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 4
Let us present the model again from
another point of view:
Plate 14: we always repeat the same
operation, creating the third "ear" of the self-intersection curve.
In the polyhedral model, the latter has the form of three squares with a vertex
in common: the triple point T .
Plate 15: by rotating the object you will find again
the polyhedral version of the Boy surface that I had presented in the Topologicon
(where you can also find an assembly plan that allows you to build it).
Last plate: I have tried to illustrate
the Steiner surface as it twists and transforms into a Boy
surface.
We see that, drawn in "rounded" form,
it takes quite a bit of practice to understand it. Our eye is very uncomfortable
when it comes to understanding an object for which, along one and the same line
of sight, more than two sheets overlap. Hence the interest of the polyhedral
model, which brings within anyone's reach, if only one tries to build
the little models oneself, transformations considered complicated in geometry.
Let us note in passing that, depending on the pairs of cuspidal points chosen,
one obtains a "right" or "left" Boy surface
(completely arbitrary definitions). The projective plane immerses itself into
space by means of two mirror "antiautomorphic" representations. We see
therefore also that one can pass from a right Boy surface to a left
Boy surface through a "central" model which is the Roman
Steiner surface.
It would undoubtedly be nice if these
drawings were published in magazines like Pour la Science or La Recherche.
But for twenty years publication in these magazines has been "forbidden" to me
because of ufological deviationism. Thank you, Messrs Hervé This and Philippe
Boulanger. I have lost count of the articles of this kind that I have proposed
to these magazines and that have been kindly rejected. One ends up getting used
to one's status of excommunicate.
As an anecdote, there exists a "d'Alembert prize" intended to reward the authors of books of mathematical
popularization. The story was told to me by a member of the commission
appointed to decide to whom the prize should be awarded (there are anyway
questions of money behind it). Dialogue:
-
In short, why don't we give the prize to Petit? He has written remarkable works
like the "Géométricon", the "Trou Noir" and the "Topologicon". -
Yes, but he hasn't done only that.
-
What are you alluding to?
-
He also wrote the "Mur du Silence".
-
Ah, well, then....
Well yes, the "Mur du Silence",
published in 1983, is an album dedicated to MHD. And, as each of us
knows, this corrosive science has as its merit, or defect, that of allowing flying saucers to move
at supersonic speed without making a Bang.
« Cachez cette science, que je ne saurais voir »
I have in my boxes a magnificent version
of the "turning inside out of the cube", which is not the polyhedral version of the
Morin variant. All my own work. One of these days....
October 22, 2003: People are not rushing too much over these pages, if I am to
believe the counter. On Monday October 13, 2003 I gave a seminar at the CMI
(Centre de Mathématiques et d'Informatique of Château-Gombert-Marseille) at the invitation
of Trotman. On that occasion I was able to bring out a collection of about
thirty cardboard models, of which one day you will be able to enjoy the first fruits,
since they were photographed by Christophe Tardy.
When one gives a seminar,
a certain atmosphere is created. In the photo below, here is a geometer expressing
his perplexity.
In the background, part of the little models
displayed with the help of my long-time collaborator, Boris Kolev, a member
of the department, also a geometer. At a certain moment I asked the question:
- How many of you have already seen a Roman Steiner surface? Raise
your hand.
No one had ever seen one. It therefore seemed
useful to me to present this object, with a virtual reality program,
on the laptop I had with me, a program made with the help of Christophe
Tardy, an engineer, and of Frédéric Descamp, of the Laue Langevin Institute of Grenoble
(ILL). Obviously this presentation disconcerts the audience, little accustomed
to seeing mathematical surfaces twirl about at will.
Two cardboard plates, visible in the
foreground, made it possible to present the whole succession of little models
in their logical order. The green and yellow models illustrate, in polyhedral form,
the essential tool for creating-dissolving a pair of cuspidal points.
The most distant white object is a polyhedral version of the Cross
Cap surface, which transforms first into a polyhedral version of the Roman
Steiner surface, then, a meter further on, at will, into a "right"
or "left" Boy surface.
The analysis of the little models brings out
various observations in the audience. One of the geometers asks:
- If it is true that, by following the little models in this order, one can pass
from the Cross Cap surface to the Boy surface, it seems that, by following the opposite procedure,
one can transform a Boy surface into a Cross Cap.
I answer in the affirmative. Emboldened,
my interlocutor adds:
- Then, if once the stage of the Roman Steiner surface is reached one stops,
it should be possible to return to a Boy surface, but reflected
with respect to the initial one.
I agree a second time. But, unfortunately,
no one will volunteer to give some clarification on this strange world in
which immersions of closed surfaces are allowed to have cuspidal points, created or dissolved in pairs, whose set constitutes a kind of
extension of the world of immersions. The term "summersion"
seems suitable to me. If a reader is capable of giving some clarification,
he is welcome.
Curvature concentrated at a cuspidal point.
We will calculate it by summing the angles
at the vertex and comparing this sum to the result obtained in the case
of the Euclidean plane: 2 p .
At the top left you can see one
of the many possible polyhedral representations of a cuspidal point. By "dismantling"
the surface one arrives at a sum of angles that exceeds the value 2p
by 2a . One deduces that the angular curvature concentrated
around this point C is - 2a. If
the angle a is equal to p/2 then the negative curvature is -p (figure at bottom left). In fact,
the curvature of a cuspidal point can assume infinitely many values. At the bottom
right we accentuate the angular sum and the curvature then becomes < -p (we have increased the negative curvature).
Operating in the opposite way, we can
reach a rather surprising situation: we can arrange
for the (angular) curvature concentrated at C to be ... zero:
Let us now start from a polyhedral representation
of the Cross Cap surface in which two cuspidal points appear,
each of curvature equal to -p
:
In this figure there are eight "posicones"
corresponding to the value + p/2. We add
four other "posicones" of curvature + p/4 and four "negacones" of curvature
- p/4.
Plus the two cuspidal points of curvature
- p .
Total: 2 p
Dividing the value of this "total
curvature" by 2p we find again the
value of the Euler-Poincaré characteristic of any representation
of the projective plane (or of the Boy surface).
During the lecture I evoked
art and the way of permuting the two cuspidal points of a Cross
Cap surface by using the turning inside out of the sphere. I no longer know
if I put this thing somewhere on my site. It is such a muddle. I will have to search,
otherwise I will add it. It is amusing. The fact is that this operation did not please
at all one of the people present at the seminar:
- I don't see why Petit makes use of so much paraphernalia to demonstrate the
symmetry that links the two cuspidal points of a Cross Cap. One can do it much
more simply.
And he drew on the blackboard the drawing of a
sphere squeezed between two rulers that touch and that effectively gives
a self-intersection set in the form of a segment at whose ends there
are two cuspidal points, like the Cross Cap surface. Unfortunately,
and the gentleman in question noticed it, this is not the Cross Cap surface.
- Devil, but what is it then? asked someone.
It is simply the immersion of a
sphere, with two cuspidal points. If one makes them converge toward a single point
one obtains a line of self-intersection that becomes a circle. And one obtains
(at the bottom right) an immersion of the sphere that we only have left to transform
into its standard embedding. We can also give a polyhedral representation
of this surface:
It is a two-sided surface
whose total curvature is 2p.
In short, one can have quite a lot of fun
with these "summersions". Let us consider an immersion of a torus
obtained by making the "infinity" symbol rotate around an axis
:
The technique of making cuspidal points
converge into a single point allows us to arrive quickly at the standard
embedding of the torus, as explained above in the successive drawings.
But things do not always turn out so
easy and obvious. Let us take for example a sphere squeezed between two segments
that, this time, are shorter than the diameter. We again obtain two cuspidal
points.
Since this surface contains a
Moebius strip, it is one-sided. We have placed next to it a polyhedral representation
of it that allows its total curvature to be calculated. We then find
zero. If I am not mistaken, it should then be a Klein bottle. Of it
generally only the classical immersion is known, in which the line of
self-intersection is a simple circle. But there are others, like this one here.
I confess that I have not yet found the way to transform it into a usual Klein
bottle. On the other hand, I do not even know whether this "immersion" and
the classical one lie in the same homotopy class (for the sphere, for
example, there is only one). A priori, it is not a given: the torus can in fact be
immersed in four different ways in three-dimensional space, which cannot
be transformed into one another by means of a regular homotopy. While waiting
to discover whether or not it is possible in this case, I have amused myself by transforming it
by creating two more cuspidal points, thus obtaining two Cross Caps connected
by a tube. By decomposing them, one finds again that the Euler-Poincaré
characteristic is zero.
This strange surface should transform
into one of the 4 possible immersions of the Klein bottle, but which one? In
any case, here is one obtained by making an 8 rotate around an axis,
while it itself meanwhile performs a half turn on itself:
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