Absolute isotropic geometry

Abstract

The term ‘absolute geometry’ was coined by Janos Bolyai to characterize the part of Euclidean geometry that does not depend on the parallel postulate. In the framework of Cayley–Klein geometries the parallel axiom characterizes three classical geometries, namely Euclidean, Galilean and Minkowskian planes. We study the part of Galilean geometry that does not depend on the parallel postulate (briefly called absolute isotropic geometry) and their models (isotropic planes). After an axiomatic foundation of absolute isotropic geometry, we develop the basic theory of isotropic planes, prove the theorem of Saccheri for the angle sum of a triangle, and construct models of the different types of planes (over fields and skew fields of characteristic $$\ne 2$$ ). Surprisingly, these models have counterparts in the theory of Hilbert planes. The article closes with a comparison between Hilbert planes and isotropic planes.