How to transform a crosscap
into a Boy surface (right or left, your choice) by way of the Roman
Steiner surface.
September 27 - October 25, 2003
page 1
All of this, as Kipling would say, is "sheer cunning
and downright magic."
I am retired but, I might say,
I still do a little research, in spite of myself, the way others line up
rows of knitting. If you have patience and get yourself
some 200 g squared Bristol board you will easily be able
to reconstruct all these models. My friend Christophe Tardy is in the process
of making an animation from this, which should be
not bad.
The Cross cap 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,
which requires that you have downloaded Cosmoplayer
onto your machine. Do it, it's really nice. Everything happens thanks
to the "cuspidal points." These points form quite naturally when
you mount a horse and squeeze your legs sharply. The horse's
body is then crushed along a segment. Its left thigh
connects with its right shoulder, while its right thigh
connects with its left shoulder. As for the cuspidal point, don't look
for it: you are simply sitting on it;
But all of this is ... rather plump. Let us move on to
a "polyhedral representation" of the cuspidal point (just
as a cube or a tetrahedron can be regarded
as polyhedral representations of a simple sphere).
The bold line represents the "self-intersection curve," finished
off, moreover, by the cuspidal point C.
Print these plates, that will be better. In what
follows you will need to recognize a cuspidal point in "different
configurations" and not confuse it with a simple vertex of the polyhedron.
Build these different objects out of
cardboard if you have the courage, you will understand them better. Below we have an essential
operation, called "creation-annihilation of a pair
of cuspidal points." The first drawing represents a sort of
cylinder that intersects itself along the bold line and whose cross-section
evokes the Greek letter gamma, upside down. This surface is then
deformed by pinching the tube, whose cross-section has the shape of an
"upside-down teardrop." This teardrop degenerates
into a point S. Then this point splits in two, giving two cuspidal
points. This is the creation of a pair. The inverse operation
causes two cuspidal points to annihilate each other. Below you will find the polyhedral version
of the operation.
Below, another polyhedral representation
of the transformation, which is close to what you will see happening in the
surface later on.
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