Foreword
Discrete Comput Geom
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© 2007 Springer Science+Business Media, Inc.
Foreword
In his book entitled The God Particle, Nobel laureate Leon Lederman repeated the
old myth about Hungarian scientists, which probably originated in Los Alamos: “ The
production of scientists and mathematicians in Budapest in the early 20th century was
so prolific that many otherwise calm observers believe Budapest was settled by Martians
in a plan to infiltrate and take over the planet Earth.” Indeed, under a fast changing and
often hostile political climate, Hungary contributed an extraordinary number of great
individuals to the history of science and mathematics, including Lipo´t Feje´r, Theodor
von Ka´rma´n, John von Neumann, Gyo¨rgy Po´lya, Frigyes Riesz, and Eugene Wigner.
They made significant discoveries in various mathematical disciplines. Yet only few of
them inspired dozens of young researchers to work in their field and left a flourishing
scientific school behind. One of these rare few, La´szlo´ Fejes To´th, passed away 2 years
ago, at the age of 90.
He was born in 1915. In his university years he was strongly influenced by Lipo´t Feje´r.
He started working in analyis but always liked geometry, and a simple and natural
geometric question by Dezso˝ La´za´r changed his mathematical interest forever. The essence
of the question was to find the densest packing of congruent circles in the plane. Unaware
of Thue’s earlier work, La´szlo´ Fejes To´th solved this problem, and, more importantly,
he fell in love with discrete geometry. This love affair lasted for well over six decades.
His central interest within discrete geometry was packings and coverings by disks,
balls, and other convex sets in two or three dimensions. Apart from some scattered results
by Kershner and Thue, all previous work in this area concerned lattice packings and lattice
coverings. Although lattices are a basic concept in the geometry of numbers and number
theory, restricting investigations to lattice arrangements is an artificial constraint from
the geometric point of view. Fejes To´th’s new idea was that the solution to many extremal
problems should remain the same without the lattice condition. He showed, for instance,
that the densest packing of congruent copies of a centrally symmetric convex set in the
plane is necessarily lattice-like. He proved, in the same spirit, that most regular polytopes
can be obtained as the unique solution to a natural extremal problem. A neat example
of this phenomenon is the fact that among all three-dimensional convex polytopes of
unit surface area with 12 facets the regular dodecahedron has maximal volume. This
point of view in geometry can be considered the “genetics” of symmetric objects. La´szlo´
Fejes To´th summarized his results in the seminal books Lagerungen in der Ebene, auf
I. Ba´ra´ny and J. Pach
der Kugel und im Raum and Regular Figures. It is hard to overestimate the impact of
these monographs. Of the first, C. A. Rogers wrote the following: “Until quite recently,
the theory of packing and covering was not sufficiently well developed to justify the
publication of a book devoted exclusively to it. After the publication of L. Fejes To´th’s
book in 1953, there would be no need for a second book on the subject. . . ”
La´szlo´ Fejes To´th was not only a superb mathematician, he was also a master of raising
beautiful and inspiring problems. He regularly came up with simply stated questions from
discrete geometry that had an aesthetic appeal and could be explained to the layman.
He also knew exactly who the right person was to ask a particular question of. He was
very generous with these problems and was always happy when one of his questions was
answered. Because of his generosity, his straightforward and modest style, and his warm
personality, he became one of the most influential geometers of the last century. Yet most
of all, it is the love of mathematics that radiated from him. This is how he described it in
an interview: “An enthusiastic mathematician is completely captivated by mathematics.
Especially as a young man, I was thinking of some mathematical problem simply all
the time. I could not get rid of it even in my dreams. Not that I wanted to. For me, this
activity has been the source of joy and happiness, inner satisfaction, and pleasure.”
The editors of this special issue started working at the Mathematical Institute of the
Hungarian Academy of Sciences at the time it was headed by La´szlo´ Fejes To´th. Their
work has been strongly influenced by Fejes To´th’s ideas, way of thinking, and beautiful
open problems. They consider themselves privileged to belong to Fejes To´th’s school of
discrete geometry, and to have enjoyed his support and help for decades.
They dedicate the present issue of Discrete & Computational Geometry to La´szlo´
Fejes To´th’s memory, as do the editors-in-chief of the journal, who are proud that La´szlo´
Fejes To´th’s final paper was published in DCG in 1999.
Imre Ba´ra´ny (...truncated)