Central (polyhedral) model of the sphere inversion
The Central Model of the Sphere Inversion
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 about?
One day, when I have time, I will install on the site a description of the sphere inversion, as I had illustrated it in the January 1979 issue of Pour la Science, that is ... 22 years ago. This would obviously require many details and an introduction. What does it mean to invert 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 located at a distance R from a fixed point O of that space. A geometer will continue to call "sphere" an object that would correspond to a "deformed sphere", a sort of "potato". To better grasp 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", one can define a tangent plane at each of its points. This already allows to imagine an infinite number of deformations of the "original sphere" into an infinite number of "potatoes", when 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 such crossings or even contacts are forbidden, one then speaks of "embeddings" of the sphere S2. But a mathematician has all the 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 a set of self-intersections (here a simple circular curve). The tangent plane must vary continuously. However, when you look at the drawings above, you can see that the operation turns part (represented in green) of the inside of the sphere to the outside. To complete such an inversion, one would have to flatten this sort of equatorial tube. This seems at first glance problematic. This flattening would break the continuity of the tangent plane. The operation would therefore include a step that is not 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 immersions of the sphere allowing to go from the "standard sphere" to its "antipodal" representation, that is to say, a sphere where all the points have been replaced by their antipodes. In short, a sphere turned inside out. Raoul Bott was Smale's advisor. While Smale's proof, purely formal, seemed impeccable, no one could see how to carry out the operation. Bott constantly told Smale, "show me how you would proceed", to which Smale, with his famous tongue-in-cheek response, answered, "I have no idea". Smale later received the Fields Medal, equivalent to the Nobel Prize, but for mathematics. By the way, you might 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 in this state for many years until an American mathematician named Anthony Phillips published in 1967 in Scientific American a first version of this inversion, 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, quite extensive, by the way. In any case, this leads us to an additional conclusion. Surfaces can be represented in polyhedral form. A cube or a tetrahedron can be considered as polyhedral representations of the sphere, inasmuch 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 invert a sphere, it is also possible to invert 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 central model with four ears". Again, I am anticipating. But just as a sphere can be represented in polyhedral form, 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 inversion, a model I invented about ten years ago. The interest of these polyhedral models is that they can be built with flat surfaces. They can even be arranged 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 on your printer in small format, not usable. **
To print this figure on
an A4 sheet
You then 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 better 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 two green cardboard sheets and two on yellow ones. You will then be able to construct the central model of the cube inversion 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 by bringing the same letters into coincidence, then assemble these facets with transparent 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, viewed 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 available 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 completely arbitrary. On the left view, the "central" point corresponds to the "double point" (where...)