Configuration (geometry)

[1] Although certain specific configurations had been studied earlier (for instance by Thomas Kirkman in 1849), the formal study of configurations was first introduced by Theodor Reye in 1876, in the second edition of his book Geometrie der Lage, in the context of a discussion of Desargues' theorem.

In the latter case they are closely related to regular hypergraphs and biregular bipartite graphs, but with some additional restrictions: every two points of the incidence structure can be associated with at most one line, and every two lines can be associated with at most one point.

Configurations having the same symbol, say (pγ ℓπ), need not be isomorphic as incidence structures.

These are called symmetric or balanced configurations[2] and the notation is often condensed to avoid repetition.

Since projective planes are known to exist for all orders n which are powers of primes, these constructions provide infinite families of symmetric configurations.

[7] However, Gropp (1990) has provided a construction which shows that for k ≥ 3, a (pk) configuration exists for all p ≥ 2 ℓk + 1, where ℓk is the length of an optimal Golomb ruler of order k. The concept of a configuration may be generalized to higher dimensions,[8] for instance to points and lines or planes in space.