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Since its discovery by Stephen Smale in 1958, Smale’s paradox has baffled more than a few mathematicians’ minds. At the time, Smale proved that there exists a way to turn a perfect sphere inside out without leaving any tears or creases on its surface. However, he did allow himself the assumption that the sphere is made out of a unique material that can smoothly pass through itself – such self-intersections, he assumed, didn’t create a tear on the sphere surface. Although such an assumption does seem to make it possible to perform a sphere eversion, it is in fact a lot harder than it seems. While it is very difficult to visualize an example of a valid Smale turning, this animated video demonstrates both the problem and the solution. |
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