What is the best way to lace your shoes?

by Burkard Polster and Marty Ross

The Age, 26 October 2009

  
Recently, we (perhaps) figured out a shortest tour of Victoria: see The Maths Masters' Tour de Victoria. Today we'll take you on a different tour.

When you lace and tie one of your shoes, the shoelace takes a tour of the eyelets. Two of the most popular such tours are the crisscross and zigzag lacings. However, there are many other possible lacings.

Which among these tours give the best lacing? To answer this we must first decide what we mean by "best". The simplest notion to capture mathematically is "best = shortest". So we'll begin by considering the lengths of lacings.

It is quite obvious that the lacing on the left is the shortest, while also being close to useless. We can preclude such short but dumb lacings by requiring that each eyelet actually contributes to pulling the two sides of the shoe together. This amounts to having one or both segments ending in the eyelet connecting to an eyelet on the other side of the shoe. This is the case for all but the first lacing above.

It turns out that, no matter the dimensions of the shoe and no matter the number of eyelets, the shortest useful lacing will always be the bowtie. As well, the lacings shown in the diagram are always in order from shortest to longest.

These are actually very surprising statements. For example, for a shoe with six pairs of eyelets there are 3,758,400 different useful lacings to compare. For God's Shoes, with 100 pairs of eyelets, the number of different useful lacings has grown astronomically, to:

1860904268074008196621915961820296614767464863477
6613609422844999556517444397765244079670698785078
8147662241595883722753596128522456064037632600625
9102537775643671738498496513186139929744880583816
8971818396349427731395000391555199195008915051014
0333395023660675502304646778723239869363765334033
2284043277107200000000000000000000000000000000000
000000000000

In most familiar lacings there are no vertical segments, with every segment connecting opposite sides of the shoe. Among all such lacings, the crisscross is always the shortest possible, and the devil lacings shown on the right are the longest possible.

There is an alternative notion of "best lacing", which is perhaps more natural. We can view a lacing as a pulley system, pulling the two sides of a shoe together. We can then compare the strength of different lacings. Here there is no clear winner: depending upon the dimensions of the shoe, either the crisscross or the zigzag is the strongest. For dimensions close to those of real shoes these two lacings are about equally strong.

So it turns out that the very popular crisscross scores well in terms of both length and strength. In addition, it is easy to remember, symmetric and pretty. It is fair to conclude that crisscross is indeed the best way to lace your shoes.

Finally, we should mention that the world's two leading shoelace experts reside in Melbourne: Maths Master Burkard, who did the maths for an article in the journal Nature  in 2002; and Ian Fieggen, who knows absolutely everything there is to know about shoelaces. Check out his amazing website

Burkard Polster teaches mathematics at Monash and is the university's resident mathemagician, mathematical juggler, origami expert, bubble-master, shoelace charmer, and Count von Count impersonator.

Marty Ross is a mathematical nomad, currently lecturing at the University of Melbourne. His hobby is smashing calculators with a hammer.



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