Produce, in virtual reality, augmented reality, or simply on-screen, a simulation of a surface S in three-dimensional space that behaves as in the context of the problem of sphere eversion. That is, S is allowed to self-intersect, but resists any creasing. The surface hangs in space, unaffected by gravity. Users may grab onto a part of S and pull it, including pulling it unresisted through other parts of itself. This will also drag along nearby points, but it will not allow S to form a cusp in the process.
Many sphere eversion procedures have been discovered since Smale proved that it was possible, including work by Shapiro and Phillips, Morin, Apéry, Sullivan and Chéritat. Much of the difficulty in finding such a procedure comes from the fact that it is difficult to experiment and visualise the effects of possible moves. We suspect that, given tools to easily and intuitively investigate movement of a sphere that follows the rules allowed for sphere eversion, even lay people will be able to find a solution given some persistence.
With this new technology, we will begin to map out and understand the space of eversions.
- Stephen Smale, A classification of immersions of the two-sphere, Trans. Amer. Math. Soc. 90 (1958), 281–290.
- Arnaud Chéritat, Yet another sphere eversion, https://arxiv.org/abs/1410.4417
- Many more at: http://www.chrishills.org.uk/ChrisHills/sphereeversion/