Geometry and gravity

The discovery of ‘non-Euclidean’ geometry in the nineteenth century came as a great surprise and was greeted by disbelief. One of the pioneers of this new geometry, Janos Bolyai, a Hungarian army officer, expressed his joy with the words:

'Euclidean’ geometry is the geometry we learn in school, with its familiar apparatus of points, straight lines, circles, ellipses and triangles. In particular, we are all brought up to believe that the three angles of a triangle add up to 180 degrees and that parallel lines never meet. Such Euclidean geometry is the geometry of the plane – technically called a ‘flat’ space. By contrast, non- Euclidean geometry describes a ‘curved’ space. What do we mean by these terms?

Some idea of a curved space can be gained by considering geometry on the surface of the Earth. The Earth is approximately spherical, and on its surface it is easy to construct triangles whose angles add up to more than I 80 degrees (Figure 9. I). Similarly, lines of longitude start out parallel at the equator but converge and cross at the poles. The surface as a whole does not obey Euclid's rules. Since such a familiar example of a surface is non- Euclidean, why are such geometries so unfamiliar to most of us?