The Ubiquitous Axiom

Abstract

This paper starts with a survey of what is known regarding an axiom, referred to as the Lotschnittaxiom, stating that the perpendiculars to the sides of a right angle intersect. Several statements are presented that turn out to rather unexpectedly be equivalent, with plane absolute geometry without the Archimedean axiom as a background, to the Lotschnittaxiom. One natural statement is shown to be strictly weaker than the Lotschnittaxiom, creating a chain of four statements, starting with the Euclidean parallel postulate, each weaker than the previous one. It then moves on to provide surprising equivalents, expressed as pure incidence statements, for both the Lotschnittaxiom and Aristotle’s axiom, whose conjunction is equivalent to the Euclidean parallel postulate. The new incidence-geometric axioms are shown to be syntactically simplest.