Hevea Project: The Folder
Flat tori finally
visualized!
The Hevea Project
The aim of the Hevea Project is to provide an implementation of the Convex Integration Theory in order to visualize isometric embeddings of flat tori in threedimensional Euclidean space. The project involves three laboratories: the Institut Camille Jordan (Lyon, France), the Laboratoire Jean Kuntzmann (Grenoble, France) and the GipsaLab (Grenoble, France).
It has received support from the Région RhôneAlpes and the CNRS.
The paragraphs below outline the results which are published in the
PNAS under the heading Flat tori in threedimensional space and convex
integration.
For a concise presentation of the project in a questions/answers lay out, click on that link.
Flat tori
A flat torus is a parallelogram whose opposite sides are identified. A
twodimensional being living in such an object can not escape from it
since each time he enters through a side of the parallelogram, he
reenters 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
threedimensional 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 and Nicolaas Kuiper (credits : Paul Halmos and 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 welldefined 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 7080’s, Mikhail Gromov puts into light the concept
of hprinciple
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 ThreeDimensional 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 threedimensional 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 antidiagonal 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 selfsimilarity
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
selfsimilarity. 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.