Harmony of the Spheres

Abstract

In the lecture on Bubbles we saw how the centres of bubbles, the centres of curvature of dividing films, and points on tangents formed various harmonic ranges. In this lecture we will start by looking at some other pretty properties of circles and spheres. But first a brief explanation of the title of this lecture.

It seems that the phrase “music of the spheres” originated from Pythagoras and it was really all about the motion of the planets. Plato said that a siren sits on each planet who carols a most sweet song, agreeing to the motion of her own particular planet, harmonizing with others. Hence Milton speaks of the “celestial syrens’ harmony that sit upon the nine enfolded spheres”. Maximus Tyrius says that the mere proper motion of the planets must create sounds and as the planets move at regular intervals, the sounds must harmonize.

There are a number of properties of circles and spheres which are simply stated but surprisingly tricky to prove unless we get the right view of the problem. So this lecture is really about how the appropriate transformation can turn a hard problem into an easy one. In a later lecture we will look at how the complex plane can easily reveal a few unexpected properties of figures involving polygons.