The Unturning of the Torus
December 9, 2004
page 6
The non-trivial unturning of the torus
J.P.Petit:
Comptes Rendus Académie des Sciences.
volume 293, session of October 5, 1981, series 1 pp. 269-272
I will content myself with presenting the
sequence of drawings, without commenting on them.
Non-trivial unturning of the torus. First part
of the transformation
Non-trivial unturning of the torus. Second part of
the transformation
When one arrives at figure v one sees that it is then easy to make the gray
structure and the pink structure coincide in order to transform this object into a
two-sheeted covering of the Klein bottle.
The unturning then takes place by
exchanging the facing sheets. Below, the same drawing with
a color coding.
Two-sheeted covering of the Klein
bottle, with color coding
( this drawing not being part of my annual report to the Cnrs.
It will be found in the Topologicon )
The different families of tori.
What Stephen
Smale had demonstrated in 1957 was that there was only a single family of immersions
of the sphere and that all of these could be connected
to one another by a homotopy. These formed a group whose neutral
element consisted in leaving the object as it was. One wondered whether it
could be the same for the torus. The mathematicians
Ioan James and Emery Thomas showed that the immersions of the torus were distributed
into four continents between which it was impossible to pass
by means of a regular homotopy.
The four families of tori
The "standard torus", drawn
at the center of the sheet, belongs to the same family as the object
depicted in b. This is what I incidentally showed
in the version of the unturning of the torus that I invented in 1980. The
family evoked in a represents a torus that has undergone a twisting
of 360°. It resembles the standard torus but the two are defined
on the basis of their mapping system, with the aid of two
families of curves. In the standard torus one uses two sets of circles
assimilated to meridians and to parallels.
On torus a one should complete the family of circles glued
onto it by a second family, twisting in the opposite direction. What one can show
then is that it is impossible, by means of a regular homotopy,
to bring the mesh of this torus a into coincidence with the mesh of the standard torus
( meridian circles plus parallel circles ). It is in this sense
that these are different objects. All these objects can obviously
be configured as a two-sheeted covering of the Klein
bottle.
The power of the geometer's tools
is to be able to predict what is possible and what is not.
Transforming the standard torus into the torus of figure b : yes.
Going from c to d : no.
It avoids wastefully losing one's time
and above all it encourages one to seek things that are in no way
obvious, such as turning a sphere inside out. It is the same
in all the sciences. It happens that people pass by
fruitful approaches, for years or even
centuries, simply because they believed them impossible to
carry out. I devoted a few years of my life to constructing
a theory of the suppression of shock waves around an object moving
at supersonic speed in a gas, with the aid of a field of Laplace forces, of " MHD ". A student
even did his thesis on this subject under my supervision and we
published this work in various peer-reviewed journals
and scientific conferences. It is a theme that is only beginning
to surface, thirty years later. It is surmised that the
Americans would have hypersonic aircraft capable of evolving
at Mach 10 without creating a shock wave (and in particular without suffering
the formidable thermal stresses linked to the recompression
of the air behind these "bangs". It is the famous myth of the Aurora,
an aircraft cruising at the altitude where the northern lights occur,
between 80 and 150 km altitude. Aurora is also the prefiguration of
future space launchers which, relying on the air, will be much more economical
than the Cnes rockets. It was impossible, in France,
to initiate such research (I had these ideas in 1975), because
people, in particular at the Cnrs, found them totally unreasonable.
The result represents thirty years of delay with respect
to the United States, in my opinion totally impossible to make up.
The tobacco pouch
To be complete one must cite the
versions of the turning inside out of the sphere having as central object a tobacco
pouch. It was an object that was common when I was
young, but that one should hardly encounter anymore today. The first
to have drawn these sequences is Georges Francis. For some
years I have been working on a polyhedral version of these versions,
which has already yielded a rather pretty central model. But,
to show it to you, I will have to manage to get my hands on it again.
Fairly quickly, I hope, because it is one of the most fascinating objects that
I have ever created.
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