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One, Two, Tie My Shoe...

Being a bibliophile, I am especially attracted to books with unusual titles. For example, consider Burkard Polster's The Shoelace Book: A Mathematical Guide to the Best [and Worst] Ways to Lace Your Shoes. My first thought....are you kidding me? Nonetheless, the 125-page book was a pleasant surprise when I got a copy.

The book is written in all seriousness, and is mathematically sound (note that it is published by the American Mathematical Society). Yet, by reputation, the author Burkard Polster is a mathematical juggler, magician, origami expert, bubble-master, shoelace charmer, and "Count von Count" impersonator. And, the article itself is an expansion of an article on the same topic that he published in 2002 in Nature.

Consider the different lacing possibilities: crisscross, zigzag, bowtie, devil, angel, or star...and then ask which are the longest, the shortest, the strongest, and the weakest lacings? Most of the mathematics needed is quite sophisticated, though stated to be at the level of combinatorics and calculus. Finally, the author claims that his "book will be enjoyed by mathematically minded people for as long as there are shoes to lace."

Here is some information gleaned from the book and other resources:

  • In 1999, British physicists Thomas Fink and Yong Mao published an article (also in Nature) about the mathematics underlying the various ways one might tie a necktie. This article led to the publication of their book The 85 Ways to Tie a Tie, which focuses on the elements of symmetry, aesthetics, and length of the tie.
  • Polster assumes that a “mathematical shoe” has "2n eyelets, arranged in the plane in two vertical columns of n each, with adjacent eyelets in a given column separated by a vertical distance of h, called the stretch of the shoe, and horizontal pairs of eyelets in each row separated by a distance of 1. An n-lacing of the shoe consists of a closed path in the plane made up of 2n line segments whose endpoints are the 2n eyelets, with two line segments sharing each eyelet." See, it has mathematics!
  • Determining the number of simple n-lacings turns is quite complicated, requiring solving for the five roots of a quintic polynomial.
  • As a specific case, if you wear a boot with eight pairs of eyelets, you can choose from a total of 52,733,721,600 possible lacing patterns.
  • As you might expect, the zigzag lacings are the longest of the simple lacings.
  • Determining the length of a shoelace path is a version of the traveling salesman problem.
  • The author includes something called the “shoelace formula,” which computes the area of a simple closed polygon in the plane.
For a more complete overview of this unusual resource or to sample sections, consider Colin Adams review in Notices of the AMS (December 2006) or the Google link .

If nothing else, get this book to put it on your shelf or coffee table....it will initiate discussions. I doubt you will sit down and read it, but I expect you will browse through it....while you retie your shoes!