Bruck nets and partial sherk planes

Abstract

In Bachmann [Aufbau der Geometrie aus dem Spiegelungsbegriff, Die Grundlehren der mathematischen Wissenschaften, Bd. XCVI (Springer, Berlin-Göttingen-Heidelberg, 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 that the converse is also true. Sherk ['Finite incidence structures with orthogonality', Canad. J. Math. 19 (1967), 1078-1083] 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 noncollinear points.