Consolidated document. 2 original pages have been brought together here; the links below open each section.
Find again: Scientific documents.
Contents
Click here to make the drawing of the 1:1 scale model appear, to print and cut out. By photocopying four copies on Bristol card of two different colours, you will be able to build the model yourself, following the instructions for assembling it
You have certainly seen a bizarre object rotating tirelessly on the left-hand part of the home page of this site. What is it?
One day, when I find the time, I will install on this site a description of the eversion of the sphere, just as I had illustrated it in the January 1979 issue of Pour la Science, that is... 22 years ago! All this will require many details and an introduction. What does "everting a sphere" mean? A sphere does not have the same meaning for the ordinary person and for the mathematician-geometer. For the ordinary person, it is nothing other than the locus of points in space situated at a distance R from a fixed point O. A geometer, however, will continue to call "sphere" even an object that corresponds to a "deformed sphere", such as a potato for example. To understand these concepts more precisely, get the Lanturlu CD containing the comic strip "Topologicon". But the mathematician goes a little further still. A surface is said to be "regular" when at each of its points a tangent plane can be defined. This already allows one to think of an infinity of possible regular deformations of the sphere, into the infinite possible shapes of a potato, while also varying the area of that surface in an arbitrary manner. That said, in our physical universe a person attempting to evert the sphere (that is, to bring its inner surface to the outside) will find themselves faced with the impossibility of managing to make its surface self-intersect. When this hypothesis is assumed, that is, when the surface is forbidden to self-intersect or even merely to "touch" itself, the mathematician speaks of an embedding of the sphere S2. But a mathematician always allows themselves everything. A sphere is, for them, a "virtual" and non-material object, in which the crossing of one sheet through another is considered possible. The sequence of drawings below shows a sphere that self-intersects. Such a representation, that is, one that allows self-intersections, is called an immersion.
An immersion therefore possesses a set of self-intersection (here it is a simple circular curve). The tangent plane, however, must vary continuously. That being so, when one looks at the drawing above, one clearly sees that the operation brings a part of the inner surface (represented in green) to the outside. To complete the eversion, one would have to flatten this kind of equatorial gut. Here there seems to be a problem: this flattening would destroy the continuity of the tangent plane, and such a transformation would therefore contain a step that is not an immersion.
One day an American mathematician, Stephen Smale, proved that "the sphere S2 possesses only one class of immersions". This sibylline sentence had as a corollary the fact that one should be able to pass, by means of a transformation containing only true immersions, from the "standard" sphere to its "antipodal" representation, that is, one in which every point is exchanged with its antipodal point: in plain words... an inverted sphere. Raoul Bott was Smale's boss. As much as the formal proof of this fact seemed correct, so much did no one seem capable of concretely carrying out this eversion operation. Bott kept asking Smale "show me how you would think of proceeding"; to which Smale, notoriously blunt, replied "I haven't the faintest idea". Smale later obtained the Field Medal, the equivalent of the Nobel for mathematics. Incidentally, you may perhaps wonder why there is no Nobel prize for mathematics. The answer is simple: his wife ran off with a mathematician.
Things stayed like that for quite a few years, until an American mathematician named Anthony Phillips published in 1967, in Scientific American, a first version of this reversal, extremely complicated. The second was invented in the early seventies by the French mathematician (blind) Bernard Morin. I was the first to draw the sequence of transformations, which will be the subject, as I announced to you, of a forthcoming article on this site, which is moreover quite abundant. Anyway, all this leads us to a consideration. Surfaces can be represented in polyhedral form. A cube or a tetrahedron can be considered polyhedral representations of the sphere, in the sense that these objects have the same topology. On this point, consult my Topologicon. Moreover, one understands that, if it is possible to turn the sphere inside out, it will be equally possible to turn a cube inside out. The transformation invented by Bernard Morin (which I illustrated in the January 1979 article in Pour la Science) passes through a central model. There is a symmetry in this sequence. It is the one I call the "central model with 4 ears". I am getting ahead of things. Anyway, just as the sphere lends itself to polyhedral representations, so do the successive steps of this transformation. What you see rotating on my home page is the polyhedral version of the central model of the sphere eversion, which I invented about ten years ago. The interest of such polyhedral models lies in the fact that they can be built with flat surfaces. They can also be built with paper and scissors. Take a look at the drawing below (I thank in parentheses my friend Christophe Tardy, who produced the elements of the right size).
It is an assembly plan of which you have an overview here. But to print it, it is preferable that you go to the découpage page. Print it. After that, equipped with this copy printed on the usual paper of your printer, photocopy 4 identical copies of it, two on green Bristol board, and two in yellow. You will be able, by means of these sheets to be cut out, to build the central model of the cube eversion.
On the elements to be cut out there are pairs of letters: a, b c, d, e, f etc.... It is enough to fold the sheet bringing the same letters to coincide, and then fix the faces with transparent adhesive tape. The drawings that follow show the way to assemble one of the four pieces. Here first of all is how one must begin to fold one of the four elements:
Here are two of the four elements, seen from different angles.
They are then arranged so as to give rise to an object with a symmetry of order four, in which green and yellow elements alternate. To see it in 3d, take a look at Tardy's realization, in the "virtual reality" section. The central model is assembled and also realized in "vrml" in this section. Here it is reproduced from various points of view:
One cannot say that one point of view corresponds to the "top" and the other to the "bottom", since these names are perfectly arbitrary. In the left image, the "central" point corresponds to the "double point" (where two sheets cross) of Morin's central model, while the central point of the right image corresponds to the "quadruple point" of the same model (where four sheets cross). I had to orient the object very carefully, so that the left figure does not evoke a swastika. Apart from that, from an architectural point of view, this polyhedral representation of Morin's central model could have constituted a fine project for a National Socialist House of Culture.
One last observation: there is no good polyhedral representation of the sphere eversion (or of the cube). By "good" I mean a sequence of sufficiently explicit models that can be described in the form of sheets to be cut out in a relatively easy way, like the model above. There would be a study to be done in this direction, within the reach of anyone, even a non-mathematician, for example a sculptor. More than twenty years ago I was a sculpture teacher at the Ecole des Beaux-Arts in Aix en Provence, at the time when my dear friend Jacques Boullier was director. It is in those premises that the first meridian representation of Boy's surface by ellipses was born, the key to the construction of the first implicit equation given by Apéry. I must say that already at the time I was surprised by the geometric imagination of art students, which often surpassed that of... geometers.
Counter installed on December 31, 2001. Number of connections:
Back to the News page Home page
Continuation
Click here to make the drawing of the model appear at 1:1 scale, to be printed and cut out. By photocopying four copies of it on Bristol board of two different colors, you will be able to build the model yourself, following the instructions for assembling it
You have certainly seen a bizarre object rotating tirelessly on the left part of the home page of this site. What is it?
One day, when I find the time, I will install on this site a description of the eversion of the sphere, just as I had illustrated it in the January 1979 issue of Pour la Science, that is... 22 years ago! All this will require many details and an introduction. What does "everting a sphere" mean? A sphere does not have the same meaning for the ordinary person and for the mathematician-geometer. For the ordinary person, it is nothing other than the locus of points in space located at a distance R from a fixed point O. A geometer, on the other hand, will continue to call "sphere" an object that corresponds to a "deformed sphere," such as a potato for example. To understand these concepts more precisely, get the Lanturlu CD containing the comic strip "Topologicon." But the mathematician goes a bit further still. A surface is said to be "regular" when at each of its points a tangent plane can be defined. This already allows one to think of an infinity of possible regular deformations of the sphere, into the infinite possible shapes of a potato, while also varying the area of such a surface in an arbitrary manner. That said, in our physical universe a person attempting to evert the sphere (that is, to bring its inner surface to the outside) will find themselves faced with the impossibility of making its surface self-intersect. When this hypothesis is assumed, that is, when the surface is forbidden to self-intersect or even merely to "touch" itself, the mathematician speaks of an embedding of the sphere S2. But a mathematician always allows himself everything. A sphere is, for him, a "virtual" and not a material object, in which the crossing of one sheet through another is considered possible. The sequence of drawings below shows a sphere that self-intersects. Such a representation, which is to say one that allows self-intersections, is called an immersion.
An immersion therefore possesses a set of self-intersection (here it is a simple circular curve). The tangent plane, however, must vary continuously. That said, when one looks at the drawing above, one can clearly see that the operation brings a part of the inner surface (represented in green) to the outside. To complete the eversion, one would have to flatten this kind of equatorial gut. Here there seems to be a problem: this flattening would destroy the continuity of the tangent plane, and such a transformation would therefore contain a step that is not an immersion.
One day an American mathematician, Stephen Smale, proved that "the sphere S2 possesses only one class of immersions." This sibylline sentence had as its corollary the fact that one should be able to pass, by means of a transformation containing only true immersions, from the "standard" sphere to its "antipodal" representation, that is, one in which each point is exchanged with its antipodal point: in plain words... an everted sphere. Raoul Bott was Smale's boss. As much as the formal proof of this fact seemed correct, so much did no one seem capable of concretely carrying out this eversion operation. Bott kept asking Smale "show me how you would go about it"; to which Smale, notoriously blunt, replied "I haven't the faintest idea." Smale later received the Fields Medal, the equivalent of the Nobel Prize for mathematics. Incidentally, you may wonder why there is no Nobel Prize for mathematics. The answer is simple: his wife ran off with a mathematician.
Things stayed like this for quite a while, until an American mathematician named Anthony Phillips published in 1967, in Scientific American, a first version of this eversion, extremely complicated. The second was invented in the early seventies by the French mathematician (blind) Bernard Morin. I was the first to draw the sequence of transformations, which will be the subject, as I announced to you, of a forthcoming article on this site, which is moreover quite abundant. Anyway, all this leads us to a consideration. Surfaces can be represented in polyhedral form. A cube or a tetrahedron can be considered polyhedral representations of the sphere, in the sense that these objects have the same topology. On this point, consult my Topologicon. Moreover, one understands that, if it is possible to evert the sphere, it will be equally possible to evert a cube. The transformation invented by Bernard Morin (which I illustrated in the January 1979 article in Pour la Science) passes through a central model. There is a symmetry in this sequence. It is what I call the "central model with 4 ears." I am getting ahead of things. Anyway, just as the sphere lends itself to polyhedral representations, so too do the successive steps of this transformation. What you see rotating on my home page is the polyhedral version of the central model of the eversion of the sphere, which I invented about ten years ago. The interest of such polyhedral models lies in the fact that they can be built with flat surfaces. They can also be built with paper and scissors. Take a look at the drawing below (I thank in parentheses my friend Christophe Tardy, who produced the elements of the right size).
It is an assembly plan of which you have an overview here. But to print it, you would do better to go to the découpage page. Print it. Then, equipped with this specimen printed on the usual paper of your printer, photocopy 4 identical copies of it, two on green Bristol board, and two in yellow. By means of these sheets to be cut out, you will be able to build the central model of the eversion of the cube.
On the elements to be cut out there are pairs of letters: a, b c, d, e, f etc.... It is enough to fold the sheet bringing the same letters to coincide, and then fix the faces with transparent adhesive tape. The drawings that follow show the way to assemble one of the four pieces. Here first of all is how one must begin to fold one of the four elements:
Here are two of the four elements, seen from different angles.
They are then arranged so as to give rise to an object with a symmetry of order four, in which green and yellow elements alternate. To see it in 3d, take a look at Tardy's realization, in the "virtual reality" section. The central model is assembled and also realized in "vrml" in this section. Here it is reproduced from various points of view:
One cannot say that one point of view corresponds to the "top" and the other to the "bottom", since these denominations are perfectly arbitrary. In the left image, the "central" point corresponds to the "double point" (where two sheets cross) of Morin's central model, while the central point of the right image corresponds to the "quadruple point" of the same model (where four sheets cross). I had to orient the object very carefully, so that the left figure does not evoke a swastika. Apart from that, from the architectural point of view, this polyhedral representation of Morin's central model could have constituted a fine project for a House of National Socialist Culture.
One last observation: there is no good polyhedral representation of the eversion of the sphere (that is, of the cube). By "good" I mean a succession of models explicit enough to be described in the form of cut-out sheets in a relatively easy way, like the model above. A study would have to be made in this direction, within the reach of anyone, even a non-mathematician, for example a sculptor. More than twenty years ago I was a sculpture teacher at the Ecole des Beaux-Arts in Aix en Provence, at the time when my dear friend Jacques Boullier was director. It is in those premises that the first meridian representation of Boy's surface by means of ellipses was born, the key to the construction of the first implicit equation given by Apéry. I must say that already at that time I was surprised by the geometric imagination of the art students, which often surpassed that of... geometers.
Counter installed on December 31, 2001. Number of connections:
Back to the News page Home page
Original pages: modele_central_polyedrique_it.htm, modele_central_polyedrique_it.htm