Warped Passages - Lisa Randall [52]
Only in the nineteenth century did mathematicians put the fifth postulate in its proper place. The great German mathematician Carl Friedrich Gauss discovered that Euclid’s fifth assumption was exactly what Euclid had claimed: a postulate that could be replaced by another. He went ahead and replaced it, discovering other systems of geometry and thereby demonstrating that the fifth postulate was independent. With that, non-Euclidean geometry was born.
A Russian mathematician, Nikolai Ivanovich Lobachevsky, also developed non-Euclidean geometry, but when he sent his work to Gauss he was disappointed to learn that the older mathematician had come up with the same idea fifty years before. But neither Lobachevsky nor anyone else had known about Gauss’s results, which the German had hidden for fear that his colleagues would ridicule him.
Gauss shouldn’t have worried. It is obvious that Euclid’s fifth postulate is not always true, because we all know about alternatives. For example, lines of longitude meet at the North Pole and at the South Pole, even though they are parallel at the equator. Geometry on a sphere is an example of non-Euclidean geometry. Had the ancients written on spheres rather than scrolls, this might have been obvious to them, too.
But there are many examples of non-Euclidean geometry which, unlike the sphere, cannot be realized physically in a three-dimensional world. The original non-Euclidean geometries of Gauss, Lobachevsky, and the Hungarian mathematician János Bolyai* dealt with such undrawable theories, which makes it less surprising that they took so long to discover them.
A few examples illustrate what makes curved geometries different from the flat geometry of this page. Figure 38 shows three two-dimensional surfaces. The first, the surface of a sphere, has constant positive curvature. The second, a section of a flat plane, has zero curvature. And the third, a hyperbolic paraboloid, has constant negative curvature. Examples of negatively curved surfaces are the shape of a horse’s saddle, the terrain between two mountain peaks, and a Pringles potato chip.
Figure 38. Surfaces of positive, zero, and negative curvature.
There are many litmus tests that will tell us which of the three possible types of curvature any particular geometric space possesses. For example, you can draw a triangle on each of the three surfaces. On the flat surface the sum of the angles of a triangle is always precisely 180 degrees. But what about a triangle on the surface of the sphere, with one vertex on the North Pole and the remaining two vertices on the equator, a quarter of the way around the equator from each other? Each of the angles of this triangle is a right angle of 90 degrees. Therefore the sum of the angles on the triangle is 270 degrees. This could never happen on a flat surface, but on a surface of positive curvature the sum of the angles of a triangle must exceed 180 degrees because the surface bulges out.
Similarly, the sum of the angles of a triangle drawn on a hyperbolic paraboloid is always less than 180 degrees, a reflection of its negative curvature. This is a bit harder to see. Draw two vertices near the top of the saddle and one down low, along one of the lower parts of the hyperbolic paraboloid, where one of your feet would go if you were sitting on a horse. This last angle is less than it would be if the surface were flat. The angles add up to less than 180 degrees.
Once it was established that non-Euclidean geometries were internally consistent—that is, their premises didn’t result in paradoxes or contradictions—the German mathematician Georg Friedrich Bernhard Riemann developed a rich mathematical structure to describe them. A piece of paper cannot be rolled into a sphere, but it can be rolled into a cylinder. You can’t flatten a saddle without having it crumple or fold back on itself. Building on Gauss’s work, Riemann created a mathematical formalism that encompassed such facts. In 1854 he found a general solution to the problem of how to characterize all