Italian
: Andrea Sambusetti, University of Rome
Click here to make
the drawing of the 1:1 scale model appear, to be printed and cut out.
By photocopying four copies of it onto Bristol board of two different
colors, you will be able to build the model yourself,
following the instructions for assembling it
You have certainly seen a bizarre object rotating tirelessly
on the left side of the home page of this site. What is it?
One day, when I find the time, I will install
on this site a description of the eversion of the sphere, just as
I had illustrated it in the January 1979 issue of Pour la Science, that is...
22 years ago! All this will require many details and an introduction.
What does "everting a sphere" mean? A sphere does not have the same
meaning for the ordinary person and for the mathematician-geometer. For the ordinary person,
it is nothing other than the locus of points in space situated at a distance
R from a fixed point O. A geometer will continue to call "sphere,"
however, even an object that corresponds to a "deformed sphere,"
like a potato for example. To understand these concepts more precisely,
get the Lanturlu CD containing the comic strip "Topologicon."
But the mathematician goes a bit further still. A surface is
called "regular" when at each of its points one can define
a tangent plane. This already allows one to think of an infinity
of possible regular deformations of the sphere, into the infinite possible shapes
of a potato, while also varying in an arbitrary manner the area of such a surface.
That said, in our physical universe a person who tries to evert
the sphere (that is, bring its inner surface to the outside) will find
himself faced with the impossibility of managing to make its
surface self-intersect. When one assumes this hypothesis, that is, one forbids the surface
from self-intersecting or even just from "touching" itself, the mathematician speaks
of an embedding of the sphere S2. But a mathematician always allows himself everything. A
sphere is, for him, a "virtual" and non-material object, in
which the crossing of one sheet is considered possible. The succession
of drawings below shows a sphere that self-intersects. A representation
of this kind, which is to say one that admits self-intersections, is called
an immersion.
An immersion therefore possesses a set of self-intersection
(here it is a simple circular curve). The tangent plane, however,
must vary continuously. That said, when one looks at the drawing
above, one clearly sees that the operation brings a part of the inner surface
(represented in green) to the outside. To complete the eversion, one would have
to flatten this sort of equatorial tube. Here there seems to be a problem:
this flattening would destroy the continuity of the tangent plane,
and such a transformation would therefore contain a step that is not an immersion.
One day an American mathematician, Stephen Smale, proved
that "the sphere S2 possesses only one class of immersions." This sibylline
sentence had as a corollary the fact that one should be able to pass,
by means of a transformation containing only true immersions, from the "standard"
sphere to its "antipodal" representation, that is, one in which every point
is exchanged with its antipode: in plain words... an everted sphere.
Raoul Bott was Smale's boss. As much as the formal proof of this fact
seemed correct, so much did no one seem capable of concretely carrying out
this eversion operation. Bott kept asking Smale "show
me how you would think of proceeding"; to which Smale, notoriously
outspoken, replied "I haven't the faintest idea."
Smale later obtained the Field Medal, the equivalent of the Nobel for mathematics.
Incidentally, you may perhaps wonder why there is no Nobel prize for
mathematics. The answer is simple: his wife ran off with
a mathematician.
Things stayed like this for quite a while,
until an American mathematician named Anthony Phillips published
in 1967, in Scientific American, a first version of this eversion,
extremely complicated. The second was invented at the beginning of the seventies
by the French mathematician (blind) Bernard Morin. I was the first to draw
the succession of transformations, which will be the subject, as I announced to you,
of a forthcoming article on this site, which is moreover quite abundant. Anyway, all
this leads us to a consideration. Surfaces can be represented
in polyhedral form. A cube or a tetrahedron can be considered
polyhedral representations of the sphere, in the sense that these objects have
the same topology. On this point, consult my Topologicon. Moreover
one understands that, if it is possible to evert the sphere, it will likewise
be possible to evert a cube. The transformation invented by Bernard Morin (which
I illustrated in the January 1979 article in Pour la Science) passes through a
central model. There is a symmetry in this succession. It is what
I call the "central model with 4 ears." I am getting ahead of things.
Anyway, just as the sphere lends itself to polyhedral representations, so too
does each successive step of this transformation. What you see rotating
on my home page is the polyhedral version of the central model
of the eversion of the sphere, which I invented about ten years ago. The interest
of such polyhedral models lies in the fact that they can be built with
flat surfaces. They can also be built with paper and scissors. Take a look
at the drawing below (I thank in parentheses my friend Christophe Tardy,
who produced the elements of the right size).
It is an assembly plan of which you have here a general
view. But to print it, it is preferable that you go to the découpage page.
Print it. Then, equipped with this copy printed on the usual
paper of your printer, photocopy 4 identical copies of it, two onto green Bristol
board, and two in yellow. You will be able, by means of these sheets to
cut out, to build the central model of the eversion of the cube.
On the elements to be cut out there are pairs of letters
: a , b c , d , e , f etc.... It is enough to fold the sheet bringing
the same letters to coincide, and then fix the faces with transparent
adhesive tape. The drawings that follow show the way to assemble one of the four
pieces. Here first of all is how one must begin to fold one of the four elements
:
Here are two of the four elements, seen from different angles.
They are then arranged so as to give rise to an object
with a symmetry of order four, in which green and yellow elements alternate.
To see it in 3d, take a look at Tardy's realization, in the
"virtual reality" section. The central model is assembled and also
realized in "vrml" in this section. Here it is reproduced from various
points of view :
One cannot say that one point of view corresponds
to the "top" and the other to the "bottom," since these
designations are perfectly arbitrary. In the image on the left, the "central"
point corresponds to the "double point" (where two sheets
cross) of Morin's central model, while the central point of the image
on the right corresponds to the "quadruple point" of the same model (in
which four sheets cross). I had to orient the object very carefully,
so that the figure on the left does not evoke a swastika. Apart from that,
from the architectural point of view, this polyhedral representation of Morin's
central model could have constituted a fine project for a National Socialist
House of Culture.
One last observation: there is no good polyhedral representation
of the eversion of the sphere (or of the cube). By "good"
I mean a succession of sufficiently explicit models that can be described
in the form of sheets to cut out in a relatively easy way, like the model
above. There would be a study to be done in this direction, within anyone's reach,
even a non-mathematician, for example a sculptor. More than twenty years ago
I was a sculpture teacher at the Ecole des Beaux-Arts in Aix en Provence, at
the time when my dear friend Jacques Boullier was director. It was in those premises
that the first meridian representation of the Boy surface was born by means
of ellipses, the key to the construction of the first implicit equation given by Apéry.
I must say that already at the time I was surprised by the geometric imagination
of the art students, which often surpassed that of... geometers.
Counter installed on December 31, 2001. Number of
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