Italian
: Andrea Sambusetti, University of Rome
Click here
to display the 1:1 scale drawing of the model, to print and cut out.
By photocopying four copies on Bristol cardboard of two different colors,
you can build the model yourself,
following the instructions for assembling it.
You have certainly seen a strange object rotating endlessly
on the left side of the homepage of this site. What is it?
One day, when I find the time, I will install
on this site a description of the sphere inversion, as
I had illustrated 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 "inverting a sphere" mean? A sphere does not have the same
meaning for the common man and for the mathematician-geometer. For the common man,
it is simply the set of points in space located at a distance
R from a fixed point O. A geometer will continue to call "sphere",
instead, even an object that corresponds to a "deformed sphere",
like a potato for example. To better understand these concepts,
get the Lanturlu CD containing the comic "Topologicon".
But the mathematician goes even further. A surface is
called "regular" when in each of its points a tangent plane can be defined.
This already allows to think of an infinity
of possible regular deformations of the sphere, in the infinite possible shapes
of a potato, varying furthermore in an arbitrary way the area of such surface.
Having said that, in our physical universe a person trying to invert
the sphere (i.e., to bring its inner surface to the outside) will
face the impossibility of making its surface self-intersect. When this hypothesis
is assumed, i.e., when self-intersections of the surface are prohibited,
or even just touching, the mathematician speaks
of embedding of the sphere S2. But a mathematician always allows everything.
A sphere is, for him, a "virtual" object and not material,
where the crossing of a surface is considered possible. The sequence
of drawings below shows a sphere that self-intersects. A representation
of this kind, which allows self-intersections, is called
an immersion.
An immersion therefore has a set of self-intersections
(here it is a simple circular curve). However, the tangent plane must vary
continuously. With this in mind, when looking at the drawing above,
it is clear that the operation brings a part of the inner surface
(represented in green) to the outside. To complete the inversion,
one would have to flatten this kind of equatorial tube. Here 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 has only one class of immersions". This cryptic sentence
had as a corollary the fact that one should be able to pass, through a transformation
containing only true immersions, from the "standard" sphere
to its "antipodal" representation, i.e., in which each point
is exchanged with its antipodal: in other words... an inverted sphere.
Raoul Bott was Smale's advisor. Although the formal proof of this fact
seemed correct, no one seemed able to actually carry out this inversion operation.
Bott kept asking Smale "show me how you would proceed";
to which Smale, notoriously without any hesitation, replied "I have no idea".
Smale later received the Field Medal, the equivalent of the Nobel Prize for mathematics.
Incidentally, you may wonder why there is no Nobel Prize for
mathematics. The answer is simple: his wife ran off with
a mathematician.
Things remained like this for many years,
until an American mathematician named Anthony Phillips published
in 1967, in Scientific American, a first version of this inversion,
extremely complicated. The second was invented in the early 1970s
by the French mathematician (blind) Bernard Morin. I was the first to draw
the sequence of transformations, which will be the subject, as I announced,
of a forthcoming article on this site, which is already quite full. Anyway, all
this leads us to a consideration. Surfaces can be represented in polyhedral form.
A cube or a tetrahedron can be considered as polyhedral representations of the sphere,
in the sense that these objects have the same topology. On this point, consult my Topologicon. Moreover
it is understood that, if it is possible to invert the sphere, it will also be
possible to invert 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 sequence. It is what
I call the "central model with four ears". I am anticipating things.
Anyway, as the sphere lends itself to polyhedral representations, the same
applies to the subsequent steps of this transformation. What you see rotating
on my homepage is the polyhedral version of the central model
of the sphere inversion, which I invented about ten years ago. The interest
of such polyhedral models lies in the fact that they can be constructed with
flat surfaces. They can also be constructed 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 correct size).
Large
It is an assembly plan, here shown in general view.
But for printing, it is preferable to go to the cutting page.
Print it. Then, with this copy printed on the usual
paper of your printer, photocopy it four identical copies, two on green
Bristol cardboard, and two in yellow. You will be able to build the central
model of the cube inversion with these cut-out sheets.
On the cut-out elements there are pairs of letters:
a, b, c, d, e, f, etc. It is sufficient to fold the sheet bringing
the same letters together, and then fix the faces with transparent tape.
The following drawings show the way to assemble one of the four pieces.
Here is first how one should begin to fold one of the four elements:
Here are two of the four elements, viewed from different angles.
They are then arranged to form an object
with four-fold symmetry, alternating green and yellow elements.
To see it in 3D, look at Tardy's realization, in the section
"virtual reality". The central model is assembled and also
realized in "vrml" in this section. Here it is reproduced from various
points of view:
It cannot be said that one viewpoint corresponds to
"above" and the other to "below", since these
designations are perfectly arbitrary. In the left image, the point
"center" corresponds to the "double point" (where two surfaces
intersect) of Morin's central model, while the central point of the image
on the right corresponds to the "quadruple point" of the same model (where
four surfaces intersect). I had to carefully orient the object,
so that the left figure does not evoke a swastika. Apart from that,
from an architectural point of view, this polyhedral representation of the central model...