How to transform a cross-cap into a Boy surface (right-handed or left-handed, your choice) via the Steiner Roman surface.
September 27 – October 25, 2003
page 2
Here is a Cross Cap (as you discovered it in the virtual reality images). It has two cuspidal points bordering a self-intersection line. You can make one by pinching a balloon with a curling iron. But you can also construct polyhedral representations of it. The one at the bottom will be of particular interest to us.
In this plate 4 lies the moment most difficult to grasp. It seems to me almost impossible for the average person to understand these figures simply by looking at the drawings. Build these models. In plain terms, we pull the cuspidal point C2 toward the "inside of the surface" (which incidentally makes no sense since, as you have probably immediately noticed, the Cross Cap is one-sided. By insisting, the surface self-intersects and the self-intersection set is completed, in "roundish" fashion, by a figure-eight shaped curve. A triple point T is created in the process.
The surface is more understandable in its polyhedral form and, at the bottom, we have enlarged certain elements to show what leads us to transform this object into a Steiner Roman Surface (see the virtual reality) whose simplest polyhedral form consists of assembling four cubes (here only three are visible).
Plate 5: The polyhedron on the left, the roundish form on the right. The arrow indicates a passage that we are going to "pinch". At the bottom, the beginning of the pinching.
Plate 6: The pinching is performed by creating a singular point B. In fact, since we pinch from both sides to save time, two singular points S1 and S2 form, followed by two pairs of cuspidal points. Without cardboard, scissors, and adhesive tape, you are in trouble.
Plate 7: We have simply migrated the various cuspidal points. If the point C2 is "obvious", you will have a bit more trouble identifying the points C3 and C4 as cuspidal points. Yet they are present at the end of a self-intersection line. 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 slightly, we arrive at a polyhedral form of the Steiner Roman surface (a 4th-degree surface 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 we impose. Surfaces that do not intersect themselves are called embeddings (of the sphere, of the torus in R3). When they do intersect themselves but the tangent plane varies continuously, they are called immersions. Example: the Klein bottle in its classic representation. There is no embedding of the Klein bottle in R3. It necessarily intersects itself. Immersions have self-intersection sets free of cuspidal points. These curves are continuous but can cross where double or triple points are located. Note: the sphere can be presented as an immersion by simply making it intersect itself. This is indeed how we manage to turn it inside out (A. Phillips, 1967, with the central step being the two-sheeted covering of a Boy surface; B. Morin and J.P. Petit, 1979, with the central model being Morin's four-eared model, of which here is a polyhedral representation I invented about ten years ago.
Plan for assembling this object using a cut-out
If we extend the rules of the game by assuming that these objects possess cuspidal points, we obtain submersions (the Cross Cap, the Steiner Roman Surface). I do not know if this is the exact term, but since I have not found any mathematician who could enlighten me, I found it amusing to invent one, provisionally, until an expert geometer comes forward. Thus the Cross Cap and the Steiner Roman Surface would be submersions of the "projective plane".
To tell you everything, after my troubles with MHD over twenty-five years, I had started 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 be confusing and lead the French Navy to think that through this research I was attempting to hide some breakthrough in submarine propulsion.
The rule of "creation-destruction" of pairs of cuspidal points allows one to pass from a submersion of one object to another, and this is what we have just done by showing that the Cross Cap and the Steiner Roman Surface were two submersions of the same object called the projective plane. Do not try to imagine what a "projective plane" looks like. This object can only be apprehended through its various representations. As for the term projective plane, it is just one among a thousand others invented by mathematicians to mislead those who would enter their closed circle. The Larousse dictionary will be of no use to you in mathematics.
We then have to move on to the Boy surface, which is an immersion of the projective plane
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