Projective plane

In mathematics, a projective plane consists of a set of "lines" and a set of "points" with the following properties:

1. Given any two distinct points, there is exactly one line incident with both of them.
2. Given any two distinct lines, there is exactly one point incident with both of them.
3. There are four points such that no line is incident with more than two of them.

The last condition simply excludes some degenerate cases.

Note that a projective plane is an abstract mathematical concept, so the "lines" need not be anything resembling ordinary lines, nor need the "points" resemble ordinary points.

Consider a sphere, and let the great circles of the sphere be "lines", and let pairs of antipodal points be "points". This is the real projective plane. It is easy to check that it obeys the rules required of projective planes: any pair of distinct great circles meet at a pair of antipodal points, and any two distinct pairs of antipodal points lie on a single great circle. If we identify each point on the sphere with its antipodal point, then we get a representation of the real projective plane in which the "points" of the projective plane really are points. The resulting surface, a two-dimensional compact non-orientable manifold, is a little hard to visualize, because it cannot be embedded in 3-dimensional Euclidean space without intersecting itself. Three self-intersecting embeddings are Boy's surface, the Roman surface, and a sphere with a cross-cap.

The smallest possible projective plane has only seven points and seven lines. It is often called the Fano plane, and is shown in the picture on the right. In this representation of the Fano plane, the seven points are shown as small blobs, and the seven lines are shown as six line segments and a circle. However, we could equally consider the blobs to be the "lines" and the line segments and circle to be the "points" -- this is an example of the duality of projective planes: if the lines and points are interchanged, the result is still a projective plane.

It can be shown that a projective plane has the same number of lines as it has points. This number can be infinite (as for the real projective plane) or finite (as for the Fano plane). A finite projective plane has n² + n + 1 points, where n is an integer called the order of the projective plane. (The Fano plane therefore has order 2.) For all known finite projective planes, the order is a prime power. The existence of finite projective planes of other orders is an open question. A projective plane of order n has n + 1 points on every line, and n + 1 lines passing through every point, and is therefore a Steiner S(2, n+1, n²+n+1) system (see Steiner system).

The definition of projective plane by incidence properties is something special to two dimensions: in general projective space is defined via linear algebra.