A sphere is a 2-dimensional surface, like an infinitely thin soap bubble or a 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.
References and links:
Anthony Phillips, Turning a surface inside out, Scientific American, May 1966
Nelson Max and William Clifford Jr, Computer Animation of the Sphere Eversion, 1975
Nelson Max and Bernard Morin, Turning a Sphere Inside Out, 1977
George Francis and Bernard Morin, Arnold Shapiro's eversion of the sphere, Mathematical Intelligencer, 1980
François Apéry, An algebraic halfway model for the eversion of the sphere, Tohoku Mathematical Journal, 1992
Outside In (1994) uses Bill Thurston's corrugation method
George Francis, John Sullivan, Rob Kusner, Ken Brakke, Chris Hartman, and Glenn Chappell, The minimax sphere eversion, Visualization and Mathematics, 1996
Erik de Neve, Sphere Eversion, 1997
John Sullivan, George Francis, and Stuart Levy, The Optiverse, also on YouTube, 1998
John Sullivan, “The Optiverse” and other sphere eversions, 1999
J. Scott Carter and Sarah Gelsinger, A Sphere Eversion, 2007
J. Scott Carter, Turning a Sphere from Red to Blue, 2008?
Iain Aitchison, The holiverse sphere eversion, The 'Holiverse': holistic eversion of the 2-sphere in ℝ³, 2010
Arnaud Chéritat, Sphere Eversion, Yet another sphere eversion, 2014
Chris Hills and Erik de Neve, The deNeve/Hills sphere eversion 2016, also on YouTube, in 3D, and Interactive model 2018
Mikami Hirasawa and Minoru Yamamoto, Sphere eversion from the viewpoint of generic homotopy, Topology and its Applications, 2017
Adam Bednorz and Witold Bednorz, Analytic sphere eversion with minimum of topological events, 2017
Adam Bednorz, Sphere eversion, 2018
Michael J. McGuffin, Sphere Eversion Program
Wikipedia, Sphere eversion
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