viii Preface
of eyelets. Here, a mathematical shoe consists of 2n eyelets which are the points of
intersection of two vertical lines in the plane that are one unit apart and n equally
spaced horizontal lines that are h units apart. An n-lacing of a mathematical shoe is
a closed path in the plane consisting of 2n line segments whose endpoints are the 2n
eyelets of the shoe. Furthermore, we require that given any eyelet E, at least one of
the two segments ending in it is not contained in the same column of eyelets as E.
This condition ensures that every eyelet genuinely contributes towards pulling the
two sides of the shoe together or, less formally, that a lacing doesn’t have “gaps”.
We introduce four important special classes of n-lacings. The dense n-lacings
are the n-lacings in which the shoelace zigzags back and forth between the two
columns of eyelets. The straight n-lacings are those n-lacings that contain all possi-
ble horizontal segments. The superstraight n-lacings are the straight n-lacings all of
whose nonhorizontal segments are verticals. Finally, if, when you trace an n-lacing,
you move exactly once from the top to the bottom of the shoe and once from the
bottom to the top and if you neither “backtrack” on the way down nor on the way
up, then the n-lacing is called simple.
We describe some families of n-lacings, representatives of which are actually
used for lacing real shoes and which pop up in the different characterizations that
this set of notes is all about. See the diagram on the next page for a quick visual
description of these and other important families of n-lacings and a summary of
the most important such characterizations. For example, the two most commonly
used n-lacings are the so-called crisscross and zigzag n-lacings, which are featured
on the left side of the diagram. As you can see, they both have very neat extremal
In Chapter 2, we consider one-column n-lacings. Imagine pulling really hard
on the two ends of the shoelace in one of your shoes that has been laced using a
straight lacing. Then, if the lacing does not get in the way and if your foot is narrow
enough, you will end up with the two columns of eyelets superimposed, one on top
of the other. This means that we do not have to distinguish any longer between
the two columns of eyelets and what we are dealing with is a one-column n-lacing,
that is, a lacing of just one column of n eyelets in which every eyelet gets visited
exactly once. We identify the shortest and longest one-column n-lacings and find
the numbers of such lacings. In a final section, we describe a simple method that
allows us to construct all straight n-lacings that contract to a given one-column
n-lacing. This method plays an important role in deriving the shortest and longest
straight n-lacings in subsequent chapters.
In Chapter 3, we derive one formula each for the numbers of those n-lacings
that belong to one of ten different classes of n-lacings considered by us: general,
dense, simple, straight, dense-and-simple, dense-and-straight, etc. The highlights of
this chapter are the formula for the number of n-lacings and the formula for the
number of simple n-lacings. Especially, the latter is a very striking example of a
simple mathematical object giving rise to a beautiful, yet surprisingly complicated,
formula. Also included in this chapter are complete lists of all 2-lacings, all 3-lacings,
and all simple 4-lacings.
In Chapter 4, we extend results by Halton [13] and Isaksen [16] by deriving the
shortest n-lacings in the different classes of n-lacings in which we are interested.
Highlights of this chapter are our proofs that the bowtie n-lacings are the shortest
n-lacings overall, that the crisscross n-lacings are the shortest dense n-lacings, and
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