Modelling finite geometries on surfaces

AbstractThe usual model of the Fano plane has several deficiencies, all of which are remedied by imbedding the complete graph K7 on the torus. This idea is generalized, in two directions: (1) Let q be a prime power. Efficient topological models are described for each PG(2,q), and a study is begun for AG(2,q). (2) A 3-configuration is a geometry satisfying: (i) each line is on exactly 3 points; (ii) each point is on exactly r lines, where r is a fixed positive integer; (iii) two points are on at most one line. Surface models are found for 3-configurations of low order, including those of Pappus and Desargues. Special attention is paid to AG(2,3). A study is begun of partial geometries which are also 3-configurations, and four general constructions are given