What is a projective plane?

Question

What is a projective plane?

Answer 1

(mathematics) The topological space obtained from the two-dimensional sphere by identifying antipodal points; the space of all lines through the origin in Euclidean space. More generally, a plane (in the sense of projective geometry) such that (1) every two points lie on exactly one line, (2) every two lines pass through exactly one point, and (3) there exists a four-point.

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In mathematics, a projective plane has two possible definitions, one of them coming from linear algebra, and another (which is more general) coming from axiomatic and finite geometry. The first definition quickly produces planes that are homogeneous spaces for some of the classical groups. The second is suitable for an exhaustive study of the simple incidence properties of plane geometry.

Combinatorial definition

According to the more general, combinatorial definition, a projective plane consists of a set of lines and a set of points, and a relation between points and lines called incidence, having the following properties:

Given any two distinct points, there is exactly one line incident with both of them.

Given any two distinct lines, there is exactly one point incident with both of them.

There are four points such that no line is incident with more than two of them.

The second condition means that there are no parallel lines. The last condition simply excludes some degenerate cases (see below).

Examples

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. For any field K, there is a projective plane with three homogeneous coordinates in K, which can also be thought of in terms of a three-dimensional vector space V over K, 'points' being one-dimensional subspaces and 'lines' two-dimensional subspaces.



The Fano planeThe smallest possible projective plane is the Fano plane. It has only seven points and seven lines. (See also finite geometry.) In the figure at right, the seven points are shown as small black balls, and the seven lines are shown as six line segments and a circle. However, one could equivalently 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. A permutation of the seven points that carries collinear points (points on the same line) to collinear points is called a "symmetry" of the plane.

Properties

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.) There exists a finite projective plane of order n, if n is a prime power, and 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, and there is nothing more known, in particular n = 12 is still open. There is a projective plane of order n if and only if there is an affine plane of order n. When there is only one affine plane of order n there is only one projective plane of order n, but the converse is not true. 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). Conversely, one can prove that all Steiner systems of this form () are projective planes.

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