Finite geometry

A finite geometry is a type of geometric system that has only a limited number of points. Unlike the geometry we see around us, called Euclidean geometry, which has endless points on a line, a finite geometry has only a set number of points. For example, the images on a computer screen can be seen as a finite geometry, where each pixel is a point.

There are many kinds of finite geometries, but the most studied ones are finite projective and affine spaces. These are picked because they are neat and simple. Other important types include finite Möbius or inversive planes, Laguerre planes, and their more complex forms called Benz planes and higher finite inversive geometries.

Finite geometries can be made using linear algebra that starts from vector spaces over a finite field. These creations often lead to what are called Galois geometries. While most finite geometries are Galois geometries, there are some special cases, like certain flat areas, that do not follow this rule and are called non-Desarguesian planes.

Finite planes

There are two main kinds of finite plane geometry: affine and projective. In an affine plane, parallel lines work like they do in regular geometry. In a projective plane, any two lines meet at one point, so there are no parallel lines. Both types follow simple rules, or axioms.

An affine plane has points and lines where each pair of points lies on exactly one line. The simplest affine plane has four points and six lines. A projective plane also has points and lines, but any two lines meet at exactly one point. The smallest projective plane, called the Fano plane, has seven points and seven lines.

Finite spaces of 3 or more dimensions

The study of spaces with three or more dimensions is important in advanced mathematics. In these spaces, points and lines follow special rules. For example, any two different points are always on one line, and each line has at least three points.

One way to create these spaces is by using math structures called finite fields. The smallest example is PG(3,2), which has 15 points, 35 lines, and 15 planes. This space can help solve problems, like organizing groups so everyone pairs up fairly.