The Central Model of the Cube Turnover
December 31, 2001
You have all seen, tirelessly turning, a strange object on the left side of the website's homepage. What is it?
One day, when I have time, I will post on the site a description of the sphere eversion, as I illustrated it in the January 1979 issue of Pour la Science, that is to say, ... 22 years ago. This would obviously require considerable detail and introduction. What does it mean to evert a sphere? A sphere does not have the same meaning for the layperson and the mathematician-geometer. For the layperson, it is defined, in three-dimensional space, as the locus of points located at a distance R from a fixed point O within that space. A geometer will continue to call "sphere" an object that corresponds to a "deformed sphere," a sort of "potato." To grasp all these concepts more precisely, obtain the CD Lanturlu, which contains the comic strip "Le Topologicon." But the mathematician goes even further. When a surface is said to be "regular," one can define a tangent plane at each of its points. This already allows one to consider an infinite number of deformations of the "starting sphere" into an infinite number of potatoes, especially when the area of said surface can be arbitrary. That being said, in a "physical universe," the person deforming this sphere would encounter the impossibility of making it pass through itself. If such self-intersections or even contacts are forbidden, we speak of embeddings of the sphere S2. But a mathematician grants himself all rights. For him, a sphere is a "virtual" object where sheet crossings become possible. The sequence of drawings below shows a sphere that has "self-intersected." This representation of the sphere is then called an immersion.
An immersion possesses a set of self-intersections (here, a simple circular curve). The tangent plane must vary continuously. That being said, when looking at the drawings above, one sees that the operation indeed turns a part (represented by the green color) of the inside of the sphere to the outside. To complete such an eversion, one would need to crush this sort of equatorial sausage. This seems problematic a priori. This crushing 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 possesses only one class of immersion." The corollary of this enigmatic statement was that one should be able to chain a sequence of immersions of the sphere allowing one to pass from the "standard sphere" to its "antipodal" representation, that is to say, one where all points have been replaced by their antipode. In short ... an everted sphere, inside out. Raoul Bott was Smale's mentor. As flawless as the latter's purely formal proof seemed, no one saw how to carry out the operation. Bott constantly said to Smale, "Show me how you would proceed," to which Smale, with his famous lisp, replied, "I haven't the faintest idea." Smale later received the Fields Medal, equivalent to the Nobel Prize, but for mathematics. Incidentally, you might wonder why Nobel never wanted to create a Nobel Prize for mathematics. The answer is simple: his wife left him for a mathematician.
Things remained as they were for quite a few years until an American mathematician named Anthony Phillips published a first version of this eversion in Scientific American in 1967, which was atrociously complicated. The second was invented in the early 1970s by the French (blind) mathematician Bernard Morin. I was the first to draw this sequence of transformations, which will, as I have already said, be the subject of a forthcoming paper on the site, quite substantial, moreover. Nevertheless, this leads us to an ancillary conclusion. Surfaces can be the subject of polyhedral representations. A cube or a tetrahedron can be considered as polyhedral representations of the sphere, insofar as these objects have the same topology. On this point, consult my comic strip Le Topologicon. Furthermore, one will understand that if it is possible to evert a sphere, it is also possible to evert a cube. The transformation invented by Bernard Morin (which I illustrated in the January 1979 article of Pour la Science) passes through a central model. There is a symmetry in this sequence. This is what is called "the four-eared central model." Here again, I am anticipating. But just as the sphere can lend itself to polyhedral representations, so too can the successive stages of these transformations. The object you see turning on my homepage is thus the polyhedral version of the central model of the sphere eversion, a model I invented about ten years ago. The interest of these polyhedral models is that they can be constructed with flat surfaces. They can even be arranged according to cutouts. Take a look at the drawing below (I thank my friend Christophe Tardy, who produced the correctly dimensioned elements).
This is a drawing that would print on your printer in a small, unusable format.
To print this figure on an A4 sheet
You then need to make four photocopies on heavy A4 paper, two sheets of one color, two sheets of another.
This is a cutout of which you have here a general view. But to print it, it is preferable that you go to the cutout page. Print it. Then, equipped with this copy printed on the normal paper of your printer, go to a photocopier and make four identical copies of this drawing, two on two sheets of green cardstock and two on yellow ones. You will be able, using this cutout, to build the central model of the cube eversion.
On these cut-out elements, you have pairs of letters: a, b, c, d, e, f, etc.... You simply need to perform the folds by bringing the same letters into coincidence, then assembling these facets with transparent adhesive tape. The following drawings show how to assemble one of the four elements. Here is first how to begin folding one of the four elements:
Here are two of these four elements, seen from different angles.
These are then arranged to give an object having a symmetry of order four, or alternating green elements and yellow elements. To see this in 3D, go take a look at Mr. Tardy's creations in "virtual reality." The fully assembled central model is also produced in "vrml" in this section. Here is this object, seen from different angles:
One cannot say that one view corresponds to the "top" and the other to the "bottom" since these designations would be totally arbitrary. In the left view, the "central" point corresponds to the "double point" (where two sheets cross) of Morin's central model, while the central point on the right is equivalent to the "quadruple point" of this same model (where four sheets cross). I had to carefully orient the object so that the left figure did not evoke a swastika. Otherwise, architecturally speaking, this polyhedral representation of Morin's central model could have constituted a very good project for a National Socialist House of Culture.
Last view:
A final remark: there is no good polyhedral representation of the sphere eversion (aka the cube eversion). By "good" one must understand a sequence of sufficiently explicit models that can be assembled in the form of cutouts relatively easily, like the model above. Research would need to be done in this direction, which is within everyone's reach, possibly a non-mathematician, an artist, for example. More than twenty years ago, I was a sculpture teacher at the School of Fine Arts in Aix-en-Provence, at a time when it was still directed by my excellent friend Jacques Boullier. It was in these premises that the first meridional representation of Boy's surface using ellipses was born, 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 art students, which very often surpassed that... of geometers.
Counter initialized on December 31, 2001. Number of connections: