Mathematical visions

The UK Institute of Mathematics and its Applications (IMA) celebrates its 50th anniversary this year. As part of the festivities, it has put together a book, 50 Visions of Mathematics, to illustrate the depth and reach of the subject. The 50 essays in the book cover both pure and applied topics, and even the most esoteric subjects are addressed in an accessible way. A good example is Yutaka Nishiyama's essay "The mysterious number 6174", which is about a process called Kaprekar's operation. Named after the Indian mathematician D R Kaprekar, the operation begins by taking a four-digit number in which the digits are not identical (say, 1964, the year the IMA was founded) and rearranging them to make the largest and smallest numbers possible (9641 and 1469 in this case). If you subtract the smaller number from the larger one, and repeat the same rearranging-and-subtraction operation for the result, you will eventually arrive at the number 6174. At this point the process reaches a dead end because 7641–1467 = 6174; hence, 6174 is the "kernel" of Kaprekar's operation for four-digit numbers. It makes a nice party trick, but Nishiyama argues that it could be more: "maybe, just maybe, an important theorem in number theory is hiding in Kaprekar's numbers". Several well-known writers (including Marcus du Sautoy, Simon Singh and Ian Stewart) have contributed essays to the collection, but the prize for effort must surely go to Thilo Gross, who wrote about applying Leonhard Euler's "Seven Bridges of Königsberg" problem to his (and Physics World's) home town of Bristol, UK. Unlike medieval Königsberg, the geography of modern Bristol is such that it is possible to devise a continuous path that crosses all of the city's major bridges exactly once. Unfortunately for Gross, who walked one such path on 23 February 2013, there are 42 major walkable bridges in Bristol, and the route that links them is 33 miles long. Now that's dedication.

  • 2014 Oxford University Press £24.99hb 208pp

Pyramid schemes

When it comes to book titles, authors of mathematics texts could learn a thing or two from the ancient Egyptians. Consider the scroll that today's scholars call, prosaically, the Rhind Mathematical Papyrus. Its authors, who lived in the second millennium BC, called this same text Accurate Reckoning, the Entrance into the Knowledge of All Existing Things and All Obscure Secrets. Despite the papyrus' fantasy-world title, though, its contents might disappoint schoolchildren: it's an introductory text on basic fractions. However, as the mathematician David Reimer vividly demonstrates in his book Count Like an Egyptian, even simple arithmetical operations were once performed in ways that now seem alien to us. Egyptian fractions, for example, had no numerator. When they wanted to express a quantity such as 2/5, they did so with a hieroglyph that "translates" literally as /3 /15; we would write it as 1/3 + 1/15. In Reimer's view, the Egyptian system has an advantage over ours because it makes estimating much easier. For example, it is immediately obvious that the Egyptian fraction 3 /2 /1310 is about 3 1/2, but the value of a fraction like 4586/1310 is much harder to evaluate at a glance. The Egyptians also had a different method of multiplication and division, and Reimer demonstrates convincingly that it beats "modern" long-division algorithms in terms of its speed, simplicity and ease of mastery. As he puts it, "somewhere in the heavens, Thoth, the scribal god, is looking down at us with a smile on his face thinking, 'Who's primitive now?' "

  • 2014 Princeton University Press £19.95/$29.95hb 256pp

Undiluted rubbish

The late Martin Gardner, who died in 2010, was one of the 20th century's greatest and most revered popularizers of mathematics. As the long-time author of the "Mathematical Games" column in Scientific American, he was responsible for introducing a wide, if quirky, range of mathematical topics to a popular audience, including Penrose tiles and John Conway's "Game of Life" simulation. A noted debunker of pseudoscience and a gifted amateur magician to boot, Gardner had lifelong interests in philosophy, theology and literature, as well as mathematics. In fact, since 1993 his work has inspired a regular series of fan-organized conferences called "Gatherings for Gardner". Unfortunately, Gardner's autobiography Undiluted Hocus-Pocus adds little to his legend. Gardner himself seems to have been aware of the book's failings, but calling his autobiography "slovenly" and "rambling" (as Gardner himself does) is insufficient. In truth it is nearly unreadable, with the chapters on his early life being particularly replete with references to now-obscure philosophical debates and anecdotes of the "maybe you had to be there" variety. Readers who stick with the book long enough may find that its disjointed style grows on them slightly. But Gardner was a prolific author. With so many better examples of his writing out there, there is little point in bothering with this one.

  • 2013 Princeton University Press £16.95/$24.95hb 288pp