Hyperbolic geometry

Abstract

One of the most remarkable discoveries of nineteenth century mathematics is that the pseudosphere discussed in Section 8.3 has a geometry that closely resembles Euclidean geometry, with geodesics playing the role of straight lines. In fact, the closest correspondence with Euclidean geometry is obtained by ‘embedding’ the pseudosphere in a larger geometry, which is called hyperbolic or non-Euclidean geometry. When this is done, we find that all the axioms of Euclidean geometry hold in hyperbolic geometry, except the so-called ‘parallel axiom’: this states that if p is a point that is not on a straight line l, there is a unique straight line passing through p that does not intersect l (i.e., which is ‘parallel’ to l in the usual sense).

Hyperbolic geometry was discovered independently and almost simultaneously by the Hungarian mathematician Janos Bolyai and the Russian Nicolai Lobachevsky, although the formulations of it, that we shall describe in this chapter, are due to Eugenio Beltrami, Felix Klein and Henri Poincaré. David Hilbert, one of the greatest mathematicians of the twentieth century, wrote that the discovery of nonEuclidean geometry was ‘one of the two most suggestive and notable achievements of the last century’. It ended centuries of attempts by Greek, Arab and later Western mathematicians to deduce the parallel axiom from the other axioms of Euclidean geometry, and it profoundly changed our view of what geometry is.