Friday Science: Do parallel lines meet?

This is my second post on Roger Penrose's The Road to Reality.

1. The second chapter is titled, "An ancient theorem and a modern question." The ancient theorem is the Pythagorean theorem (a² + b²=c²). As it turns out, the Pythagorean theorem only works if the old idea that "two parallel lines never meet" is true.

The "parallel postulate" goes back to Euclid, the old Greek (c. 300BC). Euclid was not able to prove it. In the 1700s, a man named Girolamo Saccheri devoted much of his mathematical life trying to prove the parallel postulate by pursuing its opposite. That is, he tried to show that if you assumed that parallel lines did meet, it ended with a contradiction. Perhaps he died feeling a failure, for he did not show any contradiction.

After him, another mathematician named Heinrich Lambert (1728-77) continued to develop a geometry in which Euclid's parallel lines do meet. But Carl Friedrich Gauss (1777-1855) would get the credit. Gaussian geometry, also called Lobachevskian geometry after a Russian who also explored it, is a geometry in which parallel lines do meet. It is thus a "non-Euclidean" geometry.

2. Another name for the type of geometry these individuals were developing is called "hyperbolic" geometry, and the key is that they are not exploring lines on a flat plane but on a curved one. In particular, hyperbolic geometry is the kind of geometry you end up with if you are drawing your lines on a saddle type figure. Penrose and the artist Escher (see picture) had some fruitful interchanges that resulted in several of his drawings that have "tesselations" like the one above.

From a Euclidean perspective, it looks like the fish get smaller and smaller as you approach the edges, but in fact this is an allusion caused by flattening out the saddle. In reality each fish perceives itself to be the same size as all the other fish. Similarly, the boundary is artificial because of the flattening. In reality, the fish go on forever.

3. There are two ways, if I am understanding correctly, of representing this hyperbolic saddle geometry in a flattened, Euclidean plane. The one way is called conformal and the other projective. The conformal representation has all the lines hitting the boundary circle at right angles and the lines are curved. Some sense of proportion is retained.

The other model, the projective, flattens the figure even more. The lines are straight rather than the curved lines of the conformal. The angles are distorted.

So if there is Euclidean geometry (flat), hyperbolic geometry (like a saddle), there is also spherical or elliptical geometry. Parallel lines meet on a sphere and thus defy Euclid's fifth postulate.

4. I won't pretend to understand everything in this chapter. Penrose is trying to be clear but he needs a translator. :-) The reason for the chapter is the fact that physical space exemplifies some non-Euclidean features when we get into Einstein's general relativity. He is laying some ground work for things to come.

There is debate as to whether the universe as a whole is Euclidean, hyperbolic, or elliptical. The majority I believe are currently leaning Euclidean. But I'm guessing that Penrose is a hyperbolic guy.