Have you ever had problems with your shoelaces? With broken shoelaces? With
shoelaces that constantly come undone? Tripped over your shoelaces? No, that is
not what I mean. What I have in mind are mathematical problems such as the
What is the shortest way to lace your shoes?
What is the strongest way to lace your shoes?
How many ways are there to lace your shoes?
On December 5, 2002, a short article of mine [21] that addresses these questions
appeared in the journal Nature. I don’t think anybody was more surprised than I
by the incredible amount of publicity attracted by this note on as innocent a topic
as shoelaces. In the weeks following its publication, the article was reported on by
virtually every major newspaper worldwide, and I received close to one thousand
e-mails in which people from all walks of life asked me about the mathematics
of shoelaces. This is even more remarkable since the article was not even one page
long and merely contained a summary of some of my answers to the above questions
without any proofs. This set of notes has been compiled in an attempt to provide
the comprehensive account of shoelace mathematics that many people have asked
me for.
To start with, pondering questions about the mathematics of shoelaces was not
much more than idle doodling on my part. It soon became clear to me that other
mathematicians had already thought about the shortest shoelace problem and had
come up with some very complete and neat theorems, arrived at via conceptually
appealing proofs; see [13], [14], [16], and [20]. This initial trend towards beauti-
ful results continued throughout my subsequent investigations, and, in the end, a
very complete picture emerged, consisting mainly of simple, beautiful, and often
surprising characterizations of the most common shoelace patterns, arrived at via
elementary, yet pretty and nonobvious, mathematics. I think such a picture is worth
painting in detail, as many mathematically minded people will be interested in it
for as long as they use shoelaces to tie their shoes.
Summary of Contents
In Chapter 1, we collect the most basic definitions and results about n-lacings of a
mathematical shoe, the mathematical counterparts of lacings of a shoe with n pairs
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