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Let's come back to the 2-dimensional sphere and its parallels.
Above each point of the 2-dimensional sphere,
we should imagine a Hopf circle.
Look at what is above one of the parallels of S^2,
the equator for instance.
Here is what is above another parallel
which is moving southwards.
Why does the torus seem to get thinner?
Because above the south pole,
there is of course only one circle.
and above the north pole, one sees a straight line,
actually a circle, going through infinity, this is the red line!
Well, let us spin all this around now.
Rotations, yes, but
rotations in 4-dimensional space of course.
To be honest, I have to say that some of these pictures
were already known long before me.
The existence of four families of circles on the torus
is usually attributed to the Marquis de Villarceau
but one finds earlier clues,
in a sculpture in Strasbourg cathedral for instance.
Take a torus of revolution:
this is the surface described by a circle
rotating around an axis in its plane.
Look at the section of the torus by a plane.
Notice how I chose the plane.
One says that it is bitangent to the torus,
simply because it is tangent at two points.
Now look carefully:
the plane cuts the torus along two perfect circles.
This is Villarceau's theorem:
a plane which is bitangent to the torus cuts it along two circles.
Of course, there is not just one bitangent plane.
Here is another one, cutting the torus along two other Villarceau circles.
And one can do the same for all other bitangent planes:
we just need to rotate around the axis of symmetry.
You see, through each point on a torus of revolution
one can draw four circles,
obtained by suitable slices.
One of these circles is a parallel,
another is a meridian,
then a first Villarceau circle
and a second one.
And since one can do this at any point of the torus,
we see that the torus is covered by four families of circles.
Two circles of the same family do not intersect.
A blue circle intersects a red circle in a single point.
A yellow circle and a white circle intersect in two points:
these are the Villarceau circles.
Take a good look at the yellow circles:
these are Hopf circles!
Remember when we looked at
what is above a parallel in the fibration?
We saw a torus covered with linked circles,
just like this torus covered with yellow circles.
And what about the white circles?
Well, they are the fibres of another Hopf fibration!
..the mirror image of the first one.
To finish our stroll,
we will take a torus of revolution,
with its four families of circles;
imagine it in the 3-dimensional sphere,
rotate the torus inside the 3-dimensional sphere,
and finally project it stereographically
onto 3-dimensional space.
In this way, we obtain surfaces
that are also covered by four families of circles:
the so-called Dupin cyclides.
Sometimes, when the torus passes through the projection pole;
the surface becomes infinite...
In this movement, the two faces can even be switched.
The inner face of the torus is pink and the outer one is green.
A simple rotation in the fourth dimension and …bingo!
green turns into pink and pink into green.
Isn't that magnificent?