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Relevant: How to turn a sphere inside out: https://www.youtube.com/watch?v=R_w4HYXuo9M

Perfect Shapes in Higher Dimensions: https://www.youtube.com/watch?v=2s4TqVAbfz4



I think the sphere eversion is actually not a related problem in mathematical terms.

In any case, that video is part of the excellent longer film "Outside In" made at the Geometry Center in the 90s:

http://www.geom.uiuc.edu/docs/outreach/oi/




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