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Given that this blog is currently being devoted to a rather intensive study of flows on manifolds, I thought that it might be apropos to highlight an amazing 22-minute video from 1994 on this general topic by the (unfortunately now closed) Geometry Center, entitled “Outside In“, which depicts Smale’s paradox (which asserts that an 2-sphere in three-dimensional space can be smoothly inverted without ever ceasing to be an immersion), following a construction of Thurston (who was credited with the concept for the video). I first saw this video at the 1998 International Congress of Mathematicians in Berlin, where it won the first prize at the VideoMath Festival held there. It did a remarkably effective job of explaining the paradox, its resolution in three dimensions, and the lack of a similar paradox in two dimensions, all in a clear and non-technical manner.
A (rather low resolution) copy of the first half of the video can be found here, and the second half can be found here. Some higher resolution short movies of just the inversion process can be found at this Geometry Center page. Finally, the video (and an accompanying booklet with more details and background) can still be obtained today from A K Peters, although I believe the video is only available in the increasingly archaic VHS format.
There are a few other similar such high-quality expository videos of advanced mathematics floating around the internet, but I do not know of any page devoted to collecting such videos. If any readers have their own favourites, you are welcome to post some links or pointers to them here.