Turning the sphere inside out. For those who don't like math
The great mathematician David Hilbert once said that a mathematical theory can be considered perfect only when it can be presented to the first person he meets. Hilbert's followers are in complete despair trying to live by this recipe. Mathematics is becoming more and more specialized, and now a scientist mathematician sometimes needs a lot of work, even to his colleagues, to explain the essence of the problems he is solving. However, from time to time, research in the leading and seemingly inaccessible branches of this science leads to a discovery that is interesting to the layman and at the same time can be explained without oversimplification. A striking example of this is Stephen Smale's theorem on the so-called regular mappings of the sphere, published in 1959.
The field in which Smale was then working - differential topology - one of the most abstract branches of modern mathematics. It is all the more surprising that, nevertheless, it was possible to come up with a visual explanation for one of the most striking consequences of Smale's theorem. Namely, you can demonstrate how to turn the sphere inside out.
In the usual sense, this is, of course, impossible: the sphere would have to be torn apart. But in differential topology it is allowed - mentally, of course - to drag a surface through itself - these are the "rules of the game" in this science. But then a simple solution immediately catches the eye.
It is necessary to squeeze the opposite sides towards the center until they pass through each other (I). The inner, painted surface (II) protrudes from two opposite edges. Let's continue this process of "stretching" the inner surface until the ring formed by the remaining part of the outer surface (II) completely disappears. Unfortunately, during this process, the ring forms a tight loop (III), which must be tightened. The result is a scar (IV), and this does not satisfy differential topologists, because they consider only so-called "smooth surfaces", which do not have any corners and kinks.
So, the task is to turn the sphere inside out in such a way that, getting rid of the ringlet, you do not get a scar. And here intuition again suggests that the problem is insoluble. When Smale first announced that he could prove the existence of a solution, no one believed him. But intuition was wrong: there was not a single logical error in Smale's proof. Mathematicians became convinced that it was theoretically possible to follow the proof step by step and find an explicit description of the deformation that everted the sphere. But it was so difficult that it seemed like a lost cause. For some time after Smale's discovery, it was known that, in principle, it was possible to turn a sphere without a scar inside out, but no one had the slightest idea how to do this.
But, in the end, mathematicians coped with this task. How - you will understand by looking at the pictures. They are entertaining.
Although Smale's proof did not consist of drawings alone. It is curious that in his work they do not exist at all - those figures that are implicitly contained in his abstract analytical apparatus are too complex. The most inventive artist would not have been able to portray them - the imagination of mathematicians is amazing. But perhaps even more striking is their ability to convey the most complex ideas to each other without resorting to drawings. The story with the turning of the sphere is a vivid evidence of this. She became known to the general public thanks to the French topologist Rene Thom, who learned about her from his colleague Bernard Morein, who, in turn, from the American Arnold Shapiro, the inventor of this "inversion". This is especially curious when you consider that Bernard Morin is blind.
These pictures show how you can turn a sphere inside out without violating the requirements of differential topology. First, you need to bring the opposite sides of the gray sphere (A) closer together, pushing them through each other. Then the painted surface (B) appears on both sides. Then you need to stretch one of the colored pieces (C) in such a way as to obtain a surface that resembles a saddle on two "legs" (O). These two legs are twisted counterclockwise to obtain surface E. It is shown again (P) “in section” with ribbons, which, as in the “scarred sphere”, represent cross-sections at ten different levels.
Then there is no point in depicting the resulting surfaces at each stage - they are too complex. But you can, if you like, consider the ribbons at all 10 levels and mentally finish drawing. We nevertheless decided to show one stage (H2) - just so that one could imagine what the type of the resulting figures is. The G surface appears after the saddle of the P surface is compressed and rotated 90 °.
A few more steps. Namely: between stages I and J, two legs of the same shape pass through each other. Each ribbon-like surface section in stage J has two gray sides facing each other. Between stages J and K, the inner layer expands and the outer one contracts; the surface K is obtained - exactly the same as J, but only the colors have changed places.
Then all actions are carried out in the “reverse order”. You can get an idea of them by looking at pictures I, H, C, etc. You just need to swap the colors of the ribbons in each picture. We present the end of this second row of pictures. Surface L corresponds to surface F, L2 to E, etc.
The colored sphere (surface P) corresponds to the gray sphere (surface A). So, the deformation is done, and there is no scar. The very possibility of this trick was first proved by S. Smale. And all successive stages of deformation were invented by A. Shapiro ...
P. S. What else are British scientists talking about: that the mechanism of turning the sphere inside out is sometimes no more philosophical than, say, a PDF program created by some talented programmer.
In three-dimensional space, you can turn it inside out in a diving class, that is, with possible self-intersections, but without kinks. In other words, the image of the sphere at each moment of deformation must remain smooth, that is, differentiable.
The twisting of the sphere is not at all a logical paradox, it is a theorem, only a very counterintuitive one. More accurately:
It is difficult to imagine a concrete example of such a family of dives, although there are many illustrations and films. On the other hand, it is much easier to prove that such a family exists, and this is exactly what Smale did.
Story
This paradox was discovered by Smale in 1958. According to legend, when Smale tried to publish this theorem, he received a response saying that the statement is obviously wrong, since in the process of such "eversion" the degree of the Gaussian map must be preserved. [ ] Indeed, the degree of the Gaussian map must be preserved; in particular, this shows that the circle cannot be "turned out" in the plane, but the degrees of the Gauss maps y and at v both are equal to 1. Moreover, the degree of any embedding is equal to 1.
Variations and generalizations
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Literature
- Smale, Stephen A classification of immersions of the two-sphere. Trans. Amer. Math. Soc. 90 1958 281-290.
- Francis, J. Moscow: Mir, 1991. Chapter 6. Turning the sphere inside out.
Notes (edit)
Excerpt from the Inversion of the sphere
“Again, Colonel,” the general said, “I cannot, however, leave half of the people in the forest. I ask you, I ask you, ”he repeated,“ to take position and prepare for the attack.- And I ask you not to interfere with other than your own business, - answered, ardently, the colonel. - If you were a cavalryman ...
“I’m not a cavalryman, Colonel, but I’m a Russian general, and if you don’t know that ...”
“It’s very well known, Your Excellency,” the colonel suddenly cried out, touching the horse, and turning red-crimson. “You don’t want to be in chains, and you will see that this position is nowhere worthless. I do not want to exterminate my shelf for your pleasure.
“You forget yourself, Colonel. I am not observing my pleasure and will not allow this to be said.
The general, accepting the colonel's invitation to the tournament of bravery, straightening his chest and frowning, rode with him towards the chain, as if all their disagreement was to be resolved there, in chains, under the bullets. They arrived in a chain, several bullets flew over them, and they stopped silently. There was nothing to look at in the chain, since even from the place where they had previously stood, it was clear that it was impossible to move through the bushes and ravines of the cavalry, and that the French were bypassing the left wing. The general and the colonel looked sternly and significantly, like two roosters preparing for battle, at each other, in vain expecting signs of cowardice. Both passed the exam. Since there was nothing to say, and neither one nor the other wanted to give the other a reason to say that he was the first to get out of the bullets, they would have stood there for a long time, mutually experiencing courage, if at that time in the forest, almost behind them, they heard the crackle of rifles and a dull, merging cry. The French attacked the soldiers who were in the forest with firewood. The hussars could no longer retreat with the infantry. They were cut off from the path of retreat to the left by the French chain. Now, as inconvenient as the terrain was, it was necessary to attack in order to make a way for itself.
The squadron where Rostov served, who had just managed to get on the horses, was stopped facing the enemy. Again, as on the Ensk bridge, there was no one between the squadron and the enemy, and between them, dividing them, lay the same terrible line of uncertainty and fear, like a line separating the living from the dead. All people felt this line, and the question of whether they would cross or not and how they would cross the line worried them.
Imagine that the "ordinary" two-dimensional sphere S 2 is made of elastic material that can go through itself. Is it possible to turn the sphere inside out in the usual three-dimensional space $$ \ mathbb (R) ^ 3 $$ without kinks and breaks, but with possible self-intersection (that is, in the immersion class)?
In 2000, Smale compiled a list of 18 tasks that, in his opinion, must be solved in the 21st century. This list is compiled in the spirit of Hilbert's problems, and, like the millennium problems compiled later, includes the Riemann hypothesis, the question of the equality of the classes P and NP, the problem of solving the Navier-Stokes equations, and also the Poincaré conjecture, now proved by Perelman. Smale compiled his list at the behest of Arnold, then president of the International Mathematical Union, who most likely took the idea for the list from Hilbert's list of problems.
And finally, the question: is it possible to "turn out" a circle in a plane, that is, to find a continuous family of immersions, as above?
Comments (1)
Curious. The following thing comes to mind. Let's imagine a sphere in the form of a stereographic projection - a plane with infinity. Then turning the sphere inside out looks just like rolling the plane in the other direction, i.e. with a different orientation. There is a hole in the reasoning somewhere, right?
Well, the fact is that stereographic projection implies the selection of a point on a sphere, which does not correspond to anything on the plane, and this changes the rules of the game, because according to the conditions, the sphere cannot be broken, and the point cannot be punctured exactly.
Well, in principle, I suspected that there was a weak point with an infinitely distant point. I just wanted to know an independent opinion;).
Misha, I would like to hear if K3 surfaces occur in string theory, and if so, how exactly do they arise there?
Yes, there are sometimes. In the context of compactification. K3 has the holonomy group $$ SU (2) \ subset SU (2) \ times SU (2) $$ and therefore retains half of the supersymmetries. Phenomenologically, such models are not very interesting, but people consider them anyway.
I turn the sphere without kinks even easier than in the film. You need to stick part of the surface of the sphere inward with your finger. Rotate this inner part of the sphere 180 degrees, while the hole closes without kinks. The meridians of the sphere, which were circles, will turn into "eights" with a smaller head inside a larger one. Next, we inflate the inner almost ball until it leaks out. Naturally, his appearance will turn out to be inverted. It remains what was for the most part, and now that has become smaller in comparison with the bloated, to unfold 180 degrees. The tightened hole will open, straighten the dent, and the goal is achieved!
Here it turns out a point becomes infinity, and infinity becomes a point. Or, "the sameness of the universe": what's inside, what's outside.
Therefore, a paradigm arises - the microcosm can be studied using the macrocosm and vice versa.
The question is in the radius limit =] h / 2; 2 / h [. Here h is used as the metric limit of measurement accuracy, that is, the same epsilon divided by two.
Also, the physical existence of such a sphere can be proved or disproved for various cases.
Or am I wrong?
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