Incidences

Why do we still teach geometry?  The constructions with ruler and compass were essential for the Egyptians and Greeks in order to accurately lay out large scale buildings with the only tool available (the rope). But today we have many other tools available, so there is no reason to confine ourselves to ruler and compass, unless we want to use them as a vehicle to teach the concept of proofs. That, however, is also in low demand, to the extent that graduating majors in mathematics are neither able to prove Pythagoras’ theorem nor to compute the distance of a point from a line.

I have been teaching a Geometry class twice now, and I am releasing the notes I wrote for the first part of the class into the wild. For this first part, I had set myself a few goals: I wanted to use only the most fundamental notions of geometry,  I wanted a plethora of interesting examples, and I wanted to be able to prove a substantial theorem. Finally, I needed to be able to give homework problems. 

The solution was to study incidence geometries, specializing pretty quickly to affine and projective spaces over arbitrary fields. So I did not develop axiomatic projective geometry, but rather taught the computational skills needed to quickly get to the geometry of conics in projective planes. This provided plenty of  exercises. Affine and projective transformations are intensely used in order to prove theorems or to simplify computations. The big theorem I prove at the end is Poncelet’s theorem.

The second half of the course? This deals with the two-dimensional geometries where we have circles: Möbius, Euclidean, spherical, hyperbolic. Emphasis is again on groups, and here in particular on reflection groups, proving Dyck’s theorem at the end. But I am not quite happy with the notes yet, so this part will have to wait.

So here is part I. Enjoy.

Notes on Geometry – Part I: Incidences