How to transform a Cross Cap surface
into a Boy surface (left or right, as you choose)
passing through the Roman surface of Steiner.
Italian
: Andrea Sambusetti, University of Rome
September 27
- October 25, 2003
page 1
Q
uote, as Kipling might say, this requires
"great cunning and magical strength."
I
am retired, but I feel compelled to say that
I still do a little research "against my will," just as others, in their free time, knit. If you have patience and obtain some 200 g/m² Bristol board with a grid, you can reconstruct all these models yourself. My friend Christophe Tardy is building, from these, an animation that should turn out quite well.
T
he 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,
which requires downloading the Cosmoplayer program onto your computer. Do it—it's really nice. Everything revolves around "cusp 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 attaches along that segment to the right shoulder, while the left thigh attaches to the left shoulder. As for the cusp point, don't look for it: you're simply sitting right on top of it.
B
ut all this is drawn... roundish.
Let's move on to a "polyhedral representation" of the cusp point
(such as a cube or tetrahedron can be considered polyhedral representations of a simple sphere). The bold line represents the
"self-intersection curve," ending at the cusp point
C.
P
rint out these diagrams—it's better. In the following, you'll need to recognize a cusp point in "different configurations" and not confuse it with a simple vertex of the polyhedron. If you feel up to it, build these various objects from cardboard—you'll understand better. Below is an essential operation, called "creation-destruction of a pair of cusp points." The first drawing shows a kind of self-intersecting cylinder along the bold segment, whose cross-section resembles an inverted Greek letter gamma. We then deform this surface by "pinching" the tube whose cross-section looks like an inverted teardrop. Doing so, we degenerate this "teardrop" into a point
S. Then, this point splits into two cusp points. This is the operation of creating a pair of cusp points. The reverse operation destroys two cusp points. Immediately below is the polyhedral version of this operation.
H
ere is another polyhedral representation of the transformation, resembling what you'll soon see forming on the surface.
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