Incidence geometry

Incidence geometry is an area of mathematics that studies relations of incidence between various geometrical objects such as points, lines, curves, and planes. A specific collection of such objects is called a mathematical structure. One type of mathematical structure of particular importance contains only points and lines and is called an incidence geometry.

Definition

An incidence geometry is an incidence structure for which the following axioms are true:

• Every pair of distinct points determines a unique line.
• Every line contains at least two distinct points.
• For every line, there is at least one point that does not lie on the line.

The result of this is that every incidence geometry contains at least three points and three lines. Thus, the simplest incidence geometry that can exist would look something like this:

The Fano Plane

One famous incidence geometry was developed by the Italian mathematician Fano and is known as the Fano plane:

Incidence Matrix

An incidence geometry can be modeled by an incidence matrix which serves as a visual representation of all incidence relations in the geometry. The rows of the matrix represent points, while the columns represent lines. The incidence matrix for the Fano plane looks like this:

The incidence matrix shows the sets of points and lines and which points and lines are incident. In most cases, this is sufficient information to determine the entire geometry, which is one reason why the study of incidence geometry is important.

Line-Line Matrix

The line-line matrix indicates the number of common points for each line-pair. The line-line matrix for the Fano plane is as follows:

The line-line matrix can be derived from the incidence matrix. If N is the incidence matrix and N^T is the transpose of the incidence matrix, then the line-line matrix L = N^T × N.

Point-Point Matrix

The point-point matrix indicates the number of lines common to each point-pair. The point-point matrix for the Fano plane is as follows:

The point-point matrix can also be derived from the incidence matrix. If N is the incidence matrix and N^T is the transpose of the incidence matrix, then the point-point matrix P = N × N^T.

The de Bruijn-Erdös Theorem

The de Bruijn-Erdös theorem is an important theorem in the field of incidence geometry. It was proposed by two mathematicians, Nicolaas Govert de Bruijn and Paul Erdös. The statement of the theorem is as follows:

• Every non-collinear set of n points determines at least n distinct lines.

More Examples

• Affine planes
• Projective geometries
• Möbius geometries
• Polar spaces
• Finite geometries

Related Articles

• Group theory
• Combinatorial designs
• Projective configuration
• Levi graph

References

• Buekenhout, Francis (1995), Handbook of Incidence Geometry: Buildings and Foundations, Elsevier B.V.
• Weisstein, Eric W., "de Bruijn-Erdős Theorem" from MathWorld