**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

*N*2 +*N*+ 1 points,*N*2 +*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* = *p**n*, for each prime power *p**n*. 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, *N*2 + *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.

Read more about this topic: Projective Plane

### Other articles related to "projective, finite projective planes, finite projective plane":

... Next, one can define

**projective**and quasi-

**projective**varieties in a similar way ... The most general definition of a variety is obtained by patching together smaller quasi-

**projective**varieties ...

... Kaplansky writes Early in the course I formed a one-step

**projective**resolution of a module, and remarked that if the kernel was

**projective**in one resolution it was

**projective**in all ...

**Finite Projective Planes**

... A

**finite projective plane**of order q, with the lines as blocks, is an S(2, q+1, q2+q+1), since it has q2+q+1 points, each line passes through q+1 points, and each pair of ...

... abstract mathematics called category theory, a

**projective**cover of an object X is in a sense the best approximation of X by a

**projective**object P ...

**Projective**covers are the dual of injective envelopes ...

### Famous quotes containing the words planes and/or finite:

“After the *planes* unloaded, we fell down

Buried together, unmarried men and women;”

—Robert Lowell (1917–1977)

“For it is only the *finite* that has wrought and suffered; the infinite lies stretched in smiling repose.”

—Ralph Waldo Emerson (1803–1882)