How to transform a cross-cap into a Boy surface (right or left, at will) via the Roman surface of Steiner.
September 27, 2003
page 4
We now present the model from another angle:
Plate 14: We repeat the same operation by creating the third "ear" of the self-intersection curve. In polyhedral form, this takes the shape 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 previously introduced and presented in the Topologicon (which includes a cut-out allowing its construction).
Final plate: I attempted to depict the Steiner surface (of degree 4, whereas the Boy surface is of degree 6) in the act of contorting and transforming into a Boy surface.
One sees that, in "rounded" form, it takes considerable practice to grasp the object. Our eye is very uncomfortable when trying to comprehend an object where more than two sheets superimpose along the same line of sight. Hence the value of the polyhedral model, which makes accessible to the general public transformations considered sophisticated in geometry—provided people make the effort to build the models themselves. Along the way, one notices that depending on the chosen pair of cusp points, one obtains either a "right" or "left" Boy surface (terms entirely arbitrary). The projective plane is immersed in two "enantiomorphic" representations, mirror images of each other. One sees that one can pass from a right Boy surface to a left one via a "central" model: the Roman surface of Steiner.
It would perhaps be pleasant 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 alleged UFO-related deviationism. Thank you, Messieurs Hervé This and Philippe Boulanger. I no longer count the number of articles of this kind I've sent to these journals, politely returned to me. One eventually gets used to one's status as an excommunicate.
Anecdotal note: In France, there exists 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 awarding the prize (there is, after all, some money involved). Dialogue:
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But after all, couldn't we award the prize to Petit? He has produced remarkable works like the Géométricon, the Trou Noir, and the Topologicon.
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Yes, but he hasn't only done those 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 1983, is an album devoted to MHD. And as everyone knows, this controversial science has the virtue—or the trickery—of enabling flying saucers to travel at supersonic speeds without making a sonic boom.
Hide this science, I can't bear to see it.
I have in my archives a stunning version of the "cube eversion," featuring a central model of exquisite beauty, not the polyhedral version of Morin's variant. Entirely my own creation. One of these days...
October 22, 2003
: Few people seem to be visiting these pages, judging by the counter's number. On October 13, 2003, I gave a seminar at the CMI (Centre de mathématiques et d'informatique de Château-Gombert-Marseille), invited by Trotman. On that occasion, I arranged a collection of about thirty cardboard models, which you will soon see the first images of—photographed by Christophe Tardy.
When giving a seminar, a certain atmosphere arises. In the following photo, a geometer expresses his perplexity.
In the background, part of the models on display. At one point, I asked:
- Who here has ever seen a Roman surface of Steiner? Raise your hand.
No one had ever seen it. I therefore deemed it useful to present the object—virtually—on the laptop I had brought, created with the help of Christophe Tardy, an engineer, and Frédéric Descamp from the Institut Laue Langevin in Grenoble (ILL). Clearly, this presentation bewildered the audience, little accustomed to seeing mathematical surfaces freely spinning and twisting.
Two cardboard panels visible in the foreground allowed us to present the sequence of models in logical order. The "green and yellow" models illustrate, in polyhedral form, the essential tool for creating or 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 on, and then, at will, into either a "right" or "left" Boy surface.
The analysis of the models prompted various remarks from the audience. One geometer asked:
- If, following the models in this direction, we can go from the cross-cap to the Boy surface, then going in reverse should allow us to transform a Boy surface back into a cross-cap.
I answered affirmatively. Emboldened, my interlocutor added:
- If, upon reaching the stage of the Roman surface of Steiner, we stop, it then becomes possible to continue toward a mirror-image Boy surface.
I approved once more. But alas, no one volunteered to clarify this strange world where closed surfaces are immersed with cusp points, created or annihilated in pairs, the whole constituting a kind of extension of the world of immersions. The term "submersions" seems appropriate to me. If any reader comes across clarifications, they will be most welcome.
Curvature concentrated at a cusp point
We calculate it 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 an amount 2α. We deduce that the angular curvature concentrated near this point C is -2α. If angle α equals π/2, then the negative curvature equals -π (figure at bottom left). In fact, the curvature concentrated at a cusp point can take infinitely many values. At bottom right, we increase the angular sum, so that the curvature becomes less than -2α. The negative curvature increases.
By reversing the process, we can reach a rather surprising situation: making the (angular) curvature concentrated at C... zero:
We can now start from a polyhedral representation of the cross-cap featuring two cusp points, each corresponding to a negative curvature of -π:
There are eight "posicoins" corresponding to a value of +π/2. Let us 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 recover the Euler-Poincaré characteristic of all representations of the projective plane (such as the Boy surface).
During my lecture, I mentioned the art and technique of swapping the two cusp points of a cross-cap using the sphere eversion. I no longer remember whether I included this somewhere on my website. It's such a mess. I'll have to look, otherwise I'll put it somewhere. It's quite amusing. To...