How to transform a crosscap into a Boy surface (right-handed or left-handed, at your choice) via the Roman surface of Steiner.
September 27, 2003
page 4
The model is then presented from another angle:
Plate 14: We repeat the same operation by creating the third "ear" of the self-intersection curve. In the polyhedral version, this takes the form of three squares sharing a common vertex: the triple point T.
Plate 15: By rotating the object, you recover the polyhedral version of the Boy surface that I invented and presented in the Topologicon (which includes a net allowing it to be constructed).
Last plate: I tried to depict the Steiner surface (of degree 4, whereas the Boy surface is of degree 6) as it contorts and transforms into a Boy surface.
One sees that in "rounded" form, it takes quite some habit to understand the object. Our eye is very uncomfortable when trying to comprehend an object where more than two sheets overlap along the same line of sight. Hence the interest of the polyhedral approach, which brings within reach of the general public transformations considered sophisticated in geometry, provided people make the effort to build the models themselves. Incidentally, we note that depending on the pair of cusp points chosen, one obtains a "right-handed" or "left-handed" Boy surface (totally arbitrary terms). The projective plane immerses in two "enantiomorphic" representations, mirror images of each other. We see that one can pass from a right-handed Boy surface to a left-handed one via an "intermediate" model, which is the Roman surface of Steiner.
It would certainly be nice if such drawings were published in Pour la Science or La Recherche. But for twenty years I have been "banned from publication" in these journals due to my UFO-related deviantism. Thank you, Messieurs Hervé This and Philippe Boulanger. I have lost count of the articles of this kind that I sent to these journals and which were politely returned. One eventually gets used to one's status as an excommunicated person.
As an anecdote, there exists in France an "Alembert Prize" intended to reward authors of popular mathematics books. The story was told to me by a member of the committee responsible for determining who should receive the prize (there is still some money involved). Dialogue:
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But surely, couldn't we award the prize to Petit? He has produced remarkable works such as the Géométricon, the Black Hole, and the Topologicon.
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Yes, but he hasn't only produced those albums.
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What are you alluding to?
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He also wrote The Wall of Silence.
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Ah, in that case....
Indeed, The Wall of Silence, published in '83, is an album devoted to MHD. And, as everyone knows, this dubious science has the virtue, or the malice, of allowing flying saucers to travel at supersonic speeds without producing a sonic boom.
Hide this science, which I cannot bear to see
I have in my files a superb version of the "cube turn-over," featuring a central model of great beauty, which is not the polyhedral version of Morin's variant. All of my own creation. Someday soon.....
October 22, 2003: There isn't much traffic on these pages, if I am to believe the hit counter. On Monday, October 13, 2003, I gave a seminar at the CMI (Centre de mathématiques et d'informatique in 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 soon have the first look, as they were photographed by Christophe Tardy.
When one gives a seminar, an atmosphere develops. In the following photo, a geometer expresses his perplexity.
In the background, part of the models on display. At one point, I asked the question:
- Who has already seen a Roman surface of Steiner? 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 along, an object created with the help of Christophe Tardy, an engineer, and Frédéric Descamp, from the Institut Laue-Langevin in Grenoble (ILL). Clearly, this presentation disconcerted the audience, who are not accustomed to seeing mathematical surfaces twirl at will.
Two cardboard panels, visible in the foreground, allowed the sequence of models to be presented in their logical order. The "green and yellow" models illustrate, in polyhedral form, the essential tool for creating and annihilating a pair of cusp points. The most distant white object is a polyhedral version of the Cross Cap, which first transforms into a polyhedral version of the Roman surface of Steiner, one meter further away, and then, at will, into a "right-handed" or "left-handed" Boy surface.
Analysis of the models prompted various remarks from the audience. One of the geometers asked:
- If, following the models in this direction, one can pass from the Cross Cap to the Boy surface, it seems that by doing the reverse, one should be able to transform a Boy surface into a Cross Cap.
I answered affirmatively. Emboldened, my interlocutor added:
- If, upon reaching the stage of the Roman surface of Steiner, one stops, it then becomes possible to proceed towards a mirrored Boy surface.
I agreed a second time. But alas, no one volunteered to provide clarifications on this strange world where we endow immersions of closed surfaces with cusp points, created or annihilated in pairs, the whole constituting a kind of extension of the world of immersions. The word "submersions" would seem fitting to me. If a reader comes across any clarifications, they would be most welcome.
Curvature concentrated at a cusp point
It will be calculated by summing the angles at the vertex and comparing this sum to the Euclidean sum: 2π.
At the top left, one of the multiple polyhedral representations of the cusp point is shown. The "disassembly" of the object (on the right) leads to a sum exceeding the Euclidean sum 2π by a value of 2α. We deduce that the angular curvature concentrated near this point C is -2α. If the angle α is equal to π/2, then the negative curvature is c (figure at bottom left). In fact, the curvature concentrated at a cusp point can take an infinite number of values. At the bottom right, the angular sum is increased, and the curvature must then be < 2α. The negative curvature is intensified.
By operating in the reverse manner, one can arrive at a rather surprising situation: making it so that the (angular) curvature concentrated at C is ... zero:
We can now start from a polyhedral representation of the Crosscap featuring two cusp points, each corresponding to a negative curvature equal to -π:
There are eight "positive corners" corresponding to a value of +π/2. Let us add four other "positive corners" of curvature +π/4 and four "negative corners" of curvature -π/4.
Plus the two cusp points of curvature -π.
Total: 2π
By dividing this total curvature by 2π, we recover the Euler-Poincaré characteristic of all representations of the projective plane (such as the Boy surface).
During my lecture, I discussed the art and manner of permuting the two cusp points of a Cross Cap, using the turn-over of the sphere. I no longer remember if I put this somewhere on my website. It is such a mess. I will have to look for it; otherwise, I will put it somewhere. It is quite amusing. In any case, this performance did not please one of those present at the seminar.
- I don't see why Petit uses such apparatus to demonstrate the symmetry linking the two cusp points of a Cross Cap. There is a much simpler way.
And he drew on the board the image of a sphere crushed between two rulers that are brought together, which indeed yields a self-intersection set in the form of a segment bounded by two cusp points, as is the case for the Cross Cap. Alas, and our man noticed, this is not a Cross Cap.
- Devil take it, but what is it then? Someone asked.
It is simply a sphere, equipped with two cusp points. If we make them conflate, we obtain a self-intersection line that then becomes a simple circle. And we obtain at the bottom left (in cross-section) an immersion of the sphere that we need only transform into its embedding. We can also pass to a polyhedral representation of this surface:
It is two-sided and the curvature is 2π.
One can thus have quite a bit of fun with these "submersions." Let us take an immersion of the torus which consists of rotating the "infinity" sign or a "figure eight" around an axis.
The technique of confluent cusp points will allow us to arrive very quickly at the standard embedding of the torus, as indicated in the sequence of drawings.
But things do not always prove to be as easy and obvious. For example, I take a sphere that I crush between two segments which, this time, have lengths less than the diameter. We still obtain two cusp points.
As one can inscribe a Möbius strip on it, this surface is one-sided. We have shown its polyhedral representation, which allows us to calculate its total curvature. We find zero. If I am not mistaken, this would then be a Klein bottle. Generally, only the most classical immersion is known, where the self-intersection line is a simple circle. But there are others, like this one. I admit that I have not yet found how to transform the object above into an immersion of the Klein bottle. I do not even know if the different immersions belong to the same homotopy group (the sphere has only one). A priori no, since the torus can be immersed in four different ways, which cannot be connected to each other by a regular homotopy. Meanwhile, I have amused myself by transforming this surface by creating two additional cusp points, and we then obtain two Cross Caps connected by a tube. By cutting them up, we recover an Euler-Poincaré characteristic equal to zero.
This "strange surface" should be able to transform into one of the immersions of the Klein bottle. But which one? In any case, here is one obtained by rotating the "Figure Eight" around an axis and making it perform an additional half-turn:
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