Sphere eversion and inversion

the following things are asked because of a research I am doing for a short novel that I am writing. It contains only very little mathematics, however I want it to be correct.

1. Do I understand correctly that "eversion" means turning inside out of spheres? Furthermore, it has been proven that spheres can be "turned inside out"?

2. So could a human body be "everted" like this?

3. How do you call the other way, "turning outside in", what mathemaical term would be best suitable? I thought about "inversion", but this term seems to be related only to two-dimensional objects?

4. Is it generally true that an easy object (like circle) cannot be everted whereas a more complex one (like a ball) could be?

Kind regards!

• (1) Yes, there are proven sequences of movements that turn spheres inside out. (2) (i) Eversion allows a surface to self-intersect, unlike surfaces in the physical world. (ii) Bodies aren't really spheres, as they have many holes. You'd need to decide if your model of a body is orientable or not (so that 'inside' and 'outside' are even distinguishable). (iii) Bodies are actually systems of discrete cells, not a surface. (3) "turning outside in" is the same thing as "turning inside out," it means swapping the inside and outside. (4) Not sure. – arctic tern Aug 21 '16 at 19:45
• @arctictern Thank you very much. Considering for (2)(ii) that I would take a body without holes (or, for example, a stuffed animal toy [without holes])? Is there something mathematically that would be a suitable way to express a "turning inside out" for a system of discrete cells? – Vazrael Aug 21 '16 at 19:51
• There is clearly no inside or outside for a system of discrete cells (are you just copying phrases I'm using?), so no that wouldn't make sense. And the surface of a stuffed toy cannot self-intersect in physical reality, you'd need to be some kind of magician-slash-manufacturer that built hidden zippers into it or something. – arctic tern Aug 21 '16 at 19:54

If you think of a human body (or, rather, the surface of the human body) as grosso modo a misshapen sphere, then you can perform a sphere eversion on it. (E.g. puff up it with air to make it truly round, then do a usual sphere eversion.)

But the human body has features on it (face, outlines of muscles, etc.), and I'm not sure how they end up after the eversion.

A key aspect of the sphere eversion, as was noted in comments, is that along the way the surface intersects itself, i.e. parts of the sphere pass through other parts of the sphere, and so on a physical object this would mean tearing, so that's another problem.

Another aspect of the sphere eversion is that, to achieve it, one has to first distort the surface of the sphere to be far from perfectly round --- so this would also have a pretty destructive effect on a stuffed animal or the like.

If you haven't, you should watch Outside In, whose goal is to have you physically picture the way one of the known eversions works --- e.g. it shows how regions of distortion are introduced as part of the method. (The introduction of this paper also has some interesting history and pictures.)

In the end, your question won't have a purely mathematical answer, but I think if you watch Outside In and try to imagine that the starting sphere is a round stuffed animal (like a stuffed pig or something), you will be able to find an answer of your own that will satisfy you.

1. Do I understand correctly that "eversion" means turning inside out of spheres? Furthermore, it has been proven that spheres can be "turned inside out"?

$\newcommand{\Reals}{\mathbf{R}}\newcommand{\Vec}{\mathbf{#1}}$Late to the party, but for posterity: The existence of sphere eversion is a statement about mappings from a sphere into Cartesian three-space $\Reals^{3}$, not about geometric subsets of $\Reals^{3}$. (Non-technically, sphere eversion appears to be about subsets because one usually plots images of mappings. Technically that's misleading, because the images themselves are not being deformed as point sets.)

Specifically, if $S \subset \Reals^{3}$ is the unit sphere, there is the standard embedding $I(\Vec{x}) = \Vec{x}$ and the antipodal embedding $-I(\Vec{x}) = -\Vec{x}$. Smale proved that there exists a smooth map $H:S \times [0, 1] \to \Reals^{3}$ such that

• $H(\Vec{x}, 0) = I(\Vec{x})$ for all $\Vec{x}$ in $S$;

• $H(\Vec{x}, 1) = -I(\Vec{x})$ for all $\Vec{x}$ in $S$;

• For each $t$ in $[0, 1]$, the mapping $H_{t}:S \to \Reals^{3}$ obtained by restricting $H$ to $S \times \{t\}$ is an immersion. (Loosely, the image of a sufficient small piece of $S$ acquires from $H_{t}$ a tangent plane at each point; if $H_{t}$ is not injective, i.e., "the surface crosses itself at time $t$", then a single point of the image may have multiple tangent planes.

1. So could a human body be "everted" like this?

In a sci-fi/horror novel, perhaps, but not to a main character. ;)

1. Is it generally true that an easy object (like circle) cannot be everted whereas a more complex one (like a ball) could be?

A circle can be everted in the plane by rotating it through a half-turn about the center. (The same trick works for odd-dimensional spheres sitting inside even-dimensional Cartesian spaces.)

• You seem to be using a different definition of eversion. What I have in mind would change the normal vector between inside and outside. ("Inside" is undefined for immersions in general, but is defined for $H_0$ and $H_1$.) This cannot be done with a circle. – mr_e_man Sep 7 '18 at 21:00