x Preface
ested. Most importantly, we prove that the crisscross and zigzag n-lacings are the
strongest pulley systems among all n-lacings and that the star and zigzag n-lacings
are the strongest straight n-lacings.
In Chapter 8, we derive the weakest n-lacings in some of the classes of n-lacings
in which we are interested. This results in a number of new characterizations of the
bowtie, crisscross, zigzag, zigsag, and star n-lacings.
In Appendix A, we first give a brief introduction to the so-called traveling sales-
man problems. These problems are close relatives of our shortest shoelace problems.
We also describe the so-called shoelace formula for calculating the area of polygons.
In Appendix B, we collect all kinds of curious and interesting facts about real
shoelaces and lacings.
Acknowledgements
I am grateful to Allen Offer and Hendrik Van Maldeghem, whose insightful ques-
tions prompted me to investigate simple and straight lacings. Eventually, this led
to some further nice characterizations of some popular lacing methods. I would like
to thank Ina Lindemann for her marvelous support of this project, Michael East-
wood for drawing my attention to the analysis of the relative strengths of different
suturing methods in [22], and Marty Ross for his feedback on an earlier version of
the manuscript. Finally, my most heartfelt thanks go to my wife, Anu, for accom-
panying me on countless shoelace-related expeditions.
Melbourne, Australia Burkard Polster
UNIVERSAL UCLICK. Reprinted with permission. All rights reserved.
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