What makes hyperbolic geometry particularly hard?

As I have already commented in the page on general difficulties with the course , the lack of a natural coordinate system for the hyperbolic plane forces one to think geometrically about it. There is another difficulty which makes hyperbolic geometry seem confusing when it is first introduced, which is that the question "what exactly is the hyperbolic plane?" doesn't seem to have a very clear answer. In the course you meet so-called models of the hyperbolic plane, all in their different ways somewhat complicated, but in what sense are they models? What are they models of?

To understand the answer to this question, one is forced to think not just geometrically, but abstractly , in a way that one is not with the sphere. (In fact, the set of points of norm one in R^3 is just a model of the sphere, but it such a natural model that there is not much harm in saying that it actually is the sphere.) In the abstract, hyperbolic geometry is what you get when you replace the parallel postulate with a different axiom, which I shall state informally. Given a line L and a point x not on L, then let M be the line through x perpendicular to L. (You can do this with axioms 1 to 4: take a circle about x that is large enough to meet L in two points, then construct the bisector of those two points.) The parallel postulate states that the only line through x that fails to meet L makes an angle of 90 degrees with M. The new axiom states that there is some positive theta such that a line through x fails to meet L if and only if it makes an angle with M of between 90-theta and 90+theta.

When Gauss, Bolyai and Lobatchevsky "discovered" hyperbolic geometry, what they did was to explore the consequences of this axiom. They found that, far from leading to a quick contradiction, it led to a beautiful body of theorems, different from those of Euclidean geometry and somewhat counterintuitive, but consistent with each other.

They did not, in a formal sense, prove that the parallel postulate was independent of the other Euclidean axioms, because they did not prove that their new geometry was consistent. They simply observed that they could do calculations and prove theorems, and that, however long they went on doing this, everything hung together.

One should not sneer at this: it is exactly what we do with the Zermelo-Frankel axiom system which is supposed to underlie all of everyday mathematics. Moreover, we do it of necessity, because Gödel proved that it is impossible to prove the consistency of ZF.

This does not mean that one should abandon all attempts to prove that one's axioms are consistent. Instead, one proves relative results: such and such a system is just as consistent as such and such another system that everybody trusts. In the case of hyperbolic geometry, there is an obvious other system with which to compare it, namely Euclidean geometry. Thus, the way to prove rigorously that the parallel postulate is independent of the other axioms of Euclidean geometry is to show that one can develop a new geometry which is consistent, assuming that Euclidean geometry is consistent.

How does one do this? The answer is use Euclidean geometry to construct a model of non-Euclidean geometry. If you want to know precisely what a model is, then you should go to the Part II course on Logic, Set Theory and Computation, (or read it in advance in P. T. Johnstone's book, Notes on Logic and Set Theory ). Here is an informal definition. If you are given a set of axioms (such as those for a non-Euclidean geometry) then a model for those axioms is a mathematical construction within which the axioms are true. Implicit in this idea is that you interpret the axioms within the model - that is, define what you mean by the words in the axioms solely in terms of the mathematical construction you are calling your model.

The concept of a model is not all that easy to take in in the abstract, but becomes so as soon as one sees a few examples. One of the most useful models for hyperbolic geometry is the disc model. This model consists of the open unit disc, together with a metric which is very different from the Euclidean metric. In particular, as you approach the edge of the disc, distances in the new metric become very large relative to Euclidean distances. In order to interpret axioms, one must give meanings to terms such as straight line, circle of radius r, congruent, rightangle. This can all be done and is done in the course.

In particular, a straight line, in this model, is either a Euclidean straight line segment joining two opposite points of the circle (but not including them) and hence passing through the centre of the disc, or the intersection with the disc of a Euclidean circle that meets the boundary of the disc at rightangles to it. This illustrates a very important point: when interpreting words in a model, it does not matter how you do it as long as the axioms are satisfied. Thus, it does not matter that straight lines in the disc model do not appear to be straight. Within the disc model, this is the definition of straight.

If you still find it unsettling that one should call the arc of a circle a straight line, then click here for further justification of this peculiar practice.

Returning now to the question "what is the hyperbolic plane?" the answer is that it is an abstract object, a metric space with certain symmetries and satisfying certain axioms, which can be realized in various ways (that is, has various models), none of which can be said to be the best way. This is a situation you have already met. For example, a group, in the abstract sense, is a set together with a binary operation satisfying certain axioms, but very often one is presented with a model of those axioms in a more concrete form, such as a collection of symmetries or permutations or matrices or what have you. Another example, which is not part of the Tripos, is the construction of the real numbers. The reals are defined abstractly as the unique complete ordered field. To show that this idea makes sense, one has to "construct" the reals, which means find a model for the axioms of a complete ordered field. This can be done in all sorts of ways: Dedekind cuts, equivalence classes of Cauchy sequences of rationals and so on.

With luck, if you bear these logical considerations in mind, you will feel more comfortable with the idea of hyperbolic geometry. However, be warned that all I have tried to do on this page is remove a psychological block to understanding the subject. That does not mean that it can be mastered with no work. In order to become familiar with the various models, there is no substitute for learning your lecture notes and trying problems on examples sheets.

Click here to return to the main geometry page.