Two roads diverged in a wood, and I -
I took the one less traveled by,
And that has made all the difference.
- Robert Frost

You are travelling along a dead-straight path through a wood. The path stretches as far as the eye can see in front of you and behind you. You come to a crossroads, where another straight path crosses your one. You turn to the right, turning through 90°, and travel for a short distance along this new path. Now you come to another crossroads. You turn to the right again, this time through slightly more than 90°. Now you are on a third straight path. The question is, if you continue along this path, will you ever join up with the first path again?

Think about it - the Ancient Greek geometer Euclid did. He concluded that because in the two turns you have turned through more than 180° (or two right angles as he put it), you must be pointing slightly back towards the original path and therefore you will eventually meet up with it. But he couldn't prove it. Read on to find out how this led to the contemplation of curved space and the fate of the universe...

Elements

Euclid was a Greek-speaking mathematician who lived in Alexandria, Egypt, in and around 300 BC. He wrote a series of 13 books which he called 'The Elements', in which he set down just about all the mathematics that was known at the time, including some number theory and an awful lot of geometry. We know from comments by contemporaries that this work included much that was already known by the Ancient Greeks, and also new material that Euclid himself came up with. The Elements became the basis for the teaching of mathematics in Europe for over two thousand years and was still used as the standard reference for geometry in schools as late as the early 1960s.

Euclid's grand plan was to keep everything in the book consistent, by starting with extremely basic assumptions which were obviously true and deriving everything else from them. He built a giant world of mathematics on the foundations of some rather simple ideas.

The Postulates

After getting some of the basic definitions out of the way, Euclid started by stating his five postulates. These are facts which are considered to be self-evident; they can't be proved. They are given here in a paraphrased form:

1. We can draw a straight line between any two points.

2. We can extend a straight line as far as we like at either end.

3. We can draw a circle with a given radius around any point.

4. All right angles are equal to each other.

5. If we have two straight lines and we draw a third one across them, and if the two interior angles formed on one side add up to less than two right angles, then the original two lines will meet if extended on that side.

Eh?

The first four are OK. They're easily understandable and obviously true. But that fifth one is a bit odd. In fact, it's a restating in terms of lines and angles of our problem in the wood at the start of this entry. After a bit of thought, most people will conclude that it is true, but it certainly is not immediately obvious. It is often known as the 'Parallel Postulate', because lines that never meet are parallel; the postulate gives conditions in which the lines are not parallel.

Euclid went on, armed only with these five postulates and logic¹, to prove a whole plethora of 'propositions' relating to triangles, circles, and more complicated figures, forming the foundation of the whole of geometry.

Number Five, Your Time Is Up

It is obvious that Euclid himself was not happy with his fifth postulate. He tried to use it as little as possible, proving the first 28 propositions without using it at all. Other Greeks that came soon after Euclid also weren't happy. They said that it was so complicated that it was in effect a theorem requiring a proof. They set about trying to prove it, and this task was carried on by later mathematicians, right up to the 18th century. Each person tackling the problem proved it to his own satisfaction, but later analysis of the proofs show them to be flawed, each one assuming something which is not stated in the first four postulates. Each of these 'proofs' therefore just replaced Euclid's postulate with a different equally dubious postulate, without proving it.

Three of these are particularly easy to understand:

Given a straight line and a point not on the line, it is possible to draw exactly one straight line through the point which never meets the original line, even if extended indefinitely.
- implicit in Ptolemy's² 'proof', but known as 'Playfair's Axiom'

The angles in a triangle add up to 180 degrees.
- Legendre³

If a line intersects one of a pair of parallel lines, then it intersects the other one also.
- Proclus⁴

Each of these can be shown to be equivalent to Euclid's postulate, in that if one is true or false, then the other is also true or false.

Saccheri

The most notable attempt to prove the postulate was by Saccheri⁵. This Italian priest was a master of many subjects, teaching mathematics, theology, philosophy and grammar. He tried to prove the postulate by assuming it was false and trying to show that this leads to a contradiction. This is a recognised logic technique known as reductio ad absurdum. Saccheri assumed that the angles of a triangle add up to less than 180 degrees (which would imply that the fifth postulate is false), and followed on from this proving theorems. He hoped to show that geometry based on this assumption would eventually lead to a contradiction, but try as he might, he couldn't do this. He seemed to have stumbled upon a universe with laws slightly different to our own, but consistent within itself.

Eventually Saccheri did come up with a contradiction, and published his results in a book with the wonderful title of 'Euclid Freed of Every Flaw: A geometrical work in which are established the fundamental principles of a universal geometry'. However, modern mathematicians have found the proof to be flawed, in that he assumed the existence of a 'point at infinity'. So his proof is just like all the others. It 'proves' Euclid's assumption by assuming something else which is just as much in need of proof.

Saccheri died thinking he had proved that Euclid's view of the world was the correct one. In fact, he had gone well on the road towards showing that alternative views are equally valid. With this idea floating in the air, it was only a matter of time before it was developed.

Hyperbolic Non-Euclidean Geometry

In the 19th Century, three mathematicians independently decided to see what happens if Euclid's fifth postulate is wrong. Suppose that the walker in the wood can turn through more than 180 degrees and end up on a path that never meets the original one. Gauss⁶, Bolyai⁷ and Lobachevsky⁸ each independently considered this case and discovered that it is possible to build a complete consistent world of geometry based on this assumption, but that the predictions it makes about the world don't match up with common sense.

Gauss was the first to do it, but he never published his ideas. Lobachevsky was next, and he did publish his results. Bolyai came up with the same ideas without having seen Lobachevsky's work. So the discovery of the new geometry is now generally credited to all three mathematicians. The new view of the world is known as 'hyperbolic geometry'. Some of the predictions it makes are as follows:

• Given a line and a point not on the line, it is possible to draw a number of different lines through the point, each parallel to the given line, and all intersecting each other at the point.
• The angles of a triangle add up to less than 180°. The amount less than 180° is proportional to the area of the triangle. Really big triangles differ greatly from the 180° of Euclidean geometry while really small ones are so close to 180° that you couldn't measure any difference.
• It is impossible to draw a square, with four equal sides and four equal angles.
• If you have two lines which are both perpendicular to another line, they will be parallel, which means they will never meet, but they are not a constant distance apart. As you get further from the line that crosses them, the two lines get further apart.

You might say that these are all obviously untrue for our world. But don't be hasty; because these effects only become obvious over long distances, it is possible that the world we are living in acts according to the rules of Bolyai, Gauss and Lobachevsy, rather than those of Euclid. In the small scale of our everyday lives, we would not able to tell the difference.

Curvature

As can be seen from this list, the straight lines have a lot of the properties of curved lines. They get further apart at both ends without meeting in the middle. But the lines are not curved, they are straight. One way around this dilemma is to consider the space itself to be curved. For example, if we draw apparently straight lines on the ground, they will follow the curvature of the earth, which is roughly a sphere. If we draw our lines on a weird hyperbolic surface known as a pseudo-sphere, they will behave exactly like the straight lines of hyperbolic geometry. So for this reason, we say that the lines are straight but the space is curved.

Unfortunately, the pseudosphere is too weird a surface to be drawn, but 19th Century mathematicians were able to use it to prove that hyperbolic geometry is just as internally consistent as Euclidean geometry.

Riemann and Elliptical Geometry

We've seen the case where the angles of a triangle add up to less than 180° and the case where they add up to exactly 180°. What about the case where they are more than 180°?

Saccheri considered this and dismissed it, as it contradicted Euclid's first four postulates, which nobody had ever questioned. But so what? The second postulate is rather dodgy: it says that you can extend a straight line as far as you like. But who are we to say what happens at infinity?

In 1854, Riemann⁹ decided to ignore the second postulate and build another entirely consistent non-Euclidean geometry. He did this independently of Lobachevsky, Bolyai and Gauss, and apparently unaware of their work. In this new system, there are no parallel lines at all. The lines that look parallel to us will eventually meet a long way away, at the other side of the universe. Ironically, in Riemann's geometry, Euclid's fifth postulate is true but trivial: Euclid's special lines will meet because all lines ultimately meet.

One of the consequences of all this is that if you extend a line far enough, it will loop 'through infinity' and join back up with itself at the other end. Once again the angles of a triangle depend on the size of the triangle. For small triangles they add up to just over 180°, with a greater difference as the triangle gets bigger.

Two-dimensional Riemannian geometry is easily visualised. Once again, space is curved, but the 'other way' from hyperbolic geometry, into a simple three-dimensional sphere. Lines in the 2-d Riemannian space act the same as great circles on the surface of the sphere. (A great circle is a circle on the surface of the sphere whose centre coincides with the centre of the sphere. It is the shortest distance along the surface of the sphere between any two points that it goes through).

It's worth noting that while this is a handy way of visualising Riemannian space, it is not necessary to invoke the third dimension. A 2-d Riemannian space can be completely understood in terms of itself, without resorting to embedding it in a higher dimension. Similarly, a 3-d Riemannian space can be completely described by its admittedly weird properties in three dimensions without going into a fourth one.

The Gravity of the Situation

All this would be of interest only to mathematicians except for the intervention of a guy called Albert Einstein. He showed in his General Theory of Relativity that gravity has such an effect on everything around it that space is curved and obeys Riemannian geometry. It's often said that a massive star will bend the light rays around it due to its gravity, but in reality the light continues in straight lines, following the shortest distance between two points. It is the space itself that is curved around the star.

So large masses cause little pockets of non-Euclidean space embedded in our space, which appears on the face of it to obey Euclid's laws.

The Fate of the Universe

Einstein wasn't content with just analysing stars. He applied his equations to the entire universe; he couldn't believe what he found. According to his calculations, the whole universe is expanding. Since observations of the Galaxy at the time showed it wasn't expanding, Einstein reckoned there must be something wrong with the theory. He fiddled the equations, introducting the 'cosmological constant' which was basically an unexplained term to make the equations fit the observed fact that the universe is not expanding. Einstein followed the scientific method: his theory predicted something, observation failed to reveal that something, so he changed his theory. Later, Einstein described this as the biggest blunder of his life; but it's easy to judge something in hindsight. At the time, it made sense.

A few years later, observations by Edwin Hubble revealed that the universe consists of more than just our Galaxy. There are millions of galaxies and they are all hurtling away from each other at an alarming rate. The universe is in fact expanding. Einstein's original calculations were vindicated. Not only do they predict the expansion, but they say how the universe is going to carry on. Other cosmologists have carried on Einstein's work and have come up with two conflicting predictions for the fate of the universe.

In one, the universe obeys the hyperbolic geometry of Bolyai, Lobachevsky and Gauss. It is infinite in extent, and will continue to expand for ever. The galaxies will get further apart until our own galaxy is effectively alone in the universe.

In the other, the universe is a spherical Riemannian space. It is finite in extent, albeit very large, and will stop expanding eventually, followed by a period when it will start to contract, ending in a 'Big Crunch'. The entire universe will swallow itself up in the ultimate head-on collision, disappearing into a singularity with the entire mass of the universe concentrated into a single dimensionless point. That will be the end of space and of time.

The thing that separates these two predictions is the amount of mass in the universe. If it is above a critical value, then we can expect the contraction and the Big Crunch. If it is below the value, then we can look forward to the universe ending not with a bang but with a whimper. All we need to know is the mass of the universe.

Unfortunately, this is something we don't yet know. But they're working on it. The ultimate horoscope is in preparation.