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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. The most common projective plane is the real projective plane, which is a topological surface with surprising geometric properties; after that is the complex projective plane of algebraic geometry, a topological four-dimensional manifold.

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 black balls, and the seven lines are shown as six line segments and a circle. However, we could equally consider the balls 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.

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There are two families of degenerate planes.

1) For any number of points P₁, ..., P_n, and lines L₁, ..., L_m,

L₁ = { P₁, P₂, ..., P_n}

L₂ = {P₁}

L₃ = {P₁}

...

L_m = {P₁}

2) For any number of points P₁, ..., P_n, and lines L₁, ..., L_n, (same number of points as lines)

L₁ = {P₁, P₂, P₃, ..., P_n}

L₂ = {P₁, P₂}

L₃ = {P₁, P₃}

...

L_n = {P₁, P_n}

To construct a projective plane of order N² + N + 1 (n: prime), proceed as follows:

Create N² points, which we will label P(r, c) : r, c = 0, ..., (N-1)

Create N points, which we will label P(c) : c = 0..(N-1)

Create one point P

On these points, construct the following lines:

One line L = { P, P(0), ..., P(N-1)}

N lines L(c) = {P, P(r, c}} : r, c = 0..(N-1)

N² lines L(r, i): { P(i), P((r + c*i) mod N, c) }

Note that the expression

(r + ci) mod N

will pass once through each value as c varies from 0 to N-1, but only if is N is prime.

By this construction, we have two planes

L = {P}

and

L={P, P(0)}, L(0)={P, P(0,0)}, L(0,0)={P(0), P(0,0)}

which fit the pattern but are eliminated by the third condition above.

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See also: projective geometry.