Incidence geometry

A geometric structure such as the Euclidean plane is a complicated object that involves concepts such as length, angles, continuity, betweenness, and incidence.

An incidence structure is what is obtained when all other concepts are removed and all that remains is the data about which points lie on which lines.

Even with this severe limitation, theorems can be proved and interesting facts emerge concerning this structure.

Such fundamental results remain valid when additional concepts are added to form a richer geometry.

Using geometric language, as is done in incidence geometry, shapes the topics and examples that are normally presented.

It is, however, possible to translate the results from one discipline into the terminology of another, but this often leads to awkward and convoluted statements that do not appear to be natural outgrowths of the topics.

A special case that has generated much interest deals with finite sets of points in the Euclidean plane and what can be said about the number and types of (straight) lines they determine.

Some results of this situation can extend to more general settings since only incidence properties are considered.

Very general incidence structures can be obtained by imposing "mild" conditions, such as: A partial linear space is an incidence structure for which the following axioms are true:[3] In a partial linear space it is also true that every pair of distinct lines meet in at most one point.

Further constraints are provided by the regularity conditions: RLk: Each line is incident with the same number of points.

If finite this number is often denoted by r. The second axiom of a partial linear space implies that k > 1.

A finite partial linear space satisfying both regularity conditions with k, r > 1 is called a tactical configuration.

[6] If a tactical configuration has n points and m lines, then, by double counting the flags, the relationship nr = mk is established.

This famous incidence geometry was developed by the Italian mathematician Gino Fano.

Conversely, starting with the projective plane of order three (it is unique) and removing any single line and all the points on that line produces this affine plane of order three (it is also unique).

Given an integer α ≥ 1, a tactical configuration satisfying: is called a partial geometry.

A finite Möbius plane of order m is a tactical configuration with k = m + 1 points per cycle that is a 3-design, specifically a 3-(m2 + 1, m + 1, 1) block design.

A question raised by J.J. Sylvester in 1893 and finally settled by Tibor Gallai concerned incidences of a finite set of points in the Euclidean plane.

They also mention that the Euclidean plane version can be proved from the Sylvester-Gallai theorem using induction.

A bound on the number of flags determined by a finite set of points and the lines they determine is given by: Theorem (Szemerédi–Trotter): given n points and m lines in the plane, the number of flags (incident point-line pairs) is: and this bound cannot be improved, except in terms of the implicit constants.