Circumferential Curiosities

Science  10 Dec 2004:
Vol. 306, Issue 5703, pp. 1894
DOI: 10.1126/science.1106082

π A Biography of the World's Most Mysterious Number

by Alfred S. Posamentier and Ingmar Lehmann

Prometheus, Amherst, NY, 2004. 324 pp. $26. ISBN 1-59102-200-2.

The past 15 years have seen a surge in interest among the general public in popular books on mathematics. In particular, the six most famous numbers in mathematics—zero, π, e (the base of natural logarithms, approximately 2.7183), phi (the “golden ratio,” about 0.6180), i (the “imaginary unit,” the square root of negative one), and gamma (also known as Euler's constant, about 0.5772)—have each been covered in at least one such book. That these numbers have entered the realm of popular science is reflected in the subtitles of some of these books: “the World's Most Astonishing Number” (1), “The Biography of a Dangerous Idea” (2), and now π, by Alfred Posamentier and Ingmar Lehmann.

Reuleaux triangle.

The constant width of the closed curve equals the radius of the circles used to construct it.


Any new popular book on π must inevitably be judged against the standard set by the first of the line of these biographies: Petr Beckmann's A History of π (3), which has gone through several editions since it first appeared in 1970. The two books are about the same length as far as the text goes, and they target the same audience—the layperson with an interest in the history of science and mathematics. But here the similarities end. Whereas Beckmann's account is all history, Posamentier and Lehmann (mathematicians and mathematics educators at the City University of New York and Humboldt University, Berlin, respectively) write mostly about curiosities associated with π; only one chapter is exclusively history.


Some of these curiosities can be mind-boggling. My favorite is an exotic curve known as the Reuleaux triangle, named after its discoverer, the 19th-century German engineer Franz Reuleaux. Its construction is so simple that one wonders why no one before him had come up with the idea (see the figure). What makes this curve a close relative of the circle is the fact that it has a constant width—a diameter of sorts: no matter how one measures it, the distance d between two opposite points is always the same. But that's not all. A straightforward calculation shows that the circumference of the curve is equal to πd—exactly as with the circle. (However, the familiar formula for the area of a circle, π(d/2)2, does not work for this curve.) And lest one think that this is just an interesting curiosity, the Reuleaux curve was put to practical use in Felix Wankel's internal combustion engine, invented in 1957, which powered the 1964 Mazda. Posamentier and Lehmann effectively and engagingly describe the properties of this unusual curve.

Other curiosities, however, are elevated to a status they hardly deserve. For example, the authors devote a full 20 pages to a trivial “paradox” that could have been described in half a page. Imagine a rope tightly wound around Earth's equator. Now imagine a second rope, 1 meter longer than the first and floating in space above the equator. At what height will the second rope be hanging above the ground? The answer, just under 16 centimeters, is simply a consequence of the formula for the circumference of a circle. Let R be Earth's radius and C its circumference (both measured in meters). We have C = 2πR and C + 1 = 2π(R + h), where h is the required height. Subtracting the first equation from the second, we get 2πh = 1, so h = 1/2π ∼ 0.159 meters. The surprise is that the answer does not depend on Earth's radius; only the difference between the two radii matters. A surprise it is, but to elevate this to the status of a sensational paradox is, I think, a bit excessive.

The book's many curiosities include the value of π to one hundred thousand decimal places, taking up 28 pages. (The current world's record is 1.24 trillion places, a bit too long to be printed in a book of reasonable size.) And one will find the world record for memorizing the digits of π: over 42,000 digits, a feat achieved by Hiroyuki Goto in nine hours of reciting.

I enjoyed reading the book, if not for any new deep mathematical insights, then at least for its many applications, curiosities, and anecdotes. My enjoyment was tempered, however, by the presence of more than just a few typos, including erroneous formulas (which can be rather frustrating if one tries to work through them). Hopefully, a future edition will undergo a more thorough editing, so as to make this a truly pleasurable account.

References and Notes

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