How to transform a Cross Cap surface
into a Boy surface (right or left, your choice)
by way of the Roman Steiner surface.
Italian
: Andrea Sambusetti, University of Rome
**September 27
- October 25, 2003**
page 1
This, as Kipling would say, requires
"great cunning and magical strength."
I am retired but, I feel I should say it,
I still do a little research "in spite of myself," as others, in their
free time, knit. If you have patience and get yourself some 200 gr. graph
bristol board, you will be able to reconstruct all these
models yourself. My friend Christophe Tardy is building, starting from them,
an animation that shouldn't turn out badly.
The Cross Cap surface will appear in
the drawings that follow, as will the Roman Steiner surface. But you can
discover them by going to the Virtual Reality section,
to view which it is necessary to download the Cosmoplayer program onto your
computer. Do it, it's really nice. Everything revolves around "cusp
points." These points form naturally when you mount a horse and squeeze your legs tightly. The horse's body will then find itself
flattened along a segment. The right thigh will attach along that segment
to its right shoulder, while the left thigh to the left shoulder. As for
the cusp point, don't look for it: you are simply sitting on it.
But all this is drawn... rather round.
Let us move on to a "polyhedral representation" of the cusp point
(just as a cube or a tetrahedron can be considered polyhedral representations
of a simple sphere). The bold stroke represents the
"self-intersection curve," which ends at the cusp point C.
Print these plates, it's better. In what
follows you will need to be able to recognize a cusp point in "different
configurations" and not confuse it with a simple vertex of the polyhedron. Build,
if you feel up to it, these different objects with cardboard, you will understand better.
Below we have an essential operation, called "creation-dissolution
of a pair of cusp points." The first drawing represents a kind
of cylinder that self-intersects along the segment drawn in bold,
whose cross-section recalls the Greek letter gamma, inverted. We then deform
this surface by "pinching" the tube whose cross-section resembles
an "inverted teardrop." In doing so, we make this
"teardrop" degenerate into a point S. After that, this point splits
giving rise to two cusp points. This is the operation of creating a
pair of cusp points. The opposite operation, on the contrary, destroys two
cusp points. Just below you will find the polyhedral version of this operation.
Here below is another polyhedral representation
of the transformation, which resembles what you will see forming
in the surface shortly.
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