Lecture Notes 5: Projective Planes

The 19th Century mathematicians were greatly concerned with the issue of foundations of mathematics. They set out in many disciplines to examine and strengthen the axiomatic basis. One such discipline was geometry which had reached its pinnacle in the early part of the century and which had always been considered the proto-type of axiomatically based subjects. The existence of non-Euclidean geometries had badly shaken the long held belief that the axiom system of geometry was beyond reproach. In the process of reexamining the assumptions upon which the subject is based, G. Fano looked at the problem of deciding which axioms implied the existence of an infinite number of points on a line. To his surprise he discovered that this "fact" could not be derived from the axioms and would have to be included in the axiom list. He was able to show this by producing examples of systems which satisfied all the axioms but which did not have an infinite number of points on a line. Thus was the subject of Finite Geometries born.

Before considering the finite geometries, we need to examine an important development in post-Euclidean geometry. Starting with the perspective drawings of the Renaissance artists, geometers began toying with the idea that parallel lines might meet "at infinity". To legitimize this concept the subject of projective geometry was developed. Projective geometry is a slight extension of Euclidean geometry in which no parallel lines exist (i.e., every pair of lines meet at a unique point.) To accomplish this in plane geometry, we introduce a new point on each line of a family of parallel lines - because these lines now meet in the new point they are no longer parallel. This new point, since it can not exist in the original plane is called a point at infinity. For each family of parallels we introduce a distinct point at infinity and then collecting all the points at infinity, we introduce a new line consisting entirely of these points and call it the line at infinity. This construction gives us the Real Projective Plane which is an extension of the Euclidean plane. Since we have only added things to the original plane, we have not lost any of the geometry of that plane which can be recaptured by tossing out the infinite points and line. It turns out that many statements and theorems are simplified when one views them as statements about this extended plane. The advantages of working with the Projective Geometry are so great that a number of geometers have said that "Projective Geometry is all geometry."

FINITE PROJECTIVE PLANES

A projective plane, , is a triple (,L ,I) where is a set whose elements are called points, L is a set whose elements are called lines and I is a relation between points and lines called incidence, (If A and mL we would say that A is incident with m, and write A I m; in less formal language we could say that the point A is on the line m, or the line m passes through the point A.) such that
  1. Every pair of distinct points is incident with a unique line,
  2. Every pair of distinct lines is incident with a unique point, and
  3. There exist 4 points no three of which are incident with the same line.
The first of these axioms would be included in any mathematical system which we wished to call a geometry. The second makes this geometry projective. The third axiom is a non-degeneracy condition, preventing some small and exceptional systems from being called projective planes. There are 7 systems of points and lines which satisfy the first two axioms but not the third [Homework: Find these seven degenerate cases.]

If the sets or L are finite then the projective plane is called a finite projective plane. As an example of a finite projective plane (in fact the smallest possible example), let = {1,2,3,4,5,6,7} and let L consist of the following sets:

{1,2,3} {3,4,5} {1,5,6} {1,4,7} {2,5,7} {3,6,7} {2,4,6}
if incidence is defined by set membership, this system forms a projective plane. The following diagram is a pictorial representation of this finite projective plane which is called the Fano plane.

Example: In R3, points are lines through the origin and lines are planes through the origin.

Example: Let V be a 3-dimensional vector space over some scalar field. Points are 1- dimensional subspaces of V and lines are 2-dimensional subspaces of V. If the scalars are the reals, this is just the previous example.

Sphere Model of the real projective plane.

Topological representation as a Möbius strip with a disk attached to its boundary.

The construction of the real projective plane from the Euclidean plane mentioned in the introduction is really very general and can be applied to any affine plane. Specifically, the completion of an affine plane is the result of adding one new point to each line of a parallel class of lines, for each parallel class, and adding one new line consisting of all and only these new points.

Theorem: The completion of any affine plane is a projective plane.

Example: the completion of the real affine plane (Euclidean) is the real projective plane.

Example: the completion of the smallest affine plane is the Fano plane.

Theorem: Given a projective plane, the removal of any line and all the points on that line results in an affine plane.

We should point out that if you start with an affine plane and form its completion, and then remove the line just added from the projective plane, you will of course obtain the original affine plane. However, if you remove a different line, you may obtain an affine plane which is not isomorphic to the original affine plane.

The previous two theorems show the close relationship between affine planes and projective planes. A finite projective plane which is the completion of an affine plane of order n is also said to have order n, but note that there are n+1 points on a line of a projective plane of order n. The Fano plane has order 2 and the completion of Young's geometry is a projective plane of order 3.

Theorem - In a finite projective plane of order n:

  1. every line is incident with n + 1 points,
  2. every point is incident with n + 1 lines,
  3. there are n2 + n + 1 points in , and
  4. there are n2 + n + 1 lines in L.

Proof: This follows easily from the counts proved for an affine plane of order n.

Show how the various conics are unified by this viewpoint.

Principle of Duality

A pencil of lines in a plane is the set of all lines through a point (concurrent lines).

Proposition : Let be a projective plane. Let * be the set of lines of , and define a line of * to be a pencil of lines in . Then * is a projective plane.

* is called the dual projective plane of .

The plane dual of a statement is the statement obtained by interchanging the words "point" and "line". If a statement is true in a projective plane, then the plane dual of that statement is true in the dual projective plane. If a statement is true for all projective planes, then its plane dual is also true for all projective planes. This is known as the principle of duality for projective planes.

Two projective (or affine) planes are said to be isomorphic if there exists a bijection between the points of the two planes which maps lines to lines. Such a bijection is called a collineation.

Remark: The dual of the dual of a plane is the original plane. The dual of a plane need not be isomorphic to the original plane, but this is true for the real projective plane.