Transformation of the Crosscap into a Boy surface, via the Roman surface of Steiner
How to transform a crosscap into a Boy surface (right or left, at your choice) by passing through the Roman surface of Steiner.
September 27, 2003
Page 4
At this point, the model is presented from another angle:
Plate 14: The same operation is repeated, creating the third "ear" of the self-intersection curve. In polyhedral form, it has the shape of three squares with a common vertex: the triple point T.
Plate 15: By rotating the object, you find the polyhedral version of the Boy surface that I had suggested and presented in the Topologicon (which contains a cut-out allowing its construction).
Last plate: I tried to represent the Steiner surface (of the 4th degree, whereas the Boy is of the 6th) in the act of twisting and transforming into a Boy surface.
It can be seen that, in "roundabout" fashion, it takes quite some practice to understand the object. Our eye is very uncomfortable when trying to understand an object where more than two layers overlap on the same line of sight. Hence the interest of the polyhedral version, which makes transformations considered sophisticated in geometry accessible to the general public, provided that people make the effort to build the models themselves. In passing, we note that depending on the chosen pair of cusp points, we obtain a "right" or "left" Boy surface (completely arbitrary terms). The projective plane is immersed in two "enantiomorphic" representations, mirror images. It can be seen that one can go from a right Boy surface to a left one through a "central" 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 deviationism". Thank you gentlemen Hervé This and Philippe Boulanger. I no longer count the number of articles of this kind I have sent to these journals, which have been politely returned to me. One eventually gets used to one's status as an outcast.
Anecdotaly, there is in France a "Alembert Prize" intended to reward authors of mathematical popularization books. The story was told to me by a member of the committee in charge of deciding who should receive the prize (there are some money involved). Dialogue:
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But finally, couldn't we give the prize to Petit? He has remarkable works such as the Géométricon, the Trou Noir and the Topologicon.
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Yes, but he hasn't only done these albums.
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What are you referring to?
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He also wrote the Mur du Silence.
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Ah, in that case...
Yes, the Mur du Silence, published in 83, is an album dedicated to MHD. And, as everyone knows, this controversial science has the virtue, or rather the trick, of allowing flying saucers to move at supersonic speed without making a bang.
Hide this science, I cannot see it
I have in my files a version of the "cube inversion" that is superb, with a central model of extraordinary beauty, which is not the polyhedral version of the Morin variant. All of it is my own creation. One of these days...
October 22, 2003: These pages are not very crowded, if I believe the counter's number. I gave a seminar on October 13, 2003 at the CMI (Centre de mathématiques et d'informatique de Château-Gombert-Marseille) at the invitation of Trotman. At the occasion I could display a collection of about thirty cardboard models, which you will soon have the first glimpse of, as they were photographed by Christophe Tardy.
When giving a seminar, an atmosphere emerges. On the following photo, a geometer expressing his perplexity.
In the background, part of the models displayed. At one point I asked the question:
- Who here has ever seen a Roman surface of Steiner? Raise your hand.
No one had ever seen it. I therefore judged it useful to present the object, actually virtual, on the laptop I had brought, an object realized with the help of Christophe Tardy, engineer, and Frédéric Descamp, from the Institut Laue Langevin de Grenoble (ILL). Obviously, this presentation confused the audience, not used to seeing mathematical surfaces fluttering at will.
Two cardboard panels, visible in the foreground, allowed to present the rest of the models 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, and 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 model in this direction, one can go from the Cross Cap to the Boy, it seems that by doing the opposite one should be able to transform a Boy into a Cross Cap.
I answer in the affirmative. Encouraged, my interlocutor adds:
- If, when reaching the stage of the Roman surface of Steiner, one stops, it becomes then possible to go back towards a mirrored Boy surface.
I approve a second time. But alas, no one will offer to clarify this strange world where one endows immersions of closed surfaces with cusp points, created or annihilated in pairs, the whole constituting a sort of extension of the world of immersions. The word "submersions" seems appropriate to me. If a reader finds some clarifications, they will be 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 and to the 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 2α. It can be deduced 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 the bottom and to the left). In fact, the curvature concentrated at a cusp point can take an infinite number of values. At the bottom and to the right, the angular sum is increased and the curvature then < 2α. The negative curvature is increased.
By operating in the inverse way, one can arrive at a rather surprising situation: making the curvature (angular) concentrated at C be ... zero:
Now we can start from a polyhedral representation of the Crosscap, which features two cusp points, each with a negative curvature equal to -π:
There are eight "posicoins" corresponding to a value +π/2. Let's add four more "posicoins" with curvature +π/4 and four "negacoins" with curvature -π/4.
Plus the two cusp points with curvature -π.
Total: 2π
Dividing this total curvature by 2π, we find the Euler-Poincaré characteristic...