One of the most valuable human capabilities is doubt. Education seems to contradict this, early on we are encouraged to take certain things for granted. The trust in Euclid’s axioms for geometry was certainly universal and contributed to Immanuel Kant’s confidence that time and space are given to us as a priori.
The ability to draw parallel lines, for instance, seems to be a given, and that this possibility is to a great deal responsible for being able to create realistic looking perspective drawings. What we can see with our own eyes must be true.
The discovery of hyperbolic geometry by Bolyai, Gauß, and Lobachevsky is credited with overthrowing all this. If we are willing to accept that lines are not what they appear, but only have to obey all the other axioms of Euclid, then the parallel axiom does not need to hold. As mind boggling as this entire business is for the mathematician and philosopher, as irrelevant does it seem to be to the everyday person. After all, what we see is still true, isn’t it?
That there is a hard to explain esthetic appeal to that disk with its crazily symmetric patterns doesn’t quite justify the importance of hyperbolic geometry in contemporary mathematics either. Calculus at least is useful. That hyperbolic geometry makes its inevitable appearance whenever we study very simple things like the geometry of the circle or multiplication of 2×2 matrices, doesn’t really force us to talk about it, does it?
As flawed as our universal trust in the nature of space is our trust in the linearity of things. Double the income, double the happiness? Double the pain, halve the crime rate? It sounds too easy not to be true, and is often evidenced by the linearity of Euclidean geometry. Hyperbolic geometry is the simplest geometry where linearity fails and allows for dynamical systems with chaotic behavior. We have known this for over 100 years. We experience the effects on a daily basis, but prefer to ignore it.
Below are my class notes about hyperbolic geometry and incidence geometry, taught to undergraduates. Enjoy.