Everting a Sphere

Turning a sphere inside out


A sphere is a 2-dimensional surface, like an infinitely thin soap bubble or balloon (but without the hole for inflating it). It is easy to turn a balloon inside out: cut a hole in it, pull it through the hole, and repair the hole. But can it be done without cutting the hole? In mathematical terms, everting a sphere means turning a sphere inside out in such a way that the surface is at all times continuous (without tears or holes) and smooth (no folds, creases, or kinks). The surface must be able to stretch and bend without limits, and to intersect itself, so it cannot be made of ordinary material.

In 1959 Stephen Smale proved that it is possible to evert a sphere, although it was still unclear how to actually do it. In 1961, Arnold Shapiro devised the first explicit sphere eversion, and this was published in 1979. Other people have found different eversions.

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