Projective Plane

A projective plane, sometimes called a twisted sphere (Henle 1994, p. 110), is a surface without boundary derived from a usual plane by addition of a line at infinity. Just as a straight line in projective geometry contains a single point at infinity at which the endpoints meet, a plane in projective geometry contains a single line at infinity at which the edges of the plane meet. A projective plane can be constructed by gluing both pairs of opposite edges of a rectangle together giving both pairs a half-twist. It is a one-sided surface, but cannot be realized in three-dimensional space without crossing itself.

A finite projective plane of order is formally defined as a set of points with the properties that:

1. Any two points determine a line,

2. Any two lines determine a point,

3. Every point has lines on it, and

4. Every line contains points.

(Note that some of these properties are redundant.) A projective plane is therefore a symmetric ( , , 1) block design. An affine plane of order exists iff a projective plane of order exists.

A finite projective plane exists when the order is a power of a prime, i.e., for . It is conjectured that these are the only possible projective planes, but proving this remains one of the most important unsolved problems in combinatorics. The first few orders that are powers of primes are 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, ... (OEIS A000961). The first few orders that are not of this form are 6, 10, 12, 14, 15, ... (OEIS A024619).

The smallest finite projective plane is of order , and consists of the configuration known as the Fano plane, illustrated above.

The remarkable Bruck-Ryser-Chowla theorem says that if a projective plane of order exists, and or 2 (mod 4), then is the sum of two squares. This rules out . By answering Lam's problem in the negative using massive computer calculations on top of some mathematics, it has been proved that there are no finite projective planes of order 10 (Lam 1991). The status of the order 12 projective plane remains open.

The projective plane of order 2, also known as the Fano plane, is denoted PG(2, 2). It has incidence matrix

Every row and column contains 3 1s, and any pair of rows/columns has a single 1 in common.

The projective plane has Euler characteristic 1, and the Heawood conjecture therefore shows that any set of regions on it can be colored using six colors only (Saaty 1986). The Petersen graph provides a 6-color coloring of the projective plane.