Is the intermediate point of a 2-sphere eversion a "double" Boy's surface?

Boy's surface is described here as a 'half-way model' for the eversion of a 3-sphere. If I follow a distributed set of points on a sphere up to this point, do they fall on one single surface, or is it actually to two of them? For example, the North and South poles in Jean Pierre Petit's "Le Topologicon" described here

I'm watching this, this and more so this and this video and I'm trying to recreate this visualization for myself. The image in my profile shows a surface drawn from a parameterization of Boy's surface from R. Bryant, but is this actually only a hemisphere?

There is a helpful information in this answer and a sizable collection of citations and reference material in this page linked there.

• This is my first question here and I understand it may be somewhat below-par, but this is where eversions is discussed. Please feel free to edit, suggest changes.
– uhoh
Aug 5, 2016 at 4:15
• I think it is worth that you define the main concepts precisely in the original post. What do you mean by "intermediate point in the eversion of a sphere", for example Aug 5, 2016 at 5:29
• @uhoh: Thanks for pointing out that the link to my web page was out of date. It should be working now. Aug 5, 2016 at 5:53
• @MarkGrant excellent - I hadn't seen the Optiverse video before, it may be exactly what I need. Thanks!
– uhoh
Aug 5, 2016 at 6:25
• I think you mean 2-sphere eversion. I don't think the 3-sphere admits an eversion. There are certainly some eversions that use immersions of $\mathbb RP^2$ as the `mid-point' and the Optiverse project takes that idea to the extreme. It takes the double-cover of the Boy surface as the starting point, and then it uses elastic bending-energy to find the two null-homotopies to the standard immersions on both sides. Other than using elastic bending-energy, I believe the main idea goes back to Morin. Dec 9, 2020 at 21:01

1 Answer

For a straightforward eversion, there's only a single surface. When the evolving sphere self-intersects, it's never at sheets, but only at lines ("double curves"), which in turn sometimes meet at points ("isthmus point," "triple points"). See 2:20 in the Optiverse video.

However, at 5:45, for an eversion of threefold symmetry, two sheets do overlap fully in what Prof. Francis narrates as a "doubly covered projective plane." As Boy's surface is exactly an immersion of the P.P. in 3-space, we might as well call that a doubly covered Boy's surface.

So it depends on which eversion you choose.

• Many thanks, nice to have a video handy :-)
– uhoh
Dec 9, 2020 at 21:14
• Even so, trying to trace "where a point moves to" on the surface is fiendishly tricky. I have some ideas on how to visualize that, and over the holidays maybe some time to implement them. If you'd like to discuss, let's try SE chat or outside SE. Dec 9, 2020 at 21:48
• @CamilleGoudeseune: For the eversion done by Bill Thurston, he and his sons worked out an explicit parametrization of the eversion. The software is here: geom.uiuc.edu/docs/outreach/oi/software.html You can take that software and colour the sphere any way you like, then render the images in a raytracer. That would let you keep track of where all the points are mapped to. Dec 17, 2020 at 9:27