true modern Renaissance man, with wide-ranging interests. He was a recognized authority on Lewis Carroll, and his annotated edition of Alice’s Adventures in Wonderland and Through the Looking-Glass remains his best-selling work. He was a great devotee of L. Frank Baum’s Oz books and a cofounder of the International Wizard of Oz Club. He was, with his good friend the stage magician The Amazing Randi, a founder of the Committee for the Scientific Investigation of Claims of the Paranormal, and a lifelong opponent of such pseudoscientific hogwash as UFOs, Scientology, astrology, ESP, dowsing, and creationism. He was fascinated by religious philosophy and considered himself a theist, although repulsed by most organized religion and acknowledging that his own faith was neither confirmable nor disconfirmable. The publicity-averse Gardner, after living most of his life in New York and North Carolina, returned to his native Oklahoma in 2002 after his wife’s death to be close to his son James, who teaches at the University of Oklahoma.

I remember as a teen eagerly awaiting each new volume of Martin’s endlessly intriguing problems and patient explanations, and by my twenties I had amassed a small library of his books, of which there are more than seventy. His column was one of the few things in Scientific American I could regularly wrap my poor, science-challenged brain around. (I was quite adept at mathematics, and even won the math prize in 9th grade, but I was never very good at its practical applications. To this day, I’m much better at wrestling with abstract theories than at extending them to the messy realities of everyday life.)

Gardner introduced me to many fascinating anomalies of geometry, number theory, probability, statistics, and the like. For instance, let’s take a celebrated problem in topology and graph theory, the four-color map theorem, which I first encountered in Gardner’s books. I had always been fascinated by maps and geography and spent many a lonely afternoon browsing through atlases (call me the ultimate nerd), so this topic immediately piqued my interest. The theorem states that, for any map that can be drawn on a plane surface, four colors are always sufficient to color the map so that no two contiguous areas are of the same color. The theorem was proposed in 1852 by an Englishman, Francis Guthrie, later a professor of mathematics in South Africa, who noticed while coloring a map of the counties of England that he was always able to color them using only four different colors no matter where he began to color. He took the idea to his brother, Fredrick, who in turn brought it to the attention of his teacher, Augustus De Morgan, at University College. It soon spread throughout campus and later to the entire mathematical world, and repeated attempts were made, with limited success, during the nineteenth and twentieth centuries either to produce a proof or to find a map that would require more than four colors. Though on its suface the theorem seemed simple enough, it turned out to be surprisingly difficult to prove. In 1890, it was proven that five colors were sufficient to color any map, but to bring that number down to four defeated all efforts. In fact, I can remember spending significant amouts of time in my room with my colored pencils experimenting with increasingly wild map colorings, trying in vain to become the person who would find the counterexample that would disprove the theorem and thus attain mathematical glory (as I said, I was quite the nerd).

But the proof for four colors remained elusive until 1976, when Kenneth Appel and Wolfgang Haken of the University of Illinois used a special computer program to finally yield the long-sought-after result. The announcement was not without controversy, as this was the first time computers had been employed to prove a major theorem, and some mathematicians would not accept a proof that could not be verified manually. Since then a simpler proof, though one still requiring computers, was developed, and its validity is now generally, though not universally, accepted. Gardner wrote several fascinating and informative columns on the theorem, including one celebrating its successful proof, thus elevating an obscure and, to tell the truth, not extremely significant problem in an arcane branch of mathematics to the level of art.

As someone who made mathematics learning fun for generations of Americans, Gardner was a national treasure, and we owe him a debt of gratitude in this era of intellectual midgets. •

Posted by: The Editors
Category: Hess | Link to this Entry

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