I see it, but I don’t believe it
Shocking results about spheres
Steven Smale is an eminent mathematician who has made seminal contributions to many areas of mathematics. He solved the Poincaré Conjecture for dimension ; he made fundamental contributions to the theory of dynamical systems; he, with Lenore Blum and Michael Shub, created the real number question. He is a Fields Medalist, standing alongside Michael Atiyah, Paul Cohen, and Alexander Grothendieck in 1966. His medal was awarded especially for his brilliant work on the Poincaré Conjecture—his was the first non-trivial result on it.
Today I would like to discuss mathematical shocks, which Smale and others have created.
There has been much discussion here and elsewhere on whether we tend to guess right or not. I think that there are plenty of surprises, which are results that we did not predict. See this discussion for my previous thoughts about surprises. I would characterize a shock as something that is even more than a surprise. A shock is a result that leaves us speechless, that takes us completely off-guard, that at first seems impossible.
Smale created one of the great shocks in mathematics, but he also has done some “shocking” things. Perhaps the most shocking was this statement that caused great discomfort within NSF:
He once said that his best work had been done “on the beaches of Rio.”
This statement, which was probably true, and which should have sent more top mathematicians to sit on the beaches somewhere, was not understood by politicians. The result was a ban, for a while, on Smale’s funding.
I will give four results that I think fall into the category of shocking. Each result will be about a shocking property of the sphere—that is a ball, the kind you can hold in your hand, the kind that we use to play games such as soccer or baseball or basketball. Sometimes the balls have to be in higher dimensions than three, so you cannot quite hold them in your hand.
Sphere Eversion: This shock is Smale’s result that it is possible to turn a sphere inside out, without leaving a crease or hole. Silvio Levy here gives a detailed history of this result.
My high-level understanding of the history is that in 1957 Smale proved a general abstract theorem about embeddings of spheres into Euclidean spaces. Apparently he did not notice at first that his abstract theorem implied that there had to be a way to continuously move a sphere in three space so that it turned inside out. Of course the motion must have the sphere pass through itself, but otherwise not cause any tear or crease. This cannot be done in two-dimensions, but magically can be done in three.
This was so shocking that Raoul Bott, one of the founders of differential topology, told Smale that his abstract theorem must be wrong. He had a technical point that he thought must be violated, but he was eventually convinced that Smale was correct. Still neither Smale nor Bott nor others could “see” how to actually turn a sphere inside out. Finally Bernard Morin discovered how the construction worked. One of the final shocking parts of this story is that Morin is blind, yet he was one of the first to understand and “see” how to turn a sphere inside out. A tremendous indication of Morin’s mathematical ability.
Check out this for a series of pictures that give insight into how eversion works, or watch the movie here. The “half-way” point of the construction, which is called the Morin surface looks like this:
Clearly just from this picture you can see that the construction is quite complex, and requires the sphere to be manipulated in a manner that is non-trivial.
Sphere Cardinality: This shock is not Georg Cantor’s famous diagonalization theorem: that the reals are uncountable. That did not shock him. The shock was his proof that the interval and the unit square have the same cardinality. Okay it really is not exactly about spheres, but it can be viewed as showing that the unit circle and the unit disk have the same cardinality.
When he first proved this he wrote to Richard Dedekind:
“Je le vois, mais je ne le crois pas!” (“I see it, but I don’t believe it!”)
That a line and a square could be put in a one-to-one correspondence with each other seems to have really surprised Cantor. The proof raises an interesting point too.
The way that Cantor first defined the map from square to line was by,
He interleaved the digits that defined the two numbers. This is a natural idea, but it has a problem that was pointed out to him by Dedekind: the mapping is not one-to-one. The problem is caused by the usual ambiguity of the decimal representation of reals:
Today we could easily argue around this issue, since we could use Cantor-Bernstein’s Theorem to make the interleaving argument work. But Cantor used another trick; he used that real numbers have a unique representation as continued fractions, provided they are not rational:
Theorem: Let be a number in the interval . Then it has a continued fraction representation of the form:
where are all positive integers. Further the representation is unique if is irrational.
This is a neat representation theorem, and the key part is that for irrational numbers it is unique. So there is no ambiguity issue, the problem that can make arguments about real sets messy. Of course for rationals it is easy to see that it is not in general unique.
Cantor used this theorem to do his interleaving, and since the representation is unique he could make his argument rigorous. Fernando Gouvêa has a wonderful article in this March’s Math Monthly on exactly this issue
Sphere Measure: This shock is that a solid sphere can be divided into a finite number of pieces, and these pieces can be reassembled to form another solid sphere that is twice the size. This is called the Banach-Tarski Paradox—I guess a true statement that is called a paradox might be considered a shock. Truth in blogging: this is the only result here that uses the axiom of choice, so one can, and some do, argue that the shock is due to the axiom of choice alone.
The result is due to Stefan Banach and Alfred Tarski, who proved it in 1924.
There is a very readable book, “The Pea and the Sun: A Mathematical Paradox” by Leonard Wapner on this result. He does a great job making the result understandable, and I enjoyed reading it a while ago very much.
Sphere With Exotic Structure: This shock is that there are exotic spheres. An exotic sphere is a differentiable manifold that is the same as the standard Euclidean -sphere with respect to continuous bijections, but not with respect to differential ones. The discovery of such spheres in 1956, in dimension 7, is due to the work of John Milnor, who just won the Abel prize, for this and many other great results.
Since then a great deal of work has been done on this topic, although one major open problem remains: it is unknown whether such spheres exist in four dimensions.
Michael Freedman, Robert Gompf, Scott Morrison, and Kevin Walker say that it is believed to be false, that is that such spheres do not exist. In their 2010 paper they discuss approaches to resolve the conjecture that will make heavy use of computation. Take a look at the paper, but here is a taste of what they are doing:
Computing the two-variable polynomial for K2 took approximately 4 weeks on a dual core AMD Opteron 285 with 32 gb of RAM. At this point, we haven’t been able to do the calculation for K3 . With the current version of the program, after about two weeks the program runs out of memory and aborts.
It would be cool if high-performance computation helped resolve this long standing open problem.
What is your favorite shock? Do you agree that these are really shocks? See this for another view of shocks.