Model Surface Geometry - Mathematical Models

How to transform a Cross Cap surface
into a Boy surface (right or left, as desired)
by way of the Roman surface of Steiner.

Italian: Andrea Sambusetti, University of Rome

../../Crosscap_Boy1.htm

September 27 - October 25, 2003

Page 4

We present the model from another viewpoint:

Table 14: We repeat the same operation, creating the third "ear" of the self-intersection curve. In the polyhedral model, the latter has the shape of three squares with a common vertex: the triple point T.

Table 15: By rotating the object, you will find the polyhedral version of the Boy surface that I presented at the Topologicon (where you can also find an assembly plan that allows you to build it).

Last table: I tried to illustrate the Steiner Roman surface as it twists and transforms into a Boy surface.

We see that, drawn in "rounded", it takes a lot of practice to understand it. Our eye is very uncomfortable when it comes to understanding an object for which more than two sheets overlap on the same line of sight. Hence the interest of the polyhedral model, which makes accessible to anyone, if only they try to build the models themselves, transformations considered complicated in geometry. Incidentally, note that depending on the chosen pair of cuspidal points, you obtain a "right" or "left" Boy surface (completely arbitrary definitions). The projective plane is immersed in space through two "antiautomorphic" mirror images. Thus, we can also see that one can pass from a right Boy surface to a left Boy surface through a "central" model, which is the Roman surface of Steiner.

It would certainly be nice if these drawings were published in magazines like Pour la Science or La Recherche. But for twenty years I have been "forbidden" from publishing in these magazines due to ufological deviationism. Thank you, gentlemen Hervé This and Philippe Boulanger. I have lost count of the number of articles of this kind I have submitted to these magazines and that have been kindly rejected. One ends up getting used to one's status as an outcast.

Anecdotal: there is an "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, of course, financial issues behind it). Dialogue:

• Well, why don't we give the prize to Petit? He has written notable works such as "Géométricon", "Trou Noir" and "Topologicon".

• Yes, but he didn't only do that.

• What are you implying?

• He also wrote "Mur du Silence".

• Ah, well, then...

Indeed, "Mur du Silence", published in 1983, is an album dedicated to MHD. And, as everyone knows, this corrosive science has the merit, or defect, of allowing flying saucers to move at supersonic speeds without making a bang.

« Cachez cette science, que je ne saurais voir »

I have a magnificent version of the "cube inversion" in my boxes, which is not the polyhedral version of Morin's variant. All my own work. One of these days....

October 22, 2003: These pages are not too busy, if I believe the counter. On Monday, October 13, 2003, I gave a seminar at the CMI (Centre de Mathématiques et d'Informatique de Château-Gombert-Marseille) at the invitation of Trotman. At that time, I was able to present a collection of about thirty cardboard models, which you will be able to enjoy one day, since they have been photographed by Christophe Tardy.

When giving a seminar, a certain atmosphere is created. In the photo below, here is a geometer expressing his perplexity.

In the background, a part of the models displayed with the help of my long-time collaborator, Boris Kolev, a member of the department, also a geometer. At one point I asked the question:

• How many of you have already seen a Roman surface of Steiner? Raise your hand.

None of them had ever seen it. I therefore felt it was useful to present this object, with a virtual reality program, on the laptop I had with me, a program realized with the help of Christophe Tardy, engineer, and Frédéric Descamp, from the Laue Langevin Institute of Grenoble (ILL). Clearly, this presentation bewilders the audience, not used to seeing mathematical surfaces performing acrobatics at will.

Two cardboard tables, visible in the foreground, allowed to present the whole sequence of models in their logical order. The green and yellow models illustrate, in polyhedral form, the essential tool for the creation and dissolution of a pair of cuspidal points. The white object further away is a polyhedral version of the Cross Cap surface, which first transforms into a polyhedral version of the Roman surface of Steiner, then, one meter further, at will, into a "right" or "left" Boy surface.

The analysis of the models brings out various observations from the audience. One of the geometers asks:

• If it is true that, following these models in this order, one can go from the Cross Cap surface to the Boy surface, it seems that, following the opposite procedure, one could transform a Boy surface into a Cross Cap.

I answer affirmatively. Emboldened, my interlocutor adds:

• Then, if we stop at the Roman surface of Steiner stage, it should be possible to return to a Boy surface, but reflected relative to the initial one.

I agree a second time. But, unfortunately, no one will offer to clarify this strange world where it is allowed for the immersions of closed surfaces to have cuspidal points, created or dissolved in pairs, whose set constitutes a kind of extension of the world of immersions. The term "summersion" seems appropriate to me. If a reader is able to clarify this, he is welcome.

Curvature concentrated in a cuspidal point.

We will calculate it by summing the angles at the vertex and comparing this sum to the result obtained in the case of the Euclidean plane: 2π.

In the top left, you can see one of many possible polyhedral representations of a cuspidal point. "Disassembling" the surface leads to a sum of angles that exceeds the value 2π by 2α. It is therefore deduced that the concentrated angular curvature around this point C is -2α. If the angle α is equal to π/2, then the negative curvature is -π (bottom left figure). Indeed, the curvature of a cuspidal point can take on infinitely many values. In the bottom right, we emphasize the angular sum and the curvature then becomes < -π (we have increased the negative curvature).

By working in the opposite way, we can reach a rather surprising situation: we can make the curvature (angular) concentrated in C be ... zero:

Now, starting from a polyhedral representation of the Cross Cap surface, which includes two cuspidal points, each with curvature equal to -π:

In this figure, there are eight "posiconi" of value +π/2. We add four more "posiconi" of curvature +π/4 and four "negaconi" of curvature -π/4.

Plus the two cuspidal points with curvature -π.

Total: 2π

Dividing the value of this "total curvature" by 2π, we recover the value of the Euler-Poincaré characteristic of any representation of the projective plane (or of the Boy surface).

During the conference, I mentioned art and the way to permute the two cuspidal points of a Cross Cap surface using the inversion of the sphere. I don't know anymore...