Alex's Adventures in Numberland - Alex Bellos [166]
Spherical and hyperbolic surfaces are mathematical opposites, and here is a practical example that shows why. Cut a piece out of a spherical surface, such as a basketball. When we squash the piece on the ground to make it flat it will either stretch or rip, since there is not enough material to spread out in a flat way. Now imagine we had a rubber Pringle. When we try to flatten it, the Pringle would have too much material and some of it would fold on itself. Whereas the sphere closes in on itself, the hyperbolic surface expands.
Let’s return to the parallel postulate, which provides us with a very concise way of classifying flat, spherical and hyperbolic surfaces.
For any given line and a point not on that line:
On a flat surface there is one and only one parallel line through that point.
On a spherical surface there are zero parallel lines through that point.*
On a hyperbolic surface there is an infinite number of parallel lines through that point.
We can understand intuitively the behaviour of parallel lines on a flat or on a spherical surface, because we can easily visualize a flat surface that goes on for ever and we all know what a sphere is. It is much more challenging to understand the behaviour of parallel lines on a hyperbolic surface, since it is not at all clear what such a surface might look like as it spreads to infinity. Parallel lines in hyperbolic space get further and further apart from each other. They do not bend away from each other, since for two lines to be parallel they must also be straight, but they diverge because a hyperbolic surface is constantly curving away from itself, and as the surface curves away from itself it creates more and more space between any two parallel lines. Again, this idea is totally mind-boggling, and it’s hardly surprising that, despite his genius, Riemann did not come up with a surface that had the properties he was describing.
The challenge of visualizing the hlic plane galvanized many mathematicians in the final decades of the nineteenth century. One attempt, by Henri Poincaré, caught the imagination of M.C. Escher, whose famous Circle Limit series of woodcuts was inspired by the Frenchman’s ‘disc model’ of a hyperbolic surface. In Circle Limit IV, a two-dimensional universe is contained on the circular disc in which angels and devils get progressively smaller the closer they get to the circumference. The angels and devils, however, are not aware that they are getting smaller since as they shrink, so too do their measuring tools. As far as the inhabitants of the disc are concerned, they are all the same size and their universe goes on for ever.
Circle Limit IV.
The ingenuity of Poincaré’s disc model is that it illustrates beautifully how parallel lines behave in hyperbolic space. First, we need to be clear what straight lines are in the disc. In the same way that straight lines on a sphere look curved when represented on a flat map (for example, flight paths are straight, but look curved on a map), lines that are straight in the discworld also look to us like they are curved. Poincaré defined a straight line in the disc as being a section of a circle that enters the disc at right angles. Figure 1 overleaf shows the straight line between A and B, which is made by finding the circle that goes through A and B and that enters the disc at right angles. The hyperbolic version of the parallel postulate states that for every straight line L and a point P not on that line, there is an infinite number of straight lines parallel to L that pass