42. Thomas Fink and Yong Mao, The 85 Ways to Tie a Tie.

The first third of this slim volume is the history of male neckwear, including such factoids as how the terracotta army turned necktie scholarship on its ear, what the first verse of Yankee Doodle is about, and that while the Duke of Windsor's characteristically large knot can be emulated with a so-called Windsor knot, the Duke himself wore padded ties. (That last makes me snicker every time I repeat it; I keep wanting to make a joke about him stuffing his tie with kleenex.)

From there the book goes into a fast overview of knots in the topological sense (in which we learn that the four-in-hand is a buntline hitch), and a prose description of the authors' definition of a "legal" necktie knot. (Much to my disappointment, the eighty-five knots in the title are not a cataloging of all the possible ways to put a knot in a tie, but all the possible ways to build a "respectable" knot, something you might be willing to wear to a job interview. In this case, "respectability" includes somewhat-arbitrary limits on a knot's symmetry and maximum size.) For those who are frustrated by prose descriptions of mathematics, the mathematical details are included in an appendix; for those who do not want to deal with the formal mathematics, the formal mathematics has been placed well out of your way in the appendix.

And then we get to the knots themselves. All eighty-five possible knots are described in notation and diagrams, and the history of the most-aesthetic few in each class (as defined by the mathematics) is discussed. Throughout this section there are many photos of famous people wearing neckties, but unfortunately, the particular knot used in any given photo is almost never identified. While reading all these detailed descriptions of the final forms of the twenty-odd most popular knots, I very much wanted side-by-side standardized photos. (Mostly because I doubt. Are they really as distinctive as all that?)

All in all, it's a readable little treatise that demonstrates how one can define a solution space for a not-so-abstract problem. Back in my pure-math combinatorics days, I could see handing this off to someone who was trying to get a vague sense of what math "looks" like to mathematicians (or, similarly, for those who wanted a sample of what mathematicians "do"). It could also be nice browse-through for trivia hounds. I wouldn't be surprised, though, if its most frequent usage was as a coming-of-age gift for a young person who was learning to tie his or her first necktie.