The Turning of the Torus
December 9, 2004
page 6
The Non-Trivial Turning of the Torus
J.P. Petit:
Comptes Rendus de l'Académie des Sciences.
tome 293, séance du 5 octobre 1981, série 1, pp. 269-272
I will limit myself to presenting the sequence of drawings, without commenting on them.
Non-trivial turning of the torus. First part of the transformation
Non-trivial turning of the torus. Second part of the transformation
When one reaches the figure
v
one sees that it is then easy to align the gray structure with the pink structure, transforming this object into a two-sheeted covering of the Klein bottle.
The turning then occurs by swapping the facing sheets. Below, the same drawing with a chromatic coding.
Two-sheeted covering of the Klein bottle, with chromatic coding
(this drawing is not part of my annual report to the CNRS. It can be found in the Topologicon)
Different families of tori.
What Stephen Smale demonstrated in 1957 was that there was only one family of immersions of the sphere, and that all such immersions could be connected to one another by a homotopy. These formed a group whose neutral element consisted of leaving the object unchanged. One wondered whether the same might be true for the torus. Mathematicians Ioan James and Emery Thomas showed that the immersions of the torus were distributed among four continents between which it was impossible to pass using a regular homotopy.
The four families of tori
The "standard torus," drawn at the center of the page, belongs to the same family as the object shown in
b
. This is what I demonstrated en passant in the version of the torus turning I invented in 1980. The family referred to in a represents a torus that has undergone a 360° twist. It resembles the standard torus, but the two are defined differently through their cartographic system, using two families of curves. In the standard torus, we use two sets of circles analogous to meridians and parallels. On torus a, one should complete the family of circles glued onto it with a second family, twisting in the opposite direction. What can be shown is that it is impossible, using a regular homotopy, to bring the mesh of this torus a into coincidence with the mesh of the standard torus (meridional circles plus parallel circles). In this sense, they are different objects. Of course, all these objects can be configured as a two-sheeted covering of the Klein bottle.
The power of the geometer's tools lies in their ability to predict what is possible and what is not.
Turning the standard torus into the torus shown in
b
: yes.
Going from
c
to
d
: no.
This prevents wasting time foolishly and especially encourages searching for things that are far from obvious, such as turning a sphere inside out. This applies to all sciences. It happens that people overlook fruitful approaches for years, even centuries, simply because they believed them impossible to carry out. I devoted several years of my life to developing a theory for eliminating shock waves around an object moving at supersonic speeds through a gas, using a Laplace force field, via "MHD." An undergraduate even wrote a thesis on this topic under my supervision, and we published these works in various peer-reviewed journals and scientific conferences. This subject is only now beginning to emerge, thirty years later. There is speculation that the Americans possess hypersonic aircraft capable of flying at Mach 10 without generating shock waves (and in particular without enduring the tremendous thermal stresses caused by air recompression behind these "bangs"). This is the famous Aurora myth: an aircraft cruising at the altitude where the northern lights occur, between 80 and 150 km. Aurora also foreshadows future space launchers that, relying on air, will be far more economical than CNES rockets. In France, it was impossible to initiate such research (I had these ideas in 1975), because people—especially at the CNRS—considered them utterly unreasonable. The result is a thirty-year lag behind the United States, in my opinion entirely irreparable.
The Tobacco Pouch Joke
To be complete, one must mention versions of the sphere turning that center on a tobacco pouch. This was a common object when I was young, but one rarely encounters today. The first person to draw these sequences was Georges Francis. For several years now I have been working on a polyhedral version of these, which has already yielded a rather nice central model. But to show it to you, I'll need to get my hands on it again. Soon, I hope, since it is one of the most fascinating objects I have ever created.
Previous page
Next page