Transformation of the Cross Cap into a Boy surface, via the Roman surface of Steiner
How to transform a cross cap into a Boy surface (left or right, as you choose) by passing through the Roman surface of Steiner.
September 27 - October 25, 2003
page 2
Here is a Cross Cap (as you would have discovered in the virtual reality images). It has two cusp points that border a line of self-intersection. You can make it by pinching a balloon with a curling iron. But you can also build polyhedral representations. The one at the bottom will particularly interest us.
In this plate 4 lies the most difficult moment to understand. It seems almost impossible for an average person to understand these figures by just looking at the drawings. Build these models. In short, we pull the cusp point C2 towards "inside the surface" (which, by the way, makes no sense since, as you have probably noticed immediately, the Cross Cap is a one-sided surface. By insisting, the surface self-intersects and the whole self-intersection is completed, in a "rondouillard" by a figure-eight curve. In the process, a triple point T is created.
The surface is more understandable in its polyhedral form and, at the bottom, we have enlarged some elements to show what leads us to transform this object into a Roman surface of Steiner (see the virtual reality) whose simplest polyhedral form consists of assembling four cubes (here we only see three).
Plate 5: The polyhedron on the left, the "rondouillard" on the right. The arrow takes a passage that we will "pinch". At the bottom, the beginning of the pinching.
Plate 6: The pinching is done by creating a singular point B. In fact, as we pinch from both sides, to save time; two singular points S1 and S1 are formed, then two pairs of cusp points. There, without cardboard, scissors and tape, you are in trouble.
Plate 7: We have simply moved the different cusp points. If point C2 is "obvious", you will have a bit more difficulty identifying points C3 and C4 as cusp points. They are, however, present at the end of a line of self-intersection. Above point C3 is simply what I have called a "posicoin", a point of positive curvature concentration (a point of negative curvature concentration is a "négacoin"). By deforming this object a little, we get a polyhedral form of the Roman surface of Steiner (the fourth-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 you impose. Surfaces that do not intersect themselves are called embeddings (of the sphere, the torus in R3). When they intersect themselves but the tangent plane varies continuously, they are called immersions. Example: the Klein bottle in its classic representation. There is no representation of the Klein bottle in R3 as an embedding. It necessarily intersects itself. Immersions have self-intersection sets without cusp points. These curves are continuous but can cross at points where there are double or triple points. Note: the sphere can be presented as an immersion, simply by making it intersect itself. This is indeed how one manages to turn it inside out (A. Phillips, 1967, with as a central step the two-sheeted covering of a Boy surface; B. Morin and J.P. Petit, 1979 with as a central model the four-ear model of Morin, 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 have cusp points, we obtain submersions (the Cross Cap, the Roman surface of Steiner). I don't know if this is the exact term, but as I have not found any mathematician who could enlighten me, I found it amusing to invent one, provisionally, until an expert geometer appears. Thus, the Cross Cap and the Roman surface of Steiner would be submersions of the "projective plane".
To tell you the truth, after my troubles with MHD over twenty-five years, I had started these works because they seemed as far as possible from any military application. But, as my old friend Mihn pointed out, the term "submersion" could be confusing and might suggest to the Navy that through these researches I would be trying to hide some breakthrough in underwater propulsion.
The rule of "creation-decreation" of pairs of cusp points allows us to pass from a submersion of an object to another, and this is what we have just done by showing that the Cross Cap and the Roman surface of Steiner are two submersions of the same object called projective plane. Don't try to find out what a "projective plane" looks like. This object can only be understood through its different representations. As for the word projective plane, it is just one among a thousand others invented by mathematicians to mislead those who want to enter their closed circle. The Larousse dictionary will be of no help in mathematics.
We then have to move on to the Boy surface, which is an immersion of the projective plane
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