I found Thurston's lecture The Mystery of Three- Manifolds fascinating but, well, mystifying. For those with insufficient time to watch the video, he establishes correspondence (in his very intuitive way) between sets of linked rings and different geometries. He starts by showing the Borromean rings are analgous in some way to standard three space.
I would like to explore this idea further, hopefully in a more formal exposition. So, I wuld like to ask: (1) does anyone know of a written reference to the idea Thurston presents. Did he (or someone else?) write a paper on the subject? (2) in general, what does one call this sort of idea? What subfield of geometry does it fall under?