musings of a different kind

Smale's Paradox is one of those mathematical results from differential geometry/topology that makes people do a double take when they first see it. In layman's terms, it says that it is possible to turn a sphere inside out in a 3-dimensional space with possible self-intersections but without creating any crease. I've heard of this result before and I initially thought of it as another reason why I'm glad I'm not doing a PhD in pure math. Some things in math are just too weird when viewed through the lens of common sense/everyday experience.

However, I saw a video on google that explains the paradox in slightly better than layman's terms (by precisely defining what kinds of transformations are allowed in order to bring about the sphere eversion). This is a top quality instructional video that just blew me away. It is the best demonstration of an advanced mathematical concept I have ever seen. Period. I think this would be a good video to show to students who are interested in advanced math because it explains a rather complicated topic in simple terms by providing a good way of visualizing the problem before diving headfirst into a bunch of equations. I hope you enjoy this video and the concepts it illustrates.




Incidentally, it seems Stephen Smale had proved the existence result in 1958 by using a nonconstructive proof (i.e., you could not have made any useful video based on the way he proved the result). The construction apparently came much later. This is not very unusual in modern mathematics, where people often prove the existence of things much before they can actually conjure it up for you.

Labels: math