Alex's Adventures in Numberland - Alex Bellos [165]
Gauss’s final contribution to research on the fifth postulate came shortly before he died, when, already seriously ill, he set the title for the probationary lecture of one of his brightest students, 27-year-old Bernhard Riemann: ‘On the hypotheses that lie at the foundations of geometry’. The cripplingly shy son of a Lutheran pastor, Riemann at first had some kind of breakdown struggling with what he would say, yet his solution to the problem would revolutionize maths. It would later revolutionize physics too, since his innovations were required by Einstein to formulate his general theory of relativity.
Riemann’s lecture, given in 1854, consolidated the paradigm shift in our understanding of geometry resulting from the fall of the parallel postulate by establishing an all-embracing theory that included the Euclidean and non-Euclidean within it. The key concept behind Riemann’s theory was the curvature of space. When a surface has zero curvature, it is flat, or Euclidean, and the results of The Elements all hold. When a surface has positive or negative curvature, it is curved, or non-Euclidean, and the results of The Elements do not hold.
The simplest way to understand curvature, continued Riemann, is by considering the behaviour of triangles. On a surface with zero curvature, the angles of a triangle add up to 180 degrees. On a surface with positive curvature, the angles of a triangle add up to more than 180 degrees. On a surface with negative curvature, the angles of a triangle add up to less than 180 degrees.
A sphere has positive curvature. We can see this by considering the sum of the angles of the triangle in the following diagram, which is made by the equator, the Greenwich Meridian and the line of longitude 73 degrees west of Greenwich (which goes through New York). Both angles where the longitude lines meet the equator are 90 degrees, so the sum of all three angles must be more than 180. What type of surface has negative curvature? In other words, where are there triangles whose angles add up to less than 180 degrees? Pop open a pack of Pringles, and you’ll see. Draw a triangle on the saddle part of the potato crisp (possibly with some fine French mustard) and the triangle looks ‘sucked in’ compared to the ‘puffed out’ triangle we see on a sphere. Its angles are clearly less than 180 degrees.
Triangle on a sphere: sum of angles greater than 180 degrees.
Triangle on a Pringle: sum of angles less than 180 degrees.
A surface with negative curvature is called hyperbolic. So, the surface of a Pringle is hyperbolic. The Pringle, however, is only an hors d’oeuvre in understanding hyperbolic geometry since it has an edge. Show a mathematician an edge and he or she will want to go over it.
Consider it this way. It is straightforward to imagine a surface with zero curvature and no edge: for example, this page, flattened on a desk, and extended infinitely in all directions. If we lived on such a surface and we started walking in a straight line in any direction, we would never reach an edge. Likewise, we have an obvious example of a surface with positive curvature and no edge: a sphere. If we lived on the surface of a sphere, we could walk for ever and ever in one direction and never reach an edge. (Of course, we do live on a rough approximation of a sphere. If the Earth were totally smooth, with no oceans or mountains to block our way, for example, and we started walking, we would return to our point of departure and continue going in circles.)
Now, what does a surface with negative curvature and no edge look like? It cannot lSo, theike a Pringle, since if we lived on an Earth-sized Pringle and we started walking in one direction, we would always eventually fall off it. Mathematicians have long wondered what an ‘edgeless’ hyperbolic surface might look like – one on which we could walk as far as we wished without coming to the end of it and without it losing its hyperbolic