Mathematics geometry transformation surfaces
How to transform a Cross Cap surface into a Boy surface (right or left, as you choose) passing through the Roman surface of Steiner.
Italian: Andrea Sambusetti, University of Rome
September 27 - October 25 2003
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This, as Kipling would say, involves "great cunning and magical strength".
I am retired but, I feel like saying, I still do a bit of research "against my will", as others, in their free time, knit. If you have patience and get some 200g Bristol board with squares, you can reconstruct all these models yourself. My friend Christophe Tardy is building an animation from them, which should be quite good.
The Cross Cap surface will appear in the following drawings, as will the Roman surface of Steiner. But you can discover them by going to the Virtual Reality section, for which it is necessary to download the Cosmoplayer program on your computer. Do it, it's really nice. Everything revolves around the "cuspidal points". These points form naturally when you ride a horse and squeeze your legs tightly. The horse's body will then be squashed along a segment. The right thigh will stick along that segment to its right shoulder, while the left thigh will stick to the left shoulder. As for the cuspidal point, don't look for it: you are simply sitting on it.
But all this is drawn... rounded. Let's move on to a "polyhedral representation" of the cuspidal point (just as a cube or a tetrahedron can be considered as polyhedral representations of a simple sphere). The thick line represents the "self-intersection curve", which ends at the cuspidal point C.
Print these plates, it's better. In the following, you will need to know how to recognize a cuspidal point in "different configurations" and not confuse it with a simple vertex of the polyhedron. Build, if you feel like it, these different objects with cardboard, you will understand better. Here below we have an essential operation, called "creation and dissolution of a pair of cuspidal points". The first drawing represents a kind of cylinder that self-intersects along the segment drawn in bold, whose section resembles the Greek letter gamma, upside down. We then deform this surface "pinching" the tube whose section looks like an "upside-down tear". By doing so, we make this "tear" degenerate into a point S. Then, this point splits, giving rise to two cuspidal points. This is the operation of creating a pair of cuspidal points. The opposite operation, on the contrary, destroys two cuspidal points. Just below you find the polyhedral version of this operation.
Here below is another polyhedral representation of the transformation, which looks like what you will see forming on the surface shortly.
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