We have shown that a projective plane has point and line regularity k +1 and that v = b = k² + k + 1. We then say that a projective plane has order k.

Let's try to construct a projective plane of order 3 (The Fano plane is our unique projective plane of order 2). We know that each line must contain 3 + 1 = 4 points, and each point must be incident with 4 lines. In addition, v = b = 3² + 3 + 1 = 13. Certainly it is not an easy task to construct this projective plane. Try it for yourself.

Here is the projective plane of order 3:

But this drawing does not seem to have much symmetry. From this picture, we can construct another picture by making one of our lines the line at infiinity.

The line at infinity is represented by the bold black circle around the outside. In addition, each set of antipodal points corresponds to one point at infinity: that is, we identify antipodal (opposite) points on this outer circle/line. By showing some of the symmetries of the projecive plane of order 3, we can see what some of the elements of the automorphism group of the space are.

Here is an example of a projective plane of order 4. Notice how beautiful these projective planes are!

Now that we have seen projective planes of orders 2, 3 and 4, we are curious as to what orders exist corresponding projective planes. While one may assume that projective planes exist for all orders greater than 1, this is not true. There is no projective plane of order 6 (the proof of this is quite extensive). In addition, while some orders (such as 2, 3 and 4) have unique projective planes, others do not. In the case of projective planes of order 9, there are four such non-isomorphic projective planes. We will see later that for every prime order, there exists a projective plane of that order. It is easy to imagine how complicated the figure for a projective plane becomes as the order of the projective plane increases! There are still many unanswered questions about the complicted structure and existence of projective planes.

Yet, as many questions remain, many exciting facts, theorems and lemmas about projective planes have already been proven. Recall from Prob. 12 that a projective plane has the exchange property. In addition, we have shown that projective planes (no matter how complicated they may look) are always of dimension 2.

We also know about the number of points and lines. We can only have
k² + k + 1 many lines and points. Thus, projective planes will always have an odd number of points and lines as k² + k + 1 = k(k + 1) + 1: we know that either k or k + 1 will be odd, while the other is even. An even multiplied by an odd will always yield an even. Then, the addition of 1 will always yield an odd number.

We will discover many other exciting facts about projective planes soon.

References: Batten, Bennett, Beutelspacher & Rosenbaum, Polster