If a tactical configuration has points and lines, then, by double counting the flags, the relationship is established. A common notation refers to -configurations. In the special case where (and hence, ) the notation is often simply written as . A linear space is a partial linear space such that: Every pair of distinct points determines exactly one line. Some authors add a "non-degeneracy" (or "non-triviality") axiom to the definition of a (partial) linear space, such as: There exist at least two distinct lines. This is used to rule out some very small examples (mainly when the sets or have fewer than two elements) that would normally be exceptions to general statements made about the incidence structures. An alternative to adding the axiom is to refer to incidence structures that do not satisfy the axiom as being trivial and those that do as non-trivial. Each non-trivial linear space contains at least three points and three lines, so the simplest non-trivial linear space that can exist is a triangle. A linear space having at least three points on every line is a Sylvester–Gallai design. Fundamental geometric examples Some of the basic concepts and terminology arises from geometric examples, particularly projective planes and affine planes. Projective planes A projective plane is a linear space in which: Every pair of distinct lines meet in exactly one point, and that satisfies the non-degeneracy condition: There exist four points, no three of which are collinear. There is a bijection between and in a projective plane. If is a finite set, the projective plane is referred to as a finite projective plane. The order of a finite projective plane is , that is, one less than the number of points on a line. All known projective planes have orders that are prime powers. A projective plane of order is an configuration. The smallest projective plane has order two and is known as the Fano plane. Fano plane This famous incidence geometry was developed by the Italian mathematician Gino Fano. In his work on proving the independence of the set of axioms for projective n-space that he developed, he produced a finite three-dimensional space with 15 points, 35 lines and 15 planes, in which each line had only three points on it. The planes in this space consisted of seven points and seven lines and are now known as Fano planes. The Fano plane cannot be represented