Next Wednesday we’ll begin our course Strange Spaces, a gentle introduction to topology and non-Euclidean geometry. These are interesting subjects in maths because although they can include calculation and algebraic symbol-juggling, they don’t have to. The concepts and many of the problems are qualitative in nature — they ask for answers in words, not numbers.

We cover a wide range of topics — axiomatic geometry, topological surfaces (aka “2-manifolds”), projection from the sphere to the plane (cartography) and vice versa (transferring patterns onto domes or balls), and higher-dimensional spaces. The course culminates in a look at models of hyperbolic geometry and is rounded off by a session on knot theory; an example of quipu, an Incan system of writing and calculation that uses knots, is pictured on the left.

Those in the know might be surprised that we teach geometry and topology together; after all, they’re supposed to be quite separate fields. But they come together in many places; one of the most dramatic in recent years has been the proof of Thurston’s Geometrization Conjecture, which was proved a little over a decade ago. Students on the course should have a rough idea of what the conjecture is about by the end.

Here’s Thurston talking about his work:

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When he was a young researcher, Thurston had a hard time getting his ideas accepted, because his way of forming intuitive pictures of things was notoriously difficult to communicate to others. You may get a sense of this from his talk, and he explicitly says it in the introduction.

This material is usually taught only on upper-undergraduate maths courses, so we naturally take a different approach to the usual one. The course aims to give an intuitive picture of the objects it looks at without getting technical, and where possible we make physical models to help us understand what’s going on.

In fact, this emphasis on developing visual intuition and “representational fluency” is something that’s just being caught up with in the world of maths education, where in the past it’s sometimes been perceived as “not proper maths”. Yet many successful mathematicians will tell you that the visual is extremely important to their work, even when they work with objects you can’t draw or make directly.