In a projective plane over a field F, the diagonal points of a quadrangle are collinear if and only if F has characteristic 2. Such a quadrangle together with its diagonal points and the lines connecting these points form the subplane of order 2, called a Fano plane. Using Desargues configurations and polarities, we provide a similar type of synthetic criterion and construction for characteristic 3 fields. Let F be a field with characteristic not equal to 2. From any quadrangle and one of its diagonal points V, we construct a pair of triangles Δ 1, Δ 2 in perspective from V, and the resulting Desargues configuration D such that the vertices of Δ 1 are self-conjugate under a particular polarity. For this Desargues configuration D, the vertex of perspectivity V of the pair Δ 1, Δ 2 is a fourth self-conjugate point if and only if F has characteristic 3. If F has characteristic 3, then the 10 points and 10 lines of D together with three additional points and three additional lines yield a projective subplane of order 3 of π.

Desargues configuration, Fano plane, Polarity, Projective plane
European Journal of Mathematics
School of Mathematics and Statistics

Bruen, A.A, & McQuillan, J.M. (James M.). (2018). Desargues configurations with four self-conjugate points. European Journal of Mathematics, 4(3), 837–844. doi:10.1007/s40879-018-0274-5