The folder : flat tori finally visualized !

( The paragraphs below outline the results which are published in the PNAS under the heading Flat tori in three-dimensional space and convex integration. )

Flat tori

A flat torus is a parallelogram whose opposite sides are identified. A two-dimensional being living in such an object can not escape from it since each time he enters through a side of the parallelogram, he re-enters through the opposite side. In the pictures below the parallelogram is a square. It is referred to as the square flat torus.

By stretching the fundamental square in the third dimension, it is possible to represent the square flat torus in our ambient space. The flat torus then takes the shape of a rubber ring, or of an inner tube of a bike. We say that the square flat torus has been embedded in three-dimensional space. The resulting object remains unchanged by any rotations about a vertical axis passing through its center, it is called a torus of revolution.

The torus of revolution thus represents the square flat torus in tridimensional space. But this representation is far from ideal since it distorts distances. For instance, horizontals and verticals in the square flat torus all have the same length while this is not true for the corresponding latitudes and longitudes in the torus of revolution.

Isometric embeddings

Can this defect be fixed? In other words, can we find a surface in tridimensional space representing the square flat torus without length distortion? To answer this question, we need to focus on the curvature of such a surface. The curvature of a surface can be measured in many different ways, but one of them is particularly relevant for our purpose: the Gauss curvature. This curvature has two noticeable properties. It remains unchanged if the surface is deformed while preserving all the lengths, and it contains unvaluable hints on the shape of the surface. To give an example, when the Gauss curvature is positive at a given point, the surface must be curved like a sphere as for a mountaintop or a basin. On the contrary, when the Gauss curvature is negative the surface must resemble a mountain pass or a saddle point. For a plane, the Gauss curvature vanishes everywhere.



A mountaintop (positive Gauss curvature) and a mountain pass (negative Gauss curvature)

Those properties have a very strong implication: there does not exist any way to embed the square flat torus in tridimensional space while preserving distances. We say that the square flat torus does not admit any isometric embedding in the ambient space. The reason is the following.

Let us assume for a moment that we can build such an embedding and consider a large sphere enclosing the resulting surface. We can gradually reduce the radius of the sphere until it touches the surface. On the one hand, the Gauss curvature must be strictly positive at the contact point (around that point the surface looks like a mountaintop). On the other hand, the square flat torus has an everywhere vanishing Gauss curvature precisely because it is flat like a plane. Therefore, at the contact point, the Gauss curvature must be zero and strictly positive at the same time. We have reached a contradiction.

Challenging the impossible

In 1954, John Nash while examining the isometric embedding problem in four dimensional space (or in spaces with even larger dimensions) finds an unexpected result: the obstruction to the existence of such embeddings — i.e., the curvature — can be bypassed... provided we pay the price! What price? We will shortly see. One year latter, Nicolaas Kuiper extends the work of John Nash to the case of the three dimensional space and he deduces a somewhat paradoxical consequence: there exist isometric embeddings of the square flat torus in ambient space. This is in total contradiction with what we have just seen above. How is it possible?

John Nash (credits : Paul Halmos) Nicolaas Kuiper (credits : Oberwolfach Photo Collection)

To resolve this contradiction we need to focus on the regularity of surfaces. Below are depicted three surfaces with the shape of a skateboard track. The first one is made up of a plane and a piece of a cylinder connected to each other along an edge showing a substantial angle. On that track, the continuity of the motion is certainly ensured, but it is unlikely that a skateboarder embarks on it (unless he/she performs a lipslide!) Such a surface is said to have a regularity of class C°. For the second surface, the two pieces are lined up exactly in continuation. This surface, which is more regular than the previous one, is said to be of class C¹. Yet, if a skateboarder decides to hit the slopes, he/she will feel an uncomfortable shock when passing through the connecting line. That shock is due to the discontinuity of the curvature of the track: right after the plane part, the surface is abruptly curving itself. To avoid this defect, we have to gradually curve the plane as shown in the third picture. We can now enjoy the track without feeling a shock during the descent. The third surface is said to be of class C².

A surface of class C°, a surface of class C¹ and a surface of class C².

The key point for resolving the apparent contradiction raised by John Nash and Nicolaas Kuiper is the following: if a surface is not regular enough then it becomes impossible to compute its curvature; in fact the very idea of curvature loses all meaning. That is precisely what is happening with surfaces of class C° and C¹ (in the above two first examples the curvature has no meaning along the connecting line). By contrast, the curvature is well-defined at every point of a surface of class C². For the square flat torus, its vanishing curvature prevents the existence of isometric embeddings with C² regularity. However, it does not obstruct the existence of an isometric embedding generating a surface of class C¹ only, as the curvature no longer exists for such an embedding... Indeed, John Nash and Nicolaas Kuiper show —inter alia— that isometric embeddings of the square flat torus in the ambient space do exist, but the counterpart, that is the price to pay, is that these embeddings belong to the class C¹ and can not be enhanced to belong to the class C². Surprisingly this price comes with a bonus, Nash and Kuiper prove that not only isometric embeddings in the class C¹ do exist but there are infinitely many.

Convex integration

The way John Nash and Nicolaas Kuiper demonstrate the existence of isometric embeddings is not amenable to visualization. We are faced with a frustrating situation: we know that there exist numerous surfaces with fascinating properties but we are unable to picture a single one!

Mikhail Gromov (credits : Oberwolfach Photo Collection)

In the 70-80’s, Mikhail Gromov puts into light the concept of h-principle and in the same time develops a technique, the convex integration, which dramatically systematizes and generalizes the Nash and Kuiper construction process of isometric embeddings. This technique enables him to provide a unified vision of a number of paradoxical mathematical results; for instance the existence of a sphere eversion or the existence of isometric embeddings both emerge from a unique principle.

An Isometric Embedding of the Square Flat Torus in Three-Dimensional Space

One benefit —so far practically unnoticed— of the convex integration technique is its algorithmic nature. The starting point of the Hevea Project was to take full advantage of that benefit in order to perform an implementation of that technique and to produce the first pictures of isometric embeddings of flat tori in tridimensional space. We have achieved a computer program that produces a sequence of embeddings of the square flat torus to gradually approximate an isometric embedding. This sequence starts with a short embedding, that is an embedding of the square flat torus in three-dimensional space that shortens all the lengths. Then, this embedding is warped by an infinite sequence of waves. These waves, called corrugations, serve to lengthen distances in various directions until the gap to the isometric situation is fully reduced.

Piling up the corrugations

Corrugations pile on top of each other with decreasing amplitudes and increasing frequencies, the whole process being designed to tirelessly reduce the isometric defect. The process continues indefinitely and, in the limit, builds an isometric embedding of the square flat torus. Of course, the program can only perform a finite number of tasks. We stop it at the fourth step.

Exterior and interior views of the torus of revolution after four corrugations (HD images avalaible on the PNAS site)

Indeed, for the fifth corrugations wave, the amplitudes are so small that they are not visible to the naked eye. With this in mind, the pictures obtained at the fourth step really show an isometric embedding of the square flat torus in three dimensional space.

A vertical, a horizontal, a diagonal and an anti-diagonal in the square and their corresponding curves in the image surface of an isometric embedding of the square flat torus

Because the embedding is isometric, curves corresponding to verticals and horizontals in the square have the same length in the surface. The above picture shows how a meridian and a latitude of the torus of revolution have been corrugated to reach their (equal) required lengths: the meridian being shorter than the latitude, it has undergone corrugations with stronger amplitudes.

C¹ fractals

A consequence of the C¹ regularity of isometric embeddings is the existence at every point of the resulting surface of a tangent plane. This is a plane that « looks like » the surface in the neighborhood of a given point: if we produce successive enlargements around that point, the surface will tend gradually to coincide with that plane. Surfaces do not necessarily admit tangent planes. For instance, the C° skateboard track does not have any tangent plane along its connecting line.

A surface, a point and a tangent plane

Our program generates pictures of the isometric embedding of the square flat torus that reveal some kind of self-similarity in the infinite succession of corrugations. This strongly suggests a fractal structure. This seems even more surprising because the fractal nature is incompatible with the presence of tangent planes. This seeming paradox is resolved when we look more specifically at the behavior of the corrugations at different scales. At each stage, the amplitude of oscillations decreases too quickly to ensure a perfect self-similarity. As a consequence, the limit surface is not as rough as a fractal. Since the limit surface is C¹ regular, we call it a C¹ fractal . The pictures below show the differences of roughness between a fractal curve and a C¹ fractal.

A fractal, the Koch snowflake, and a C¹ fractal, an indefinitely corrugated meridian of an isometric embedding of the square flat torus in the ambient space

A detailed analysis of that C¹ fractal structure has revealed an unexpected link between isometric embeddings generated by the convex integration technique and a specific kind of infinite product, namely the Riesz products. These products belong to the realm of analysis and are better understood than isometric embeddings of flat tori. They offers an immediate understanding of the geometry of our embeddings; it is composed of an infinite number of elementary pieces whose analytic expression is similar to the one found in a Riesz product. Ultimately, these products prove to be a key opening the door to the understanding of the paradoxical surfaces thought up by Nash and Kuiper almost sixty years ago.