Bruck nets and partial Sherk planes

TL;DR

This paper extends the characterization of finite geometries by showing that Bruck nets of even degree, which generalize Sherk's affine planes of odd order, can be obtained through weakened axioms allowing non-collinear points.

Contribution

It introduces a broader class of finite geometries, Bruck nets of even degree, by relaxing Sherk's axioms to include non-collinear points.

Findings

Bruck nets of even degree are a natural extension of Sherk's affine planes.

Weakening Sherk's axioms allows for the inclusion of non-collinear points.

The paper provides a new geometric framework for finite geometries.

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.