Combinatorial Geometry in the Plane (translation of Kombinatorische Geometrie in der Ebene, with a new chapter supplied by the translator)
by: Hugo Hadwiger, Hans Debrunner (translated by V. Klee)
There are various mathematical subjects in which elementary exercises
lead at once to more advanced and partially unsolved problems, so that the
simplest matters of school mathematics are closely tied to those that are of
scientific interest and are studied by specialists. A key point in this is that the
two professional levels are not, as is usual, separated from each other by
highly developed advanced theories and stratified scales of ideas.
Such a subject is combinatorial geometry, which has an especially simple
character when restricted to the plane. Its problems are directly connected
with the basic ideas of elementary plane geometry and are based on various
primitive relations and processes, such as those of inclusion, intersection,
decomposition, and so forth, and on the combinatorial possibilities associated
with these relations and processes.
The subject is related to combinatorial topology; however, genuine
topological considerations remain well j n the background, and the difficulties
are those of elementary geometry. As was more fully explained by H. Hopf
[47], there is a certain reciprocal relationship between the metric and topological
viewpoints in combinatorial geometry.
The gathering of numerous special problems that we have undertaken is
not, however, totally restricted to the methods of combinatorial geometry;
these form only a tiny nucleus of a complex of questions that has exerted a
special attraction because of the simple and basic nature of its subject matter,
and the purely combinatorial aspect of the necessary inferences.
How a person familiar only with elementary concepts can pose problems
so as to develop this taste, and accustom himself to a change that leads from
the methods and topics of the familiar classical domain to those of a more
currently oriented research area with exciting new possibilities — this will be
brought home to the reader by the examples collected here.
Except for basic material from elementary geometry and the theory of
real numbers, little is needed in the way of previous knowledge; a certain
familiarity with set-theoretical reasoning is useful, and the notion of a plane
point set is important. Most of the notation will be briefly explained.
In Part I, selected theorems are assembled, arranged into groups of
related propositions without proof but with rather detailed comments and
with references to the literature. The proofs, often only briefly indicated,
follow in Part II. Thus the reader will have the opportunity to practice the
search for and execution of his own ideas of proof. Through the numerous
references, especially interested readers may also find their way to the current
research literature and may even pursue the unsolved problems that are
suggested.
With these selected special problems we hope to stimulate an intensive
study of the fascinating questions of combinatorial geometry and to bring into
active being the close contact that exists between school mathematics and
scientific research in this domain.
Part I
1. Incidence of Points, Lines, and Circles
2. Integral Distances. Commensurable Angles
3. Hull Formation. Separability
4. Helly's Theorem. Transversal Problems for Ovals
5. Covering Problems
6. Point Set Geometry and Convexity
7. Realization of Distances
8. Simple Paradoxes Involving Point Sets
9. Pure Combinatorics. Graphs
10. Additional Theorems of Helly Type
11. Further Development of Combinatorial Geometry (by the translator)
Part II: Proofs
Bibliography
Index
http://ifile.it/djeqv8n/hadwiger_debrunner.djvu
