Every World in a Grain of Sand: John Nash’s astonishing Geometry

As has been widely reported, John Forbes Nash Jr died tragically in a car accident on May 23 of this year. Many tributes have been paid to this great mathematician, who was made famous by Sylvia Nasars biography A Beautiful Mind and the subsequent movie based on that book.

Much has been said about Nashs work on game theory . But less has been said about Nashs other mathematical achievements. Many mathematicians who understand Nashs work would agree, I think, that although his work in game theory had the most impact on other fields, Nash made other breakthroughs which were even more impressive.

Apart from game theory, Nash worked in fields as diverse as algebraic geometry , topology , partial differential equations and cryptography .

But perhaps Nashs most spectacular results were in geometry . To honour Nashs life, I would like to try to give a flavour of some of this work.

John Nash and pure mathematics

A great deal of Nashs work was in the field of geometry. But this kind of geometry differential geometry is very different from the geometry learned at high school. It is not about trigonometry or Pythagoras, as found in secondary maths textbooks. Rather, it is about topics like surfaces, curvature and smoothness.

Like all pure mathematicians, Nash proved theorems: logical statements that are rigorous, precise and absolutely true, with no tolerance for vagueness. The world of pure mathematics is austere and often abstruse, but its claims to truth are eternal and absolute.

Well, thats the theory at least. Breakthroughs in pure mathematics are often at the very limits of human understanding. It takes time, even for those in the field, to fully comprehend new developments.

Nashs work was an extreme case. His papers could be chaotically presented, hard to follow and his approaches to problems were often unlike anything that had come before him, bamboozling students and experts alike. But he was almost otherworldly in his creativity.

While mathematical arguments are tightly constrained by the rigorous requirements of logic, Nashs constructions and methods were wild. And nowhere was this more so than in his work on geometry.

Nashs geometry

Take a flat sheet of paper. You can bend it, but without ripping it or creasing it, what shapes can you make? You cant make a sphere, or even a section of a sphere, because a sphere is curved , while the paper is flat .

But you can make a cylinder. And even a cone, as youll know if youve ever seen a dunces hat. (This fact is also useful for making waffle cones, as shown below.)

Waffle cones start off as flat surfaces. Gotham3/ingur

As it turns out, even though a cylinder or a cone looks curved, it is intrinsically flat . In an undergraduate course on differential geometry (such as the one I teach at Monash), one studies this intrinsic curvature, and it turns out that there are lots of flat surfaces.

This surface might not look flat, but it is. Richard Morris/Wikipedia

These ideas were around for hundreds of years before Nash, but Nash took them much further.

The embedding problem

Nash took up the idea of embedding a surface: placing it into space without tearing, creasing or crossing itself. An embedding which does not distort the surfaces intrinsic geometry is isometric. In other words, the surfaces above are isometric embeddings of the plane into 3-dimensional space.

The isometric embedding question can be asked not just for the plane, but for any possible surface: spheres, donuts (which mathematicians call tori to try to sound respectable) and many others.

As it turns out, there are surfaces that are so strongly curved or tangled up that they cannot be embedded into 3-dimensional space at all. In fact, they cant even be embedded into 4-dimensional space.

But Nash showed that any surface can be embedded into 17-dimensional space. Extra dimensions, far from making the problem even more difficult, actually make it easier giving you more room to embed your surface! Later on, Nashs work was improved by others, and we now know that any surface can be embedded into 5-dimensional space.

However, surfaces are only 2-dimensional. And Nash was interested in surfaces of any possible dimension. These higher dimensional analogues of surfaces are known as manifolds.

Nash proved that you can always embed a manifold into space of some dimension, without distorting its geometry. With this momentous result, he solved the isometric embedding problem.

Nashs proof of the isometric embedding problem came as a complete surprise to much of the mathematical community. His methods were revolutionary. The great mathematician Mikhail Gromov said that Nashs work on the embedding problem struck him to be as convincing as lifting oneself by the hair. But after great effort, Gromov finally understood Nashs proof: at the end of Nashs lengthy argument, Gromov said, Nash miraculously, did lift you in the air by the hair!

Isometric embedding in action

Gromov went on to develop his own ideas, inspired by Nashs work. He wrote a book similarly renowned among mathematicians for its incomprehensibility, just like Nashs work in which he developed a method called convex integration.

Gromovs method had several advantages. One is that it is easier to draw pictures of an embedding made with his convex integration method. Prior to Gromov, we knew isometric embeddings existed, and had wonderful properties, but had a very tough time trying to visualise them, not least because they were often in higher dimensions.

In 2012, a team of French mathematicians produced computer graphics of isometric embeddings using Gromovs convex integration methods. They are extremely intricate, almost fractal-like, yet smooth. Some are shown below.

The world in a grain of sand

Nashs work on the isometric embedding problem has many facets and has led to huge amounts of subsequent research.

One particularly amazing aspect is how isometric embeddings are constructed. Nashs work, combined with subsequent work by Nicolaas Kuiper , showed that if you wanted to isometrically embed a surface in 3-dimensional space, its enough to be able to shrink it.

If you have a shrunken embedding of your surface that is, with all lengths decreased then Nash and Kuiper show how you can obtain an isometric embedding of your surface just by adjusting your shrunken version a bit.

This sounds ridiculous. For instance, take a sphere say the surface of a tennis ball and imagine shrinking it down to have a nanometre radius. Nash and Kuiper show that by ruffling the surface sufficiently (but always smoothly; no creasing or folding or ripping or tearing allowed!) you can have an isometric copy of your original tennis ball, all contained within this nanometre radius. This type of ruffling of the surface was reproduced in the French teams computer graphics.

The French team considered taking a flat square piece of paper. Glue the top side to the bottom side, to get a cylinder. Now glue the left side to the right side. If you think about it, you might be able to see that you get a donut. But youll find the paper is now creased or distorted.

Can you embed it into 3-dimensional space without distortion? Nash and Kuiper say yes. Gromov says use convex integration. And the French mathematicians say this is what it looks like!

Isometric embedding of the square flat torus in ambient space. Hevea Project , CC BY-SA

More pictures are available at the Projects website .

But the mathematical theorem doesnt just apply to tennis balls or donuts: the theorem holds for any manifold of any dimension. Any world can be contained in a grain of sand.

How did he do it?

Nash had a rare combination of genius and hard work. In her biography of Nash, Sylvia Nasar details his formidable intensity and effort spent working on the problem.

As is well known from the movie, Nash came to believe in outlandish conspiracy theories involving aliens and supernatural beings, as a result of his schizophrenia. When later asked why he, an extremely intelligent scientist, could believe in such things, he said those ideas came to me the same way that my mathematical ideas did. So I took them seriously.

And frankly, if my head told me ideas as accurate and as insightful as those needed to prove the isometric embedding theorem, Id likely trust it on aliens and the supernatural too.