For most of this chapter (sections 3.1–3.8) we shall be dealing with the notion of projective plane, before introducing the more general notion of projective space.

Projective planes

Recall (section 2.5) that a projective plane is a linear space in which

1. PP1 any two lines meet,

2. PP2 there exists a set of four points no three of which are collinear.

It follows from L2 that any two lines meet in a unique point.

So far we have seen two examples: the Fano plane (figure 1.1.1) and the extended real plane (example 2.1.3). We present one other example in this section, but will see many more examples later in the chapter, along with examples of more general projective spaces.

The linear space of figure 3.1.1 is a projective plane with thirteen points {1,2,…,13} and thirteen lines {{1,2,3,11}, {4,5,6,11}, {7,8,9,11}, {1,4,7,13}, {2,5,8,13}, {3,6,9,13}, {1,5,9,12}, {2,6,7,12}, {3,4,8,12}, {1,6,8,10}, {2,4,9,10}, {3,5,7,10}, {10,11,12,13}}.

We also recall two lemmas from section 2.5 (lemmas 2.5.3 and 2.5.5) reproducing them here as lemmas 3.1.1 and 3.1.2.

Lemma 3.1.1. A projective plane has the exchange property.

Lemma 3.1.2. A projective plane has dimension 2.

Lemma 3.1.2 gives us the ‘right’ to call a projective plane a plane.

Recalling the labelling of points and lines from chapter 2, we are able to prove the next lemma.