Preface ix

that the star n-lacings are the shortest among all those n-lacings that are both

dense and straight.

In Chapter 5, we consider variations of the shortest shoelace problem. For ex-

ample, we derive the solution of the shortest shoelace problem for lacings that are

not closed, for lacings consisting of more than one shoelace, and for lacings of shoes

in which the eyelets are not perfectly aligned. In particular, generalizing a result

by Misiurewicz [20], we show that the crisscross n-lacing of a general n-shoe is the

shortest dense n-lacing of this shoe. This last result demonstrates that the solution

of the shortest shoelace problem in the class of dense n-lacings is very robust, as

it stays unchanged even when the underlying array of eyelets is perturbed quite

radically. On the other hand, we also show that the same cannot be said about the

bowtie n-lacings as the shortest n-lacings overall.

In Chapter 6, we derive the longest n-lacings in most of the different classes

of n-lacings in which we are interested. One of the most surprising and interesting

results of this chapter is the characterization of the zigzag n-lacings as the longest

simple n-lacings.

In Chapter 7, we consider n-lacings as pulley systems and succeed in identifying

the strongest pulleys in all the different classes of n-lacings in which we are inter-