Bruck nets and partial Sherk planes

Abstract

In Bachmann's Aufbau der Geometrie aus dem Spiegelungsbegriff (1959), it was shown that a finite metric plane is a Desarguesian affine plane of odd order equipped with a perpendicularity relation on lines, and conversely. Sherk (1967) generalised this result to characterise the finite affine planes of odd order by removing the 'three reflections axioms' from a metric plane. We show that one can obtain a larger class of natural finite geometries, the so-called Bruck nets of even degree, by weakening Sherk's axioms to allow non-collinear points.

We have removed the condition from our main theorem that there is a constant number of lines on any point. Instead, we have replaced it with the much weaker condition that there is a line all of whose points are thick (incident with more than 2 lines)