Non-Euclidean Geometry*

Abstract

Abstract This chapter focuses on the connections between non-Euclidean geometries and the complex numbers. Euclid began with just five axioms, the first four of which never aroused controversy. However, the status of the fifth axiom (the so-called parallel axiom) was less clear, and it became the subject of investigations that ultimately led to the discovery of non-Euclidean geometry. Carl Friedrich Gauss was the first person to ever conceive of the possibility that physical space might not be Euclidean. Quite correctly, Gauss did not conclude that physical space is definitely Euclidean in structure, but rather that if it is not Euclidean then its deviation from Euclidean geometry is extremely small. The chapter then looks at spherical geometry and hyperbolic geometry. Of these two non-Euclidean geometries, hyperbolic geometry is by far the more intriguing and important: it is an essential tool in many areas of contemporary research.