The Sphere Eversion
December 7, 2004
page 1
Introduction.
We will consider in what follows closed surfaces, such as the sphere, the torus, and a few others. These are surfaces in the sense understood by the layperson, meaning they are two-dimensional objects represented in a three-dimensional Euclidean space, R³, which is our mental space of representation. These surfaces can be represented in several ways. If they do not intersect themselves, we will say they are
immersed
(in R³). If they do intersect, we will speak of
immersions
and this intersection will then be represented by an
self-intersection set
(self-intersection).
In our immersions we will assume that the tangent plane varies continuously and that the surface has no singularities, such as the tip of a cone. Our surfaces will be
regular.
In the case of immersions, we will require that along the self-intersection curves the two tangent planes of the intersecting sheets be distinct.
The world of geometry, as conceived by the mathematician, is quite different from the physical world. The fact that surfaces can self-intersect does not bother it in the least. The physical world does not allow such things. But this becomes possible in the metaphysical world. Thus, in the Bible it is written that when the dead rise again, they will do so in the form of "glorious bodies." They will then be able to pass through anything, and in principle be capable of passing through themselves. Thus, when the time of the Last Judgment comes, if you stroll through Rome in the form of a glorious body, lost and searching for Piazza Navona, you might be tempted to ask directions from another resurrected human being, who looks exactly like you. Suppose the person you ask is walking in the opposite direction from the square. In ordinary physical space, he would have to turn around to point his finger in the right direction. But if he walks in the form of a glorious body, this rotation would no longer be necessary. He could point his index finger at his navel and pass through himself. When his hand re-emerges from his back, he would only need to say to you, "That's the way." By pushing his arm through his belly, he would have created within his bodily envelope a self-intersection set consisting of two circles, which would disappear when he returns to his normal configuration.
If a human being closes his mouth, puts a clothespin on his nostrils to seal them, and we abstract from his other natural orifices, his bodily envelope then takes on the topology of the sphere S². Imagine a resurrected being in the form of a glorious body whose natural orifices are thus sealed. We know that he can pass through himself, meaning his bodily envelope can transition from an embedded state to an immersed state. One of the metaphysical problems that then arose was whether a resurrected person in the form of a glorious body could turn itself inside out without creating wrinkles.
A small aside. Magicians know how to use "magic circles" that can interpenetrate "magically." One could imagine representing surfaces using a kind of "magic mesh," such that the two sheets, shown here in black and pink respectively, could pass through each other without difficulty.
Magic mesh
In any case, one must admit that there is often little difference between mathematics and magic. Twenty years ago I created a comic strip: The Topologicon. It is now out of print and unobtainable, except as a collector's item. On one of its pages, one could see this:
It is truly a pity that Belin decided to abandon this collection. One must admit that with a production cost of barely over one euro, selling albums for 13 euros (plus shipping), through mail order, while leaving a profit margin of 12 euros—meaning a profit exceeding 92 percent of the selling price—does not represent a very obvious commercial strategy, especially for black-and-white comics.
Consider a sphere S² embedded in R³. We assume its outer surface is gray and its interior is a faded pink. We can press two antipodal points—arbitrarily called the "north pole" and "south pole"—together until they touch at a single point. One can do this, for example, with a donut. When dealing with a mathematical donut (we don’t know whether donuts resurrect or not in the form of glorious bodies), the two polar regions, after touching at a point, can self-intersect along a self-intersection curve forming a circle. Anticipating, we will say that this surface has undergone a catastrophe of type Do.
One might then be tempted to try to turn the donut, the sphere, inside out by continuing the operation. But a bulge will form, which will degenerate into an unsightly fold, or more precisely a fold surface (figure d).
At the end of the 1950s, the serious question of whether one could turn metaphysical donuts inside out without creating folds remained unresolved. To be honest, everyone thought it was strictly impossible. But in 1957, a mathematician, Stephen Smale (who later received the Fields Medal, but for an entirely different work), proved that all the immersions of the sphere S² in R³ form a single connected set, and that it is always possible to find a continuous sequence of immersions (also called regular homotopy) allowing one to pass from one configuration to another. The corollary was that one should be able to pass, via a continuous sequence of immersions, from the standard embedding of the sphere S² to the antipodal embedding. Put more simply: one should be able to turn a sphere inside out without creating folds, provided one allows it to turn itself inside out.
Smale’s advisor was Raoul Bott. Bott asked his student how one should go about it, and Smale replied that he had no idea, but that his theorem was completely unassailable. Smale did not visualize things in space at all, but he didn’t care (as is the case with many geometers). And, if we are completely honest, after proving his theorem, he carelessly dismissed the question of how one might actually carry out the process, and quickly moved on to another subject, leaving his fellow mathematicians utterly perplexed. I find it rather unsympathetic to create such problems and then leave others to figure out the solution ten years later.
It must be admitted that it is quite difficult to imagine immersions in one’s mind. Yet we know of surfaces that can only be represented in R³ in this way. The Klein bottle, for example.
Klein bottle
We have represented it here using a mesh-coordinate system composed of two sets of closed curves, just like the torus. Thus, one can mesh a Klein bottle without creating any singularities...