Finale

Abstract

These lectures have had a number of goals. One was simply to present some pretty or surprising configurations. Another was to reveal that despite the fact that we live in a three-dimensional world, nevertheless our understanding of three dimensions is fairly poor. Thus few people know that if you cut through a cube in a certain way, then the cross-section is a hexagon. Another goal, and perhaps the most important in a philosophical sense, was that solving and understanding a problem depends very much upon getting the right point of view.

The lecture on inversion and its use to prove Steiner's porism and explain Soddy's hexlet is perhaps the most intriguing example of getting the right point of view that we have encountered. Another example is that of the train marking the track in special relativity where it is important to analyse the situation from the point of view of the correct observer.

In this final lecture we will look at some more examples where getting the right point of view is so important. These are about certain curious properties of triangles, squares and other rectilinear figures and how the use of the Argand plane can provide very simple explanations in many cases.