It is famously possible to continuously deform a sphere until it’s inside out. You have to imagine the sphere is something like a bubble, which can ‘pass through’ itself. This transformation is called an eversion.
The fact is established by a theorem due to Stephen Smale (1959 – JSTOR),
which is not constructive — Smale was not able to tell us exactly how we can carry out this deformation. However, recent computer-minimization techniques have allowed us to describe many (constructive) ways to turn a sphere inside out.
The process is difficult for most of us to visualize. Fortunately, three University of Illinois mathematicians have made a 7-minute computer animation that illustrates how it’s done. (Warning: this video is trippy.)
Smale’s theorem says that any two immersions of a sphere a real n-dimensional manifold are regularly homotopic. Thus, in particular, it follows that a sphere sitting in 3-space according to the immersion,
is regular homotopic to the same sphere turned inside out (that is, to inverse immersion). Smale is also well known for having proven higher-dimensional versions of the Poincaré conjecture in the 60’s.
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