Chapter 1
Beginnings
1.1. Introduction
The word “configuration” has many meanings in both the colloquial and
technical settings. In the present work, however, it will be used with one
meaning only, although with several nuances which will be explained soon.
By a k-configuration, specifically an (nk) configuration, we shall always mean
a set of n points and n lines such that every point lies on precisely k of these
lines and every line contains precisely k of the points. The variants of the
meaning will concern the interpretation of “point” and “line”, with addi-
tional distinctions regarding the space in which the points and lines are
taken. However, in this Introduction it is best to interpret the words at
their most basic meaning—points and lines in the Euclidean plane. It is
probably surprising that even with this simple interpretation there is suffi-
cient material to consider writing a book and that there are many problems
that are easily stated but still unsolved. For a quick orientation, in Figure
1.1.1 we give three examples of well-known configurations (n3), about which
much has been written and which are known by the names of specific math-
ematicians. Each will appear several times in our discussions. Much less
known are 4-configurations; three examples are shown in Figure 1.1.2.
A configuration (505) is illustrated in Figure 1.1.3.
It is both clear and natural that, with increasing k, the images of k-
configurations become more complicated. In fact, the smallest n for which
a configuration (n6) is known to exist has a value of n = 110. (This topic
will be discussed in detail in Chapter 4.) One concern that can be answered
easily is whether for an arbitrary integer k there exists a configuration (nk).
Indeed, taking in the k-dimensional Euclidean space a “box” consisting of
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http://dx.doi.org/10.1090/gsm/103/01
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