# Mathematical Field Notes

## Cone eversion

Last year, around the time Chris Wendl was running the h-principle learning seminar at UCL, I set my second years an exercise from Eliashberg-Mishachev as a difficult challenge problem: to find an explicit cone eversion. In other words, find a path in the space of functions on $\{(r,\theta)\in\mathbf{R}^2\ :\ r\in[1,2]\}$ connecting $r$ to $2-r$ such that none of the intermediate functions has a critical point. One of these students, Tom Steeples, got hooked on the problem, almost solved it, and afterwards used Mathematica to produce some beautiful computer animations of a solution given by Tabachnikov in American Mathematical Monthly (1995) Vol 102, Issue 1, pp 52–56. Here, below the fold because it’s quite a large file, is one of his images. Reproduced with Tom’s kind permission (the copyright is his).