Projective Plane - Finite Projective Planes

Finite Projective Planes

It can be shown that a projective plane has the same number of lines as it has points (infinite or finite). Thus, for every finite projective plane there is an integer N ≥ 2 such that the plane has

N2 + N + 1 points,

N2 + N + 1 lines,

N + 1 points on each line, and

N + 1 lines through each point.

The number N is called the order of the projective plane. (See also the article on finite geometry.)

Using the vector space construction with finite fields there exists a projective plane of order N = pn, for each prime power pn. In fact, for all known finite projective planes, the order N is a prime power.

The existence of finite projective planes of other orders is an open question. The only general restriction known on the order is the Bruck-Ryser-Chowla theorem that if the order N is congruent to 1 or 2 mod 4, it must be the sum of two squares. This rules out N = 6. The next case N = 10 has been ruled out by massive computer calculations. Nothing more is known; in particular, the question of whether there exists a finite projective plane of order N = 12 is still open.

Another longstanding open problem is whether there exist finite projective planes of prime order which are not finite field planes (equivalently, whether there exists a non-Desarguesian projective plane of prime order).

A projective plane of order N is a Steiner S(2, N + 1, N2 + N + 1) system (see Steiner system). Conversely, one can prove that all Steiner systems of this form (λ = 2) are projective planes.

The number of mutually orthogonal Latin squares of order N is at most N − 1. N − 1 exist if and only if there is a projective plane of order N.

While the classification of all projective planes is far from complete, results are known for small orders:

• 2 : all isomorphic with PG(2,2)
• 3 : all isomorphic with PG(2,3)
• 4 : all isomorphic with PG(2,4)
• 5 : all isomorphic with PG(2,5)
• 6 : impossible as the order of a projective plane, proved by Tarry who showed that Euler's thirty-six officers problem has no solution
• 7 : all isomorphic with PG(2,7)
• 8 : all isomorphic with PG(2,8)
• 9 : PG(2,9), and three more different (non-isomorphic) non-Desarguesian planes. (All described in (Room & Kirkpatrick 1971)).
• 10 : impossible as an order of a projective plane, proved by heavy computer calculation.
• 11 : at least PG(2,11), others are not known but possible.
• 12 : it is conjectured to be impossible as an order of a projective plane.