Consolidated document. 9 original pages have been gathered here; the links below open each section.
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Contents
- How to transform a crosscap into a Boy surface (right or left, your choice) by way of the Steiner Roman surface.
- How to transform a crosscap into a Boy surface (right or left, your choice) by way of the Steiner Roman surface.
- How to transform a crosscap into a Boy surface (right or left, your choice) by way of the Steiner Roman surface.
- How to transform a crosscap into a Boy surface (right or left, your choice) by way of the Steiner Roman surface.
- How to transform a crosscap into a Boy surface (right or left, your choice) by way of the Steiner Roman surface.
- How to transform a crosscap into a Boy surface (right or left, your choice) by way of the Steiner Roman surface.
- How to transform a crosscap into a Boy surface (right or left, your choice) by way of the Steiner Roman surface.
- How to transform a crosscap into a Boy surface (right or left, your choice) by way of the Steiner Roman surface.
- How to transform a crosscap into a Boy surface (right or left, your choice) by way of the Steiner Roman surface.
How to transform a crosscap into a Boy surface (right or left, your choice) by way of the Steiner Roman surface.
CrossCap_Boy/italiano/Crosscap_Boy2_it.htm
September 27 - October 25, 2003
**Page 1 **Page 2 **Page 3 **Page 4 **
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How to transform a crosscap into a Boy surface (right or left, your choice) by way of the Steiner Roman surface.
September 27 - October 25, 2003
page 1
All this, as Kipling would say, is "great cunning and mighty magic".
I am retired but, I might say, I still do a little research, in spite of myself, the way others line up rows of knitting. If you have patience and get hold of sheets of 200 g squared Bristol board you will easily be able to reconstruct all these models. My friend Christophe Tardy is in the process of making an animation from this, which should be quite good.
The Cross cap will appear in the drawings that follow, as will the Steiner Roman surface. But you can discover them by going to the Virtual Reality section, which requires that you have downloaded Cosmoplayer onto your machine. Do it, it's really nice. Everything happens thanks to the "cuspidal points". These points form quite naturally when you mount a horse and squeeze your legs sharply. The horse's body is then crushed along a segment. Its left thigh then connects with its right shoulder, while its right thigh connects with its left shoulder. As for the cuspidal point, don't look for it: you are simply sitting on it;
But all that is ... rather plump. Let us move on to a "polyhedral representation" of the cuspidal point (just as a cube or a tetrahedron can be regarded as polyhedral representations of a simple sphere). The bold line represents the "self-intersection curve", which moreover ends at the cuspidal pointC.
Print these plates, that will be better. In what follows you will need to recognize a cuspidal point in "different configurations" and not confuse it with a simple vertex of the polyhedron. Build these different objects out of cardboard if you have the courage, you will understand them better. Below we have an essential operation, called "creation-annihilation of a pair of cuspidal points". The first drawing represents a sort of cylinder that intersects itself along the bold line and whose cross-section evokes the Greek letter gamma, upside down. This surface is then deformed by pinching the tube whose cross-section has the shape of an "upside-down teardrop". This teardrop degenerates to a pointS. Then this point splits in two, giving two cuspidal points. This is the creation of a pair. The reverse operation causes two cuspidal points to annihilate each other. Below you will find the polyhedral version of the operation.
Below is another polyhedral representation of the transformation which is close to what you will see happen in the surface later on.
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How to transform a crosscap into a Boy surface (right or left, your choice) by way of the Steiner Roman surface.
September 27 - October 25, 2003
page 1
All this, as Kipling would say, is "great cunning and mighty magic".
I am retired but, I might say, I still do a little research, in spite of myself, the way others line up rows of knitting. If you have patience and get hold of sheets of 200 g squared Bristol board you will easily be able to reconstruct all these models. My friend Christophe Tardy is in the process of making an animation from this, which should be quite good.
The Cross cap will appear in the drawings that follow, as will the Roman Steiner surface. But you can discover them by going to the Virtual Reality section, which requires that you have downloaded Cosmoplayer onto your machine. Do it, it's really nice. Everything happens thanks to the "cuspidal points". These points form quite naturally when you mount a horse and squeeze your legs sharply. The horse's body is then crushed along a segment. Its left thigh then connects with its right shoulder, while its right thigh connects with its left shoulder. As for the cuspidal point, don't look for it: you are simply sitting on it;
But all this is ... chubby. Let's move on to a "polyhedral representation" of the cuspidal point (just as a cube or a tetrahedron can be considered polyhedral representations of a simple sphere). The bold line represents the "self-intersection curve", ending moreover at the cuspidal pointC.
Print these plates, it will be better. In what follows you will have to recognize a cuspidal point in "different configurations" and not confuse it with a simple vertex of the polyhedron. Build these different objects out of cardboard if you have the courage, you will understand them better. Below we have an essential operation, called "creation-annihilation of a pair of cuspidal points". The first drawing represents a sort of cylinder that intersects itself along the bold line and whose cross-section evokes the Greek letter gamma, upside down. This surface is then deformed by pinching the tube whose cross-section has the shape of an "upside-down tear". This tear degenerates into a pointS. Then this point splits in two, giving two cuspidal points. This is the creation of a pair. The reverse operation causes two cuspidal points to annihilate each other. Below you will find the polyhedral version of the operation.
Below is another polyhedral representation of the transformation which is close to what you will see happen in the surface afterwards.
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September 27 - October 25, 2003
page 2
Here is a Cross Cap (as you will have discovered it in the virtual reality images). It has two cuspidal points that border a self-intersection line. It can be made by pinching a balloon with a curling iron. But you can also build polyhedral representations of it. The one at the bottom will interest us particularly.
In this plate 4 lies the most difficult moment to grasp. It seems to me more or less impossible for the first comer to understand these figures by simply looking at the drawings. Build these models. Clearly, the cuspidal pointC2 is pulled toward "the inside of the surface" (which incidentally makes no sense since, you will doubtless have immediately noticed, the Cross Cap is one-sided. By insisting, the surface passes through itself and the self-intersection set is completed, in "chubby" fashion, by a figure-eight-shaped curve. In the process a triple pointT is created.
The surface is more comprehensible in its polyhedral form and, at the bottom, we have enlarged certain elements to show what prompts us to transform this object into a Roman Steinder surface (see the virtual reality) whose simplest polyhedral form consists in assembling four cubes (here only three are visible).
Plate 5: The polyhedral on the left, the chubby on the right. The arrow takes a passage that will be "pinched". At the bottom the beginning of the pinching.
Plate 6: the pinching is carried out by creating a singular pointB. In fact, since we pinch on both sides, to save time, two singular pointsS1 andS1 form and then two pairs of cuspidal points. There, without cardboard, scissors and adhesive tape, you are in trouble.
Plate 7: we have simply made the different cuspidal points migrate. If the pointC2 is "obvious" you will have a little more difficulty identifying the pointsC3 andC4 as cuspidal points. They are nevertheless present at the end of a self-intersection line. Above the pointC3 is simply what I have called a "posicorner", a point of concentration of positive curvature (a point of concentration of negative curvature is a "negacorner"). By deforming this object a tiny bit one arrives at a polyhedral form of the Roman Steiner surface (a surface of the 4th degree invented by Steiner in Rome. See its presentation in virtual reality).
So, the trick is done. There are different types of surfaces, depending on the rules one sets for oneself. Surfaces that do not intersect themselves are called embeddings (of the sphere, of the torus in R3). When they intersect themselves but the tangent plane varies continuously, they are called immersions. Example: the Klein bottle in its classical representation. There exists no representation of the Klein bottle in R3 in the form of an embedding. It necessarily intersects itself. Immersions possess self-intersection sets free of cuspidal points. These curves are continuous but can cross at points where there are double or triple points. Remark: the sphere can present itself in the form of an immersion, simply by making it intersect itself. That is moreover how one manages to turn it inside out (A. Phillips, 1967, with as its central stage the two-sheeted covering of a Boy surface; B. Morin and J. P. Petit, 1979 with as their central model Morin's four-eared model, of which here follows a polyhedral representation that I invented about ten years ago.
Plan for assembling this object using a cut-out
If one extends the rule of the game by supposing that these objects possess cuspidal points, one obtains submersions (the Cross Cap, Steiner's Roman Surface). I don't know if that's the exact word, but since I haven't found any mathematician who could enlighten me, I found it amusing to invent one, provisionally, until a chartered surveyor comes forward. Thus the Cross Cap and Steiner's Roman Surface would be submersions of the "projective plane".
To tell you everything, after my setbacks in the matter of MHD over twenty-five years I had begun this work because it seemed to me as far removed as possible from any military application. But, as my old friend Mihn pointed out, the term submersion could lend itself to confusion and let it be implied to the National Navy that through this research I was trying to hide some breakthrough in the matter of submarine propulsion.
The rule of "creation-decreation" of pairs of cuspidal points allows one to pass from one submersion of an object to another and that is what we have just done by showing that the Cross Cap and Steiner's Roman Surface were two submersions of a same object called the projective plane. Do not try to find out what a "projective plane" looks like. This object can only be grasped through its different representations. As for the words projective plane, they are only one among a thousand others invented by mathematicians to mislead those who would like to penetrate their closed circle. The Larousse will be of no use to you in mathematics.
It then remains for us to move on to the Boy surface, which is an immersion of the projective plane
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page 2
Here is a Cross Cap (as you will have discovered it in the virtual reality images). It has two cuspidal points bordering a line of self-intersection. One can make it by pinching a balloon with a curling iron. But you can also construct polyhedral representations of it. The one at the bottom will interest us particularly.
In this plate 4 is located the most difficult moment to grasp. It seems to me more or less impossible that the first comer would understand these figures by simply looking at the drawings. Build these models. In plain terms one pulls the cuspidal point C2 towards the "interior of the surface" (which incidentally has no meaning since, you will doubtless have noticed immediately, the Cross Cap is one-sided. By insisting, the surface passes through itself and the self-intersection set completes itself, in a "plump" way, with a figure-eight-shaped curve. In the process a triple point T is created.
The surface is more comprehensible in its polyhedral form and, at the bottom, we have enlarged certain elements to show what induces us to transform this object into a Steiner Roman Surface (see the virtual reality) whose simplest polyhedral form consists in assembling four cubes (here only three are visible).
Plate 5: The polyhedral on the left, the plump one on the right. The arrow takes a passage that will be "pinched". At the bottom the beginning of the pinching.
Plate 6: the pinching is carried out by creating a singular point B. In fact since one pinches on both sides, to save time, two singular points S1 and S1 are formed, then two pairs of cuspidal points. There, without cardboard, scissors and adhesive tape, you are in trouble.
Plate 7: one has simply made the different cuspidal points migrate. If the point C2 is "obvious" you will have a little more difficulty identifying the points C3 and C4 as cuspidal points. They are nevertheless present at the end of a line of self-intersection. Above the point C3 is simply what I have called a "posicoin", a point of concentration of positive curvature (a point of concentration of negative curvature is a "negacoin"). By deforming this object ever so slightly one falls upon a polyhedral form of Steiner's Roman surface (a surface of the 4th degree invented by Steiner in Rome. See its presentation in virtual reality).
So, the trick is done. There are different types of surfaces, depending on the rules one sets for oneself. Surfaces that do not intersect themselves are called embeddings (of the sphere, of the torus in R3). When they intersect themselves but the tangent plane varies continuously, they are called immersions. Example: the Klein bottle in its classical representation. There is no representation of the Klein bottle in R3 in the form of an embedding. It necessarily intersects itself. Immersions possess self-intersection sets free of cuspidal points. These curves are continuous but can cross where there are double or triple points. Remark: the sphere can present itself in the form of an immersion, simply by making it intersect itself. That is moreover how one manages to turn it inside out (A. Phillips, 1967, with as its central stage the two-sheeted covering of a Boy surface; B. Morin and J. P. Petit, 1979 with as their central model Morin's four-eared model, of which here follows a polyhedral representation that I invented about ten years ago.
Plan for assembling this object by means of cutting out
If one extends the rule of the game by supposing that these objects possess cuspidal points, one obtains submersions (the Cross Cap, the Steiner Roman Surface). I do not know if that is the exact word, but as I have found no mathematician who could enlighten me, I found it amusing to invent one, provisionally, until some expert geometer makes himself known. Thus the Cross Cap and the Steiner Roman Surface would be submersions of the "projective plane".
To tell you everything, after my setbacks in the matter of MHD over twenty-five years, I had begun this work because it seemed to me as far removed as possible from any military application. But, as my old friend Mihn pointed out, the term submersion could give rise to confusion and let it be implied to the Navy that through this research I was trying to hide some breakthrough in the matter of submarine propulsion.
The rule of "creation-decreation" of pairs of cuspidal points makes it possible to pass from one submersion of an object to another, and that is what we have just done by showing that the Cross Cap and the Steiner Roman Surface were two submersions of one and the same object called the projective plane. Do not try to find out what a "projective plane" looks like. This object can only be grasped through its different representations. As for the word projective plane, it is only one among a thousand others invented by mathematicians to mislead those who would like to enter their closed circle. The Larousse will be of no use to you in mathematics.
It then remains for us to move on to the Boy surface, which is an immersion of the projective plane
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page 3
Plate 8: One begins by making two cuspidal points migrate (C2 and C4) which approach the triple point T. Dashed lines have then been laid down on portions of the surface which we are going to "punch out from the inside" with a "pyramidal punch" (for pity's sake, build these models or you are headed for the psychiatric hospital. As they develop, the tips of these pyramids are none other than the cuspidal points C2 and C4 which "migrate" and go to meet each other.
Plate 9: the cuspidal points meet at S and "annihilate" each other. The self-intersection curve loses two cuspidal points and gains .. a loop (in polyhedral terms: a closed polygonal contour).
Plate 10 this "square-section tube" is formed.
Plate 11: one turns this object to see it from another angle and makes two new cuspidal points migrate, then one punches out the dashed parts "from the inside" (which is silly since a Steiner Roman surface, a surface of the 4th degree) is one-sided). One continues this migration-confluence of this second pair of cuspidal points.
In the last image the points are close to meeting. Plate 12: the passage is open. There remain only two cuspidal points.
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Plate 8: One begins by making two cuspidal points migrate (C2 and C4) which approach the triple point T. Dashed lines have then been laid down on portions of the surface which we are going to "punch out from the inside" with a "pyramidal punch" (for pity's sake, build these models or you are headed for the psychiatric hospital. As they develop, the tips of these pyramids are none other than the cuspidal points C2 and C4 which "migrate" and go to meet each other.
Plate 9: the cuspidal points join together in anSand "annihilate" each other. The self-intersection curve loses two cuspidal points and gains .. a loop (in polyhedral form: a closed polygonal contour).
Plate 10 this "square-section tube" is formed.
Plate 11: we turn this object to see it from another angle and we migrate two new cuspidal points, then we punch out the dotted parts "from the inside" (which is silly since a Roman Steiner surface, a surface of the 4th degree) is one-sided). We continue this migration-confluence of this second pair of cuspidal points.
In the last image the points are close to joining. Plate 12: the passage is open. Only two cuspidal points remain.
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page 4
The model is then presented from another angle:
Plate 14: The same operation is repeated by creating the third "ear" of the self-intersection curve. In polyhedral form this has the shape of three squares having a common vertex: the triple pointT.
Plate 15: by rotating the object you find again the polyhedral version of the Boy surface that I had invented and presented in the Topologicon (where there is a cutting pattern allowing it to be built).
Last plate: I have tried to depict the Steiner surface (of the 4th degree, whereas the Boy is of the sixth) contorting itself and transforming into a Boy surface.
One sees that in "round" form it takes a hell of a lot of practice to understand the object. Our eye is very uncomfortable when it comes to understanding an object where, along a single line of sight, more than two sheets are superimposed. Hence the interest of the polyhedral form, which brings within everyone's reach transformations considered sophisticated in geometry, insofar as people make the effort to build the models themselves. In passing one notices that depending on the pairs of cuspidal points chosen one obtains a "right" or "left" Boy surface (totally arbitrary words). The projective plane immerses itself according to two "enantiomorphic" representations, in mirror image. One sees that one can pass from a right Boy to a left Boy by way of a "central" model which is the Roman Steiner surface.
It would doubtless be nice for such drawings to be published in Pour la Science or La Recherche. But for twenty years I have been "banned from publication" in these journals on account of UFOlogical deviationism. Thank you Messrs Hervé This and Philippe Boulanger. I have lost count of the articles of this kind that I have sent to these journals and that have been politely returned to me. One ends up getting used to one's status as an excommunicate.
As an anecdote, there exists in France a "prix d'Alembert" intended to reward authors of popular mathematics books. The story was told to me by a member of the commission responsible for determining to whom the prize should go (there is still some money in it, after all). Dialogue:
-
But come now, couldn't we award the prize to Petit? He has produced remarkable works like the Géométricon, the Trou Noir and the Topologicon.
-
Yes, but he hasn't done only those albums.
-
What are you alluding to?
-
He also wrote le Mur du Silence.
-
Ah, in that case....
Indeed, le Mur du Silence, published in 83, is an album devoted to MHD. And, as everyone knows, this sulphurous science has the virtue, or the mischief, of allowing flying saucers to travel at supersonic speed without making a Bang.
Hide this science, that I cannot bear to see
I have in my drawers a superb version of the "turning of the cube" with a central model of great beauty, which is not the polyhedral version of Morin's variant. All of my own making. One of these days.....
October 22, 2003: People are not exactly rushing to these pages, if I believe the counter reading. On Monday October 13, 2003 I gave a seminar at the CMI (Centre de mathématiques et d'informatique de Château-Gombert-Marseille) on Trotman's invitation. On that occasion I was able to line up a collection of about thirty cardboard models, of which you will have the first look one of these days, these having been photographed by Christophe Tardy.
When one gives a seminar, an atmosphere emerges. In the following photo, a geometer expressing his perplexity.
In the background, part of the models on display. At one point I asked the question:
*- Which of you have already seen a Roman Steiner surface? Raise your hand. *
No one had ever seen one. I therefore judged it useful to present the object, in virtual reality, on the laptop I had brought, an object made with the assistance of Christophe Tardy, engineer, and of Frédéric Descamp, of the Laue Langevin Institute of Grenoble (ILL). Visibly this presentation disconcerts the audience, little accustomed to seeing mathematical surfaces whirl about at will.
Two cardboard panels, visible in the foreground, had made it possible to present the rest of the models in their logical order. The "green and yellow" models illustrate, in polyhedral form, the essential tool of creation-destruction of a pair of cuspidal points. The most distant white object is a polyhedral version of the Cross Cap, which first transforms into a polyhedral version of the Roman Steiner surface, a meter further away, then, at will, into a "right" or "left" Boy surface.
The analysis of the models brings out various remarks from the audience. One of the geometers asks:
*- If, by following the models in this direction, one can go from the Cross Cap to the Boy surface, it seems that by doing the reverse one must be able to transform a Boy surface into a Cross Cap. *
I answer in the affirmative. Emboldened, my interlocutor adds:
*- If, on arriving at the stage of the Roman Steiner surface one stops, it then becomes possible to head back toward a mirror Boy surface. *
I approve a second time. But alas no one will volunteer to give clarifications on this strange world where immersions of closed surfaces are endowed with cuspidal points, created or annihilated in pairs, the whole constituting a sort of extension of the world of immersions. The word "submersions" would seem fitting to me. If a reader gets hold of some clarifications, they will be welcome.
Curvature concentrated at a cuspidal point
It will be calculated by summing the angles at the vertex and comparing this sum to the Euclidean sum: 2 p .
At the top left one of the many polyhedral representations of the cuspidal point has been shown. The "dismantling" of the object (on the right) leads to a sum exceeding the Euclidean sum 2 p by a value 2 a . We deduce from this that the angular curvature concentrated in the neighborhood of this point C is - 2 a. If the angle a is equal to p/2 then the negative curvature is worthc(figure at the bottom left). In fact the curvature concentrated at a cuspidal point can take an infinite number of values. At the bottom right the angular sum is accentuated and the curvature must then be < 2 a. The negative curvature is accentuated.
By operating in the opposite way one can arrive at a rather astonishing situation: making sure that the (angular) curvature concentrated at C is ... zero:
One can now start from a polyhedral representation of the Crosscap in which two cuspidal points appear, each corresponding to a negative curvature equal to**- p**:
There are eight "posicoins" corresponding to a value + p/2. Let us add four other "posicoins" of curvature + p/4 and four "negacoins of curvature - p/4
Plus the two cuspidal points of curvature - p .
Total: 2 p
By dividing this total curvature by 2 p one recovers the Euler-Poincaré characteristic of all the representations of the projective plane (like the Boy surface).
During my lecture I mentioned the art and manner of swapping the two cuspidal points of a Cross Cap, by using the eversion of the sphere. I no longer know if I put that somewhere on my site. It is such a mess. I will have to look, otherwise I will put it somewhere. It is quite amusing. Be that as it may, this performance hardly pleased one of those present, during the seminar.
- I don't see why Petit makes use of such a rigmarole to demonstrate the symmetry that unites the two cuspidal points of a Cross Cap. There is much simpler.
And he drew on the blackboard the image of a sphere crushed by two rulers brought together, which indeed gives a self-intersection set in the form of a segment bordered by two cuspidal points, as is the case of the Cross Cap. Alas, and our man realized it, it is not a Cross Cap.
- Good heavens, but then what is it? Asked someone.
It is quite simply a sphere, provided with two cuspidal points. If one makes them merge one obtains a self-intersection line which then becomes a simple circle. And one obtains at the bottom left (in cross-section) an immersion of the sphere which one has only to transform into its embedding. One can moreover move on to a polyhedral representation of this surface:
It is two-sided and the curvature is 2 p.
One can therefore amuse oneself quite a bit with these "submersions". Let us take an immersion of the torus which consists in rotating the "infinity" sign or a "figure eight", around an axis.
The technique of confluence of cuspidal points will allow us to arrive very quickly at the standard embedding of the torus, as indicated in the series of drawings.
But things sometimes turn out not to be so easy and obvious. Let us take for example a sphere that I crush between two segments which, this time, have lengths shorter than the diameter. Two cuspidal points are again obtained.
Since a Möbius strip can be inscribed on it, this surface is one-sided. Its polyhedral representation has been shown, which allows its total curvature to be calculated. The result is then zero. If I am not mistaken, it would then be a Klein bottle. Generally, only the most classical immersion is known, in which the self-intersection line is a simple circle. But there are others, such as this one. I admit that I have not yet found how to transform the above object into an immersion of the Klein bottle. Moreover, I do not know whether the different immersions belong to the same homotopy group (the sphere has only one). A priori not, since the torus can be immersed in four different ways, which cannot be connected to one another by a regular homotopy. In the meantime, I amused myself by transforming this surface by creating two additional cuspidal points, and one then obtains Two Cross Caps connected by a tube. By decomposing them, one again finds an Euler-Poincaré characteristic equal to zero.
This "strange surface" should be able to be transformed into one of the immersions of the Klein bottle. But which one? In any case, here is one obtained by rotating the "Eight" around an axis and, as a bonus, making it perform a half-turn:
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The model is then presented from another angle:
Plate 14: The same operation is repeated by creating the third "ear" of the self-intersection curve. In polyhedral form, this has the shape of three squares having a common vertex: the triple pointT.
Plate 15: by rotating the object you find again the polyhedral version of the Boy surface that I had invented and presented in the Topologicon (where there is a cutting pattern allowing it to be constructed).
Last plate: I tried to depict the Steiner surface (of the 4th degree, whereas the Boy surface is of the sixth) twisting and transforming itself into a Boy surface.
One sees that in "plump" form it takes a hell of a lot of practice to understand the object. Our eye is very uncomfortable when it comes to understanding an object where, along a single line of sight, more than two sheets are superimposed. Hence the interest of the polyhedral form, which brings within everyone's reach transformations considered sophisticated in geometry, insofar as people make the effort to build the models themselves. Incidentally, one notices that depending on the pairs of cuspidal points chosen, one obtains a "right" or "left" Boy surface (totally arbitrary words). The projective plane immerses itself according to two "enantiomorphic" representations, in mirror image. One sees that one can pass from a right Boy to a left Boy by way of a "central" model which is the Roman Steiner surface.
It would doubtless be nice for such drawings to be published in Pour la Science or La Recherche. But for twenty years I have been "banned from publication" in these journals because of UFOlogical deviationism. Thank you Messrs Hervé This and Philippe Boulanger. I have lost count of the articles of this kind that I have sent to these journals and that have been politely returned to me. One ends up getting used to one's status as an excommunicate.
As an anecdote, there exists in France a "Prix d'Alembert" intended to reward authors of popular mathematics books. The story was told to me by a member of the commission responsible for determining to whom the prize should go (there is still some money in it, after all). Dialogue:
-
But come now, couldn't we award the prize to Petit? He has produced remarkable works such as the Géométricon, the Trou Noir and the Topologicon.
-
Yes, but he hasn't done only those albums.
-
What are you alluding to?
-
He also wrote le Mur du Silence.
-
Ah, in that case....
Indeed, le Mur du Silence, published in 83, is an album devoted to MHD. And, as everyone knows, this sulphurous science has the virtue, or the mischief, of allowing flying saucers to travel at supersonic speed without making a Bang.
Hide this science, that I may not see it
I have in my drawers a superb version of the "turning the cube inside out", with a central model of great beauty, which is not the polyhedral version of Morin's variant. All of my own making. One of these days.....
October 22, 2003: People are not exactly rushing to these pages, if I believe the counter reading. On Monday, October 13, 2003, I gave a seminar at the CMI (Centre de mathématiques et d'informatique de Château-Gombert-Marseille) at Trotman's invitation. On that occasion I was able to line up a collection of about thirty cardboard models, of which you will one day have the first look, these having been photographed by Christophe Tardy.
When one gives a seminar, an atmosphere emerges. In the following photo, a geometer expressing his perplexity.
In the background, part of the models exhibited. At one point I asked the question:
*- Which of you have already seen a Roman Steiner surface? Raise your hand. *
No one had ever seen one. I therefore deemed it useful to present the object, in virtual reality, on the laptop I had brought, an object made with the help of Christophe Tardy, an engineer, and Frédéric Descamp, from the Laue-Langevin Institute in Grenoble (ILL). Visibly this presentation disconcerts the audience, little accustomed to seeing mathematical surfaces twirl at will.
Two cardboard panels, visible in the foreground, had made it possible to present the series of models in their logical order. The "green and yellow" models illustrate, in polyhedral form, the essential tool of creation-destruction of a pair of cuspidal points. The most distant white object is a polyhedral version of the Cross Cap, which first transforms into a polyhedral version of the Roman Steiner surface, a meter further on, then, at will, into a "right" or "left" Boy surface.
The analysis of the models brings out various remarks from the audience. One of the geometers asks:
*- If, by following the models in this direction, one can go from the Cross Cap to the Boy surface, it seems that by doing the reverse one must be able to transform a Boy surface into a Cross Cap. *
I answer in the affirmative. Emboldened, my interlocutor adds:
*- If, on arriving at the stage of the Roman Steiner surface one stops, it then becomes possible to head back toward a mirror Boy surface. *
I approve a second time. But alas no one will volunteer to shed light on this strange world where closed-surface immersions are endowed with cuspidal points, created or annihilated in pairs, the whole constituting a sort of extension of the world of immersions. The word "submersions" would seem fitting to me. If a reader gets hold of some clarifications, they will be welcome.
Curvature concentrated at a cuspidal point
It will be calculated by summing the angles at the vertex and comparing this sum with the Euclidean sum: 2 p .
At the top left one of the many polyhedral representations of the cuspidal point has been shown. The "dismantling" of the object (on the right) leads to a sum exceeding the Euclidean sum 2 p by a value 2 a . We deduce from this that the angular curvature concentrated in the neighborhood of this point C is - 2 a. If the angle a is equal to p/2 then the negative curvature equalsc(figure at the bottom left). In fact the curvature concentrated at a cuspidal point can take an infinite number of values. At the bottom right the angular sum is accentuated and the curvature must then be < 2 a. The negative curvature is accentuated.
By operating in the opposite way one can arrive at a rather astonishing situation: making the (angular) curvature concentrated at C ... zero:
One can now start from a polyhedral representation of the Crosscap in which two cuspidal points appear, each corresponding to a negative curvature equal to**- p**:
There are eight "posicoins" corresponding to a value + p/2. Let us add four other "posicoins" of curvature + p/4 and four "negacoins" of curvature - p/4
Plus the two cuspidal points of curvature - p .
Total: 2 p
By dividing this total curvature by 2 p we recover the Euler-Poincaré characteristic of all the representations of the projective plane (like the Boy surface).
During my lecture I mentioned the art and manner of swapping the two cuspidal points of a Cross Cap, using the eversion of the sphere. I no longer know whether I put that somewhere on my site. It is such a mess. I will have to look, otherwise I will put it somewhere. It is quite amusing. Be that as it may, this performance hardly pleased one of those present, during the seminar.
- I don't see why Petit makes use of such a rigmarole to demonstrate the symmetry that unites the two cuspidal points of a Cross Cap. There is much simpler.
And he drew on the blackboard the image of a sphere flattened by two rulers brought together, which indeed gives a self-intersection set in the form of a segment bordered by two cuspidal points, as is the case of the Cross Cap. Alas, and our man noticed it, it is not a Cross Cap.
- Good heavens, but then what is it? Asked someone.
It is quite simply a sphere, furnished with two cuspidal points. If one makes them merge one obtains a self-intersection line which then becomes a simple circle. And one obtains at the bottom left (in cross-section) an immersion of the sphere which one has only to transform into its embedding. One can moreover move on to a polyhedral representation of this surface:
It is two-sided and the curvature is 2 p.
One can therefore amuse oneself quite a bit with these "submersions". Let us take an immersion of the torus which consists in rotating the "infinity" sign or a "figure eight" around an axis.
The technique of confluence of cuspidal points will allow us to arrive very quickly at the standard embedding of the torus, as indicated in the series of drawings.
But things sometimes turn out not to be so easy and obvious. Let me take for example a sphere that I flatten between two segments which, this time, have lengths shorter than the diameter. Two cuspidal points are again obtained.
Since a Möbius strip can be inscribed on it, this surface is one-sided. Its polyhedral representation has been shown, which allows its total curvature to be calculated. One then finds zero. If I am not mistaken, it would then be a Klein bottle. In general, only the most classical immersion is known, in which the self-intersection line is a simple circle. But there are others, such as this one. I admit that I have not yet found how to transform the above object into an immersion of the Klein bottle. Moreover, I do not know whether the different immersions belong to the same homotopy group (the sphere has only one). A priori not, since the torus can be immersed in four different ways, which cannot be connected to one another by a regular homotopy. In the meantime, I amused myself by transforming this surface by creating two additional cuspidal points, and one then obtains Two Cross Caps connected by a tube. By decomposing them, one again finds an Euler-Poincaré characteristic equal to zero.
This "strange surface" should be able to be transformed into one of the immersions of the Klein bottle. But which one? In any case, here is one obtained by rotating the "Eight" around an axis and, as a bonus, making it perform a half-turn:
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Original pages: crosscap_boy0.htm, crosscap_boy1.htm, crosscap_boy1.htm, crosscap_boy2.htm, crosscap_boy2.htm, crosscap_boy3.htm, crosscap_boy3.htm, crosscap_boy4.htm, crosscap_boy4.htm