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