3. Non-Euclidean geometry

‘Non-Euclidean geometry’ begins with a discussion on spherical geometry, which is the study of objects on the sphere and has lines that are defined as great circles. Spherical geometry is an example of a non-Euclidean geometry, as the lines do not satisfy Euclid’s parallel postulate. Hyperbolic geometry is another example of a non-Euclidean geometry, as it violates the parallel axiom and cannot be embedded in ordinary space. Hyperbolic geometry can be introduced as an abstract surface wherein lines are singled out and the distance which makes these lines the shortest are shown. With hyperbolic geometry, the apparent paradoxes of M. C. Escher’s angels and devils can be revealed.