The mathematics of shoelaces (by Nuno Crato, Portugal) ...

There are no problems that mathematicians like more than those that are realistic and easy to formulate. Often they turn out to be the most difficult, and therefore even more interesting. This creates great enthusiasm for apparently trivial questions such as finding the best way to lace your shoes!

The possible lacing patterns have been studied by the mathematician John Halton, who considered them to be individual cases of the famous Traveling Salesman problem. This is a difficult mathematical problem, inspired by a real-life situation: a salesman wants to pass through a fixed number of towns, visiting each one once only, and with his starting and ending points fixed. After all, the pathway of a shoelace is equivalent to the salesman’s route, with the eyes (the holes through which the lace passes) representing the towns. The shortest pathway for the shoelaces is equivalent to determining the shortest route between all the towns.

The problem was approached anew by the Australian mathematician Burkard Polster and his work was published in Nature, one of the most prestigious scientific journals in the world. Polster systematically studied the different ways of lacing shoes.

It is a question that surprises people. For most of us there is only one acceptable method of lacing shoes. However, the reality is that people in different cultures use different patterns to lace their shoes. To give only two examples, there are the methods used in the USA and in Europe, which are the most common ways. In the former, the shoelaces are threaded in opposing zigzags, and when seen from above they seem to be crossed. In the latter, they are threaded in alternating zigzags in such a way that the eyes of the shoes seem joined horizontally by the shoelaces when viewed from above. There is also the “ex-factory” method in which the shoelace makes a continual zigzag from top to bottom and then returns in a diagonal line. Which do you think is the most efficient method?

The first curious fact is that there are an astronomical number of options. For shoes with two rows of five eyes, Polster verified that there are 51,840 different ways to thread the laces. This number increases to millions when the number of eyes rises.

Polster restricted himself to ways of threading the laces that necessarily occupy all the eyes and allow the eyes to be pulled together by applying pressure to the laces: for example, this implies that the laces may not pass through three successive eyes on the same side, as this would not exert any individual pressure on the eye in the middle. Then he defined the efficiency criteria. He stated that the security of the binding should be maximized and the compression of the laces should be minimized. Comparing the three above-mentioned systems, he verified that the most economical is always the American method, with the second-best system depending on the number of eyes. If there are four or more pairs of eyes, the European method is superior to the ex-factory method. In the case of three pairs of eyes they are equal. And in the case of only one or two pairs of eyes, the problem is trivial, as all three methods are equally good. If you try to verify this, you will see that it is not difficult.

American method	European method	Ex-factory method	“Bow-tie” method
Four different ways to lace your shoes

However, Polster did not restrict himself to studying only these three methods. He analyzed the problem from a general perspective, only taking the above-mentioned restrictions into account. He discovered that the most economical system is not one of the three commonly used methods. He discovered a less common pathway that he called the “bow-tie” method.

As far as the criterion of maximum security of the binding was concerned, he did not find any esoteric method, which is comforting. After all, the American and ex-factory methods are the best. When the rows of eyes are farther apart, the ex-factory method is the strongest. When the rows are close together, the American system is preferable.

Polster was inspired to write his mathematical work on shoelaces by an equally curious study that the computational physicists Thomas Fink and Yong Mao had published several years previously. It considered the various ways to tie a tie, a subject that gave rise to a book that they wrote in 1999, The 85 Ways to Tie a Tie: The Science and Aesthetics of Tie Knots,

In this study, which begins with a brief history of ties and then explains the mathematical theory of knots, the two physicists search for every possible type of tie knot, but are obliged to limit themselves to those that can be tied in less than ten moves. Even so, they find 85 ways to tie a tie. The simplest method requires only three moves. You start by placing the tie with the right side on your shirt, and the odd number of turns ensures that it ends up with the right side facing outwards, as is normal. This is called the oriental knot and is seldom used for occidental ties. It is followed by a four-move knot, which is more usual. Things become more complicated when the number of moves increases. One of the more impressive eight-move knots is the Windsor, which the eponymous Duke did not use, but it comes in handy when a larger-volume knot is desired. Many other knots are described, and perhaps one of them will come into fashion. Thanks to mathematics.