Constructions for large spatial point-line (nk) congurations

Abstract

Highly symmetric figures, such as regular polytopes, can serve as a scaffolding on which spatial (n_k) point-line configurations can be built. We give several constructions using this method in dimension 3 and 4. We also explore possible constructions of point-line configurations obtained as Cartesian products of smaller ones. Using suitable powers of well-chosen configurations, we obtain infinite series of (n_k) configurations for which both n and k are arbitrarily large. We also combine the method of polytopal scaffolding and the method of powers to construct further examples. Finally, we formulate an incidence statement concerning a (100₄) configuration in 3-space derived from the product of two complete pentalaterals; it is posed as a conjecture.