# Ivars Peterson's MathTrek

December 23, 2002

## Lacing Shoes, Revisited

What's the best way to lace your shoes? This seemingly simple question, rooted in everyday life, can provoke passionate argument—and invite a mathematical response.

There are at least three common ways to lace shows, as illustrated (right): American (or standard) zigzag, European straight, and quick-action shoe store. Which lacing style you use depends on a variety of factors, ranging from aesthetic appeal to tying efficiency (or it may simply depend on the particular lacing style you learned as a kid).

Three common lacing styles.

Lacing patterns can be quite complex, and different patterns require different lengths of lace. Over the years, mathematicians and other researchers have studied these patterns, using various criteria to try to come up with the "best" lacing.

You might wonder which lacing pattern requires the shortest laces, for example. When computer scientist John H. Halton tackled the problem, he considered the shoelace question as a special, restricted instance of the classic traveling salesman problem, in which a salesman must visit customers in a number of cities scattered across the country and then return home, following the shortest possible route visiting each city only once.

In the shoelace version of the problem, you have to find the shortest path from the top eyelet (or lacehole) on one side to the top eyelet on the other side, passing through every eyelet just once. Having ventured into the realm of mathematical modeling, you can idealize the shoelace to be a mathematical line of zero thickness and the eyelets to be equally spaced points arranged in two columns.

It's then possible to calculate the lace length in terms of the number n of pairs of eyelets, the distance d between them, and the gap g between corresponding left and right eyelets. Of the three lacings that Halton considered, it turned out that if n is at least four, the shortest laces are always American, followed by European, then shoe store. For n = 3, American remains shortest, but European and shoe-store lacings are of equal length.

The American style also wins when the eyelets are irregularly spaced instead of being arranged in two neat rows. However, shorter lacings are possible if the lace doesn't have to pass alternately through the eyelets on the left and right side of the shoe.

Here are some alternative lacings you could try. The first two work only if your shoes have an even number of eyelet pairs. Watch out, though. You might find that by saving shoelace length, you end up with shoes that slip off your feet more easily or laces that break more often.

Now, mathematician Burkard Polster of Monash University in Australia has revisited the shoelace problem. In the Dec. 5 Nature, he argued that the lacing that uses the least amount of lace is a rarely used and unexpected type that he describes as a "bowtie" lacing.

Bowtie lacing.

In his model, Polster considered an arrangement of 2n eyelets, situated at the points of intersection of two vertical lines and n equally spaced horizontal lines. He specified that the shoelace visit all eyelets and that every eyelet contribute toward pulling the two sides of the shoe together. Polster then defined a "dense" lacing to be one that zigzags back and forth between the two columns of eyelets, as in the standard American lacing.

This model reveals that there are several other "reasonable" ways of lacing shoes. Of these, the bow-tie method is the most efficient in terms of requiring the shortest lace yet using all the eyelets. However, the two traditional "dense" styles win when you're looking for the strongest lacing—that is, the one that gives the maximum tension on both sides of the shoe. Which of the two is stronger depends on the distance between the two rows of eyelets: zigzag when the eyelets are close together and straight when they are farther apart.

Hundreds of years of trial and error have led to the strongest—if not the most efficient—way of lacing our shoes, Polster concluded. That's in the face of a staggering 51,840 possible lacings for a shoe with just five eyelets on each side, and millions more for shoes with a larger number of eyelet pairs!

References:

2002. A knotty problem—what's the best way to lace our shoes? Monash University press release. Dec. 5. Available at http://www-pso.adm.monash.edu.au/news/Story.asp?ID=829&SortType=1.

Clarke, T. 2002. Laces high. Nature Science Update (Dec. 5). Available at http://www.nature.com/nsu/021202/021202-4.html.

Gale, D. 1998. Tracking the Automatic Ant and Other Mathematical Explorations. New York: Springer-Verlag.

Halton, J.H. 1995. The shoelace problem. Mathematical Intelligencer 17(No. 4):36-41.

Isaksen, D.C. 2000. Shortest shoelaces. Mathematics Magazine 73(February):60-61.

Misiurewicz, M. 1996. Lacing irregular shoes. Mathematical Intelligencer 18(No. 4):32-34.

Peterson, I. 1999. How to lace like an ace. Muse 3(October):33. Available at http://home.att.net/~mathtrek/muse1099.htm.

______. 1999. The shoelace problem. MAA Online (Feb. 8).

Polster, B. 2002. What is the best way to lace your shoes? Nature 420(Dec. 5):476. Summary available at http://dx.doi.org/10.1038/420476a.

Stewart, I. 1996. Arithmetic and old lace. Scientific American 275(July):94-97. Feedback in Scientific American 275(December):118.