SUMS: Large -- Inversion

Speaker: Tim Large (Cambridge undergrad) 

Abstract: We all know what it means to reflect about a line.  But what about reflection
in more general objects - in particular, what about a circle? The idea of ’reflecting in
a circle’ sounds nonsensical, but it gives rise to the notion of geometric inversion.
And unlike normal reflection, this does really strange things to the plane - lines can
become circles and circles can become lines! But it turns out this can actually be an
incredibly powerful tool in normal Euclidean geometry, even providing a neat proof of
Pythagoras’ theorem.  But what will seem at first a rather bizarre and unusual notion
turns out to be not so unusual at all when we look at the plane in a slightly different
way - with the aid of the stereographic projection - thus entering the world of
non-Euclidean geometry.  It turns out this even provides a crucial link between simple
and elementary geometry, and the deep world of the complex plane - something hitherto
almost entirely algebraic.