Projective plane

A projective plane is a geometric structure that extends the standard concept of a flat plane. In the ordinary Euclidean plane, two lines usually intersect at a single point. However, some pairs of lines are parallel and never meet. A projective plane changes these rules so that any two distinct lines intersect at exactly one point. This structure is a two-dimensional projective space. It is defined as a rank 2 incidence structure. This means it consists of a set of points, a set of lines, and a relationship called incidence. Incidence describes how points and lines connect to one another.

To be a true projective plane, the structure must follow three specific rules. First, any two distinct points must have exactly one line incident with both of them. Second, any two distinct lines must have exactly one point incident with both of them. This second rule ensures that parallel lines do not exist in this space. Third, there must be at least four points where no single line contains more than two of them. This final rule prevents the structure from being a degenerate case. Because incidence is a symmetric relation, the number of points on any line is equal to the number of lines passing through any point. This value is known as the order of the plane.

One way to build a projective plane is through a process called projective completion. We can start with the extended Euclidean plane and add new elements. First, we look at every set of parallel lines. We associate each set with a single new point called a point at infinity. These new points are all distinct from each other. Next, we add a new line called the line at infinity. This line is incident with all the points at infinity. This process, also called projectivization, turns a standard plane into a projective plane.

Mathematicians also use vector spaces to construct these planes. We can start with a three-dimensional vector space over a division ring, which we call K. The points of the projective plane are the one-dimensional subspaces of K3. These are essentially lines through the origin. The lines of the projective plane are the two-dimensional subspaces of K3. These are essentially planes through the origin. We can describe the coordinates of these points using homogeneous coordinates. This algebraic method allows us to create many different types of planes.

Projective planes can be infinite or finite. The real projective plane, denoted as RP2, is a famous infinite example. It arises when we use real numbers for our construction. In topology, RP2 is a closed, non-orientable real 2-manifold. You can model it by taking a unit sphere and identifying antipodal points. This means every point on the sphere is matched with the point directly opposite it. The lines in this model appear as great circles. Another major example is the complex projective plane, CP2. This plane uses complex numbers and is a closed complex 2-manifold.

Finite projective planes are also very important in mathematics. These are often called field planes because they are built using finite fields. A well-known small example is the Fano plane. Another example is the projective plane of order three. This plane contains exactly thirteen points and thirteen lines. We can track the connections between them using an incidence matrix. In this matrix, a 1 shows that a point is incident with a line. A 0 or a blank cell shows they are not connected. This specific plane of order three follows all the required rules of incidence.

These mathematical structures connect to many different fields. In algebraic geometry, planes over fields serve as fundamental examples. The study of these planes is also linked to Desargues's theorem. If a projective plane follows Desargues's theorem, it can be embedded in a three-dimensional projective space. However, not all projective planes share this property. Some unusual structures, like the Moulton plane, do not follow this theorem. The Moulton plane is an affine plane where some lines appear bent. By studying these different rules, mathematicians can explore the deep connections between geometry, algebra, and topology.