An incidence geometry consists of a set of points and a collection of blocks such that for . The number of blocks containing a point is called the degree, denoted by . Blocks are also called lines and is called the length of the block . A pair with is called a flag. In this case, the point is said to lie on the line . The line is said to pass through . Additionally, in an incidence geometry any pair of points is joined by at most one line, i.e., for all , .
An incidence geometry is called configuration of type if (1) for and (2) for . A configuration with is called symmetric. Its type is simply denoted by which is the same as because in this case. A configuration is called decomposable if it can be written as the union of two configurations , on distinct point sets: , , and and . Indecomposable configurations are also called connected. The isomorphisms are structure preserving mappings.
A triangle of a configuration consists of three points, say , , and , such that the three pairs , , are contained in different blocks. A configuration that has no triangles is called a triangle-free configuration. An isomorphism is called an anti-automorphism. A configuration which admits an anti-automorphism is called self-dual. An anti-automorphism of order 2 is called a polarity. A configuration which admits a polarity is called self-polar.
The authors give tables of configurations for and triangle-free configurations for together with some statistics about some properties of structures like transitivity, self-duality or self-polarity.