This final chapter is quite different from the earlier ones in style and intention: I let my hair down with a number of informal fairy stories on different topics, tying together loose strands in the historical and mathematical argument of the book, and opening up some new directions. In particular, I give a ‘popular science’ discussion of some of the surprising and amazingly fertile links between the geometry, topology and Lie group theory discussed in this book and different aspects of twentieth century physics.

There are many other topics closely related to the main text, both frivolous and serious, that I would have liked to write about. But life is short, and I confine myself to a brief list of a few directions and developments. Several of these topics can form the basis for undergraduate essays or projects.

• The classification of locally Euclidean geometries in the style of Nikulin and Shafarevich.

• Spherical trig and geometry in the history of navigation. Modern developments: GPS (global positioning system) devices.

• Spherical geometry and cartography (map making): Mercator's and other projections, as discussed for example in.

• Plane and spherical geometry and plate tectonics, following for example, Chapter 2. Why South America and West Africa fit together like pieces of a spherical jigsaw puzzle; Euler's theorem and the classification of fault types.

• SO(3) and Euler angles, mechanics in moving frames, Coriolis forces.

• […]