The Central Model of the Sphere's Turnover
December 31, 2001
You have all seen, endlessly, a strange object rotating on the left side of the homepage of the site. What is it?
One day, when I have time, I will install on the site a description of the sphere's turnover, as I had illustrated in the January 1979 issue of Pour la science, that is, 22 years ago. Obviously, this would require many details and an introduction. What does it mean to turn a sphere? A sphere does not have the same meaning for the average person and the mathematician-geometer. For the average person, it is defined as, in a three-dimensional space, the set of points at a distance R from a fixed point O. A geometer will continue to call "sphere" an object that would correspond to a "deformed sphere," a sort of "potato." To better understand all these concepts, get the Lanturlu CD containing the comic strip "Le Topologicon." But the mathematician goes even further. When a surface is said to be "regular," a tangent plane can be defined at each of its points. This already allows considering an infinite number of deformations of the "original sphere" into an infinite number of "potatoes," when, additionally, the area of the said surface can be arbitrary. However, in a "physical universe," the person deforming this sphere would face the impossibility of making it pass through itself. If these crossings or even these contacts are forbidden, one then speaks of
sphere S2 embeddings
. But a mathematician has all rights. A sphere is, for him, a "virtual" object where the crossings of surfaces become possible. The following drawings show a sphere that has "self-crossed." This representation of the sphere is then called an
immersion
.
An immersion has an
set of self-intersections
or
self-intersections
(here a simple circular curve). The tangent plane must vary continuously. However, when looking at the above drawings, one can see that the operation turns a part (represented in green) of the sphere's interior to the outside. To complete such a turnover, one would have to flatten this kind of equatorial tube. This seems at first problematic. This flattening would break the continuity of the tangent plane. The operation would therefore include a step that would not be an immersion.
One day, an American mathematician, Stephen Smale, proved that "the sphere S2 had only one class of immersion." The corollary of this cryptic sentence was that one should be able to chain a sequence of sphere immersions allowing to go from the "standard sphere" to its "antipodal" representation, that is, one where all the points had been replaced by their antipode. In short, a turned sphere, front and back. Raoul Bott was Smale's advisor. While Smale's purely formal proof seemed flawless, no one could see how to perform the operation. Bott constantly told Smale, "Show me how you would proceed," to which Smale, with his famous tongue-in-cheek, replied, "I have no idea." Smale later received the Fields Medal, equivalent to the Nobel Prize, but for mathematics. By the way, you may wonder why Nobel never wanted to create a Nobel Prize for mathematics. The answer is simple: his wife left him with a mathematician.
Things remained as they were for many years until an American mathematician named Anthony Phillips published in Scientific American in 1967 a first version of this turnover, extremely complicated. The second was invented in the early 1970s by the French mathematician (blind) Bernard Morin. I was the first to draw this sequence of transformations, which, as I have already said, will be the subject of a forthcoming article on the site, rather substantial, by the way. In any case, this leads us to an additional conclusion. Surfaces can be represented as
polyhedral representations. A cube or a tetrahedron can be considered as polyhedral representations of the sphere, insofar as these objects have the same
topology
. For this point, consult my comic strip "Le Topologicon." Moreover, it becomes clear that if it is possible to turn a sphere, it is also possible to turn a cube. The transformation invented by Bernard Morin (which I illustrated in the January 1979 issue of Pour la science) goes through a
central model
. There is a symmetry in this sequence. This is called "the four-ear central model." Again, I anticipate. But just as the sphere can be represented as a polyhedron, so can the successive steps of these transformations. The object you see rotating on my homepage is thus the polyhedral version of the central model of the sphere's turnover, a model I invented about ten years ago. The interest of these polyhedral models is that they can be constructed with flat surfaces. You can even arrange them according to cutouts. Take a look at the drawing below (I would like to thank my friend Christophe Tardy, who produced the correctly dimensioned elements).
This is a drawing that would come out of your printer in small format, not useful.
To print this figure on an A4 sheet
you need to make four copies on thick A4 paper, two sheets of one color, two sheets of another
This is a general view of the cutout. But for printing, it is preferable that you go to the
cutout
page. Print it. Then, with this printed copy on the normal paper of your printer, go to a photocopier and make four identical copies of this drawing,
two on green cardboard sheets
and two on yellow ones. You will then be able to construct the central model of the cube's turnover using this cutout.
On these cut-out elements, you have pairs of letters: a, b, c, d, e, f, etc. You just need to fold them so that the same letters coincide, then assemble these facets with transparent tape. The following drawings show how to assemble one of the four elements. Here is first how to start folding one of the four elements:
Here are two of these four elements, seen from different angles.
These then fit together to form an object with four-fold symmetry or alternating green and yellow elements. To see this in 3D, take a look at Tardy's "virtual reality" realizations. The fully assembled central model is also produced in "vrml" in this section. Here is this object, viewed from different angles:
It cannot be said that one view corresponds to the "top" and the other to the "bottom," since these denominations would be entirely arbitrary. On the left view, the "central" point corresponds to the "double point" (where two layers cross) of Morin's central model, while the central point on the right corresponds to the "quadruple point" of the same model (where four layers cross). I had to carefully orient the object so that the left figure does not evoke a swastika. Otherwise, architecturally speaking, this polyhedral representation of Morin's central model could have been a very good project for a National Socialist Cultural House.
Final view:
Final remark: there is no good polyhedral representation of the sphere's turnover (also known as the cube's turnover). By "good," we mean a sequence of models sufficiently explicit that can be assembled as cutouts relatively easily, like the model above. A research would be needed in this sense, which is within the reach of anyone, possibly even a non-mathematician, a sculptor, for example. More than twenty years ago, I was a sculpture teacher at the École des Beaux-Arts in Aix-en-Provence, when it was still directed by my excellent friend Jacques Boullier. It was in these premises that the first southern representation of the Boy surface was created using ellipses, the key to the construction of the first implicit equation by Apéry. I must say that at the time, I was always surprised by the geometric imagination of the art students, which often surpassed that of... the geometers.
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