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science & techanalysisWikipedia

A sphere can be turned inside out without tearing or creasing it, if it is allowed to pass through itself

the gist

A sphere can be turned inside out in three-dimensional space without cutting, tearing or creasing it, as long as the surface is allowed to pass through itself during the process. Mathematicians call this sphere eversion, and it is possible because the surface can be moved smoothly from one shape to its mirror version through a continuous family of immersions. The result is surprising because intuition suggests a sphere's inside and outside cannot be swapped without a crease. It was proven by Stephen Smale in 1957 using ideas from regular homotopy, and later explicit constructions were found. It is a veridical paradox: true, but it seems false at first glance.

how it works

The process allows the surface to pass through itself, which is impossible for a physical object but permitted for a mathematical immersion in three-dimensional space. The sphere is moved through a sequence of smooth shapes, each one continuous with the last, so no tearing or sharp fold is needed.

so what

It shows that topology cares about continuity rather than shape in the everyday sense, and it changed how mathematicians think about deforming surfaces. Early constructions were found by Bernard Morin and later refined by others, making it a famous example of geometric intuition failing.

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