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BRANKO GRUNBAUM

Chapter 2, Arrangements of lines

2.1. Isomorphism-types of arrangements.

Arrangements of straight lines are among the simplest objects one may study in the real

projective plane R It is easily seen that similar investigations could be made in the Euclidean

plane E2; however, since the projective variant is somewhat simpler, and since most of the

Information from the projective case may be utilized in the Euclidean by the simple ex-

pedient of adjoining the "line at infmity", we shall concentrate our attention on arrange-

ments in the projective plane. Such arrangements (or their Euclidean counterparts) have

been studied» with varying emphasis, since Steiner [1826], von Staudt [1847], Schläfli

[1852], Wiener [1864], and Sylvester [1867]. Considering the large number of papers

written on different aspects of the subject it is rather surprising that no systematic exposition

is available. (A first attempt in that direction was made in Chapter 18 of Grünbaum [1967].)

By an arrangement of lines Á we mean a finite family of ç = w(A) lines Ll, *·*»£„

in the real projective plane R If there exists a point common to all lines Li we shall call

the arrangement trivial. Unless the opposite is explicitly stated we shall in the sequel assume

that all arrangements we are dealing with are non-trivial; therefore also ç 3. If no point

belongs to more than two of the lines Lt the arrangement is called simple.

With an arrangement Á there is associated the 2-dimensional cell complex into which

the lines of Á decompose Ñ (see, for example, Veblen-Young [1918, Chapter 9], Carver

[1941]). The vertices, edges, and cells (or polygons) of that complex are also said to belong

to the arrangement, and their numbers are denoted by /0(A), /|(A), and /2(A). Two

arrangements are said to be isomorphic provided the associated cell complexes are isomorphic;

that is, if and only if there exists an incidence-preserving one-to-one correspondence between

the vertices, edges and cells of one arrangement and those of the other. The totality of all

mutually isomorphic arrangements forms an isomorphism-type of arrangements. If all the

cells of an arrangement are triangles we shall say that the arrangement and its isomorphism

type are simplicial Clearly the notions of simple and simplicial arrangements, and the num-

bers w(A) and fj(K) are isomorphism invariants of A. (Some other equivalence classes of

arrangements shall be discussed later, but for most of the exposition the appropriate and

natural notion is that of isomorphism.) It is of some interest to note that although simple

arrangements have been studied frequently, simplicial arrangements appear only in Melchior

[1940]. This is rather surprising since the simplicial arrangements frequently occur as Solu-

tions of extremal problems (see, for example, Theorem 2.5 below).

One of the first problems concerning arrangements is the determination of the numbers

c(n)t c*(n) and

eA(n)

of different isomorphism-types of arrangements, simple arrangements,

or simplicial arrangements of ç lines. Analogous enumeration problems for polytopes, com-

plexes, and graphs are notorious for their* difficulty, but it seems that even less is known con-

cerning the enumeration of types of arrangements. The few available results are as follows:

THEOREM 2.1. c(3) = 1; e(4) = 2; e(5) = 4; c(6) = 17.

http://dx.doi.org/10.1090/cbms/010/02