Applied Geometry
HOW ROUND IS YOUR CIRCLE?: Where Engineering and Mathematics Meet. John Bryant and Chris Sangwin. xxii + 306 pp. Princeton University Press, 2008. $29.95.
The great power of computers to model various aspects of geometry and mechanics has made it possible to visualize things quickly and in useful and innovative ways. But nothing beats the construction of a physical model. And when the model conforms exactly to the mathematical prediction, it is very satisfying. How Round Is Your Circle?, by John Bryant and Chris Sangwin, is a guide to making physical models of various phenomena of geometry that are related to serious applications, both historical and contemporary. The mathematics required is elementary: standard geometry and trigonometry, with occasional bits of calculus.
Let's start with the book's title, which is connected to the problem of how to determine whether a roundish object is exactly round, to a certain tolerance. This turns out to be much trickier than one would expect. For a start, there are the curves of constant width (such as the Reuleaux triangle, which is made by drawing three 60degree arcs of a circle centered at the vertices of an equilateral triangle). Because one can make such curves with many bumps, a device that just checks several diameters for equality can be fooled. The authors describe various ways in which one might try to confirm roundness, but they all have drawbacks, and when it comes to the definitive answer, Bryant and Sangwin admit that it takes very complicated machinery to perform a proper check (basically by rotating the given object around an axis).
The discussion of roundness leads naturally to a discussion of curves of constant width, and that is very well done here, with lots of detail. The authors describe two applications—the design of the British 50pence coin (a 7sided curve of constant width) and the design of the rotary (Wankel) engine used to power some cars. Bryant and Sangwin, who are British, are probably not aware of a beautiful American application: In San Francisco there are manhole covers in the shape of Reuleaux triangles (see image at www.drainspotting.com/view_photo.php?photoid=2662), which are easily distinguished from round covers, yet will not fall through the hole.
The authors do discuss in detail a littleknown application: a device that can drill square holes. I was aware that Reuleaux triangles could be used to make almostsquare holes, but they describe an extremely elegant device—based on a variation of the classic Reuleaux triangle—that makes exact square holes. Moreover, they have made a working model of such a drill, shown in the photograph at the bottom of the preceding page; for a short video clip of the drill in action, go to www.howround.com and click on "Applications of nonroundness." It appears to be quite a beautiful device.
Another chapter I enjoyed discusses shapes with odd balancing properties. Calculus teachers know that one can stack congruent bricks so that the overhang is arbitrarily large; this is done in a harmonic pattern, with the uppermost brick having an overhang of 1/2 of a bricklength, the next one 1/4 of a bricklength, the third 1/6, the fourth 1/8, and so on (see photo, which shows approximations to the harmonic pattern). To get the top brick to fully overhang the bottom one requires only five bricks; to get overhangs of two or more full bricks requires exponentially more bricks in the stack. But the book presents several variations, my favorite being a selfrighting stack: When knocked over in the direction of the overhang, it automatically rights itself to the overhung position! The authors also explain how to construct a polyhedron that is stable on only one face. I would have liked to see mention of the Gömböc (www.gomboc.eu/site.php), a recently discovered selfrighting body that has additional unusual properties. One error is the assertion on page 263 that the recurrence relation δ_{n+1}=(1+nδ_{n})/(2(n+1)) with δ_{1}=1/2 "does not seem to have a simple closed form solution." It does (δ_{n}=(1–2^{–n})/n), as I found out in a second using Mathematica's RSolve command.
In a chapter on the catenary (the curve formed by a chain hanging between two supports) and related topics, some standard material on arches and cables on bridges is discussed. Here, as happens throughout the book, there are gems mixed in with the standard material. For example, one learns the curious fact that when a parabola rolls along a flat surface, the locus of the focus is a catenary.
Several other chapters concern items that might be of little more than historical interest, such as how to use linkages to draw a straight line, how to make a ruler or protractor when you don't already have one, and how to construct slide rules of various types (Sangwin's grandfather is the inventor of a special slide rule used by ships' captains to help load freighters safely). Nevertheless, I found fascinating the chapter on the planimeter, a device that can measure area (on a map, say) by just tracing a needle around the boundary. The mathematics of this method is lovely, and the authors discuss both the math and the history in detail. So even though today one would just solve such a problem electronically (by counting pixels in a stored image), the cleverness of the devices is immortal.
There are many books that include ideas or instructions for making mathematical models. What is special about this one is the emphasis on the relation of model or toolbuilding with the physical world. The authors have devoted themselves to making wood or metal models of most of the constructions presented; 33 color plates nicely show off their success in this area.
Stan Wagon, a professor of mathematics and computer science at Macalester College in St. Paul, Minnesota, is the designer of a squarewheeled bicycle (pictured at www.stanwagon.com). He is the author or coauthor of a number of books, including The SIAM 100Digit Challenge: A Study in HighAccuracy Numerical Computing, with Folkmar Bornemann, Dirk Laurie and Jörg Waldvogel (SIAM, 2004); Which Way Did the Bicycle Go?, with Joseph D. E. Konhauser and Dan Velleman (Mathematical Association of America, 1996), and The BanachTarski Paradox (Cambridge University Press, 1985).
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