Modelling 3-configurations on Surfaces

A 3-configuration is a finite geometry satisfying axioms: (i) each line contains exactly 3 points; (ii) each point is on exactly r lines, where r is a fixed positive integer; and (iii) each pair of distinct points are on at most one common line. Such geometries correspond to K3− decompositions of their Menger graphs, and hence to bichromatic dual surface (or pseudosurface) imbeddings of these graphs where one color class consists of triangular regions modelling the lines of the geometry. Such imbeddings have been found for all 320 pairs (v, r), where v is the number of points in the geometry, satisfying 2r + 1 ≤ v ≤ 50 and vr ≡ 0 (mod 3). Here we discuss some of the more interesting among these.