Configuration (geometry) - Constructions of Symmetric Configurations

Constructions of Symmetric Configurations

There are several techniques for constructing configurations, generally starting from known configurations. Some of the simplest of these techniques construct symmetric (p_γ) configurations.

Any finite projective plane of order n is an (n2 + n + 1)_{n + 1} configuration. Let π be a projective plane of order n. Remove from π a point P and all the lines of π which pass through P (but not the points which lie on those lines except for P) and remove a line l not passing through P and all the points that are on line l. The result is a configuration of type (n2 - 1)_n. If, in this construction, the line l is chosen to be a line which does pass through P, then the construction results in a configuration of type (n2)_n. Since projective planes are known to exist for all orders n which are powers of primes, these constructions provide infinite families of symmetric configurations.

Not all configurations are realizable, for instance, a (43₇) configuration does not exist. However, Gropp (1990) has provided a construction which shows that for k ≥ 3, a (p_k) configuration exists for all p ≥ 2 l_k + 1, where l_k is the length of an optimal Golomb ruler of order k.