Circles in space that are arranged so as to create beautiful ornaments. In order to understand better the three-dimensional sphere in four-dimensional space,

I will show you how to fill the space with circles and thus create what mathematicians call a fibration. By the way, my name is Heinz Hoppe and I am one of the main contributors to the development of topology

during the first half of the 20th century. Look at this toric surface, filled with circles that appear to be linked. Let me explain this picture to you.

Circles, spheres and tori are among the simplest objects studied by topologists. A topologist tries to understand the connections between these objects. I worked in Berlin, Princeton and Zurich and one still comes across my name often in contemporary mathematics.

Poincare-Hoppe theorem, Hoppe fin variant, Hoppe algebra, Hoppe fibration. Let me paint my portrait for you. I published the discovery of my fibration in 1931.

But as always, I have to say that I relied upon many predecessors like Clifford for instance, who you see here and who worked in England during the 19th century.

Let's begin with some explanations on a blackboard. Well, a whiteboard this time. What do you see? A two-dimensional plane?

Well, yes and no. This is indeed a two-dimensional plane but it is a plane of complex dimension too. Or in other words, a space with real dimension for. Go on, make an effort. Each point in this plane is determined by two coordinates but each of these two coordinates is a complex number

which, remember, is itself defined by two real numbers. Each of the axes is a complex line so that each point on these axes has one coordinate which is a complex number. For instance, here you see the point 2-i on the first axis.

The same is true for the other axis, the y-axis. Here we can see the point 1-2i on this axis.

Now our whiteboard is magical but not enough to be able to show us the two planes simultaneously. If we try to depict them in three-dimensional space they will intersect along a line but in four-dimensional space they intersect only at the origin.

After all, they are axes. Now what do you see? A circle? Yes and no. What you see, or rather what you should imagine is the set of points in four-dimensional space

that are at distance 1 from the origin. In other words, this is nothing other than the 3-sphere S3. Well, of course you need to have a little imagination.

Let's try to see at least how this sphere intersects the first axis. The 3-sphere intersects the first axis in a set of points on this axis which are at distance 1 from the origin.

You see, the 3-sphere intersects the first axis in a circle. The same is true for the second axis which intersects the 3-sphere in a circle as well

the blue circle. Now what is true for the horizontal line and the vertical line is equally true for all lines going through the origin.

Here you can see the line with equation z2 equal to minus 2 z1 but we could do the same with any line z2 equal to a times z1

for any complex number a. In this manner the 3-sphere in four-dimensional space is filled with circles one for each complex line going through the origin in our plane of complex dimension 2. Careful though, in the picture you get the impression

that the red circles intersect each other but this is not the case in the reality of dimension 4. Lines only meet at the origin so their intersections with the unit sphere don't intersect at all in fact. I was the one who discovered this decomposition of the sphere into circles

and ever since it is known as the Hopf vibration. Why vibration? Well you should think of the fibres of fabric. We are going to look at all that using stereographic projection. Imagine that we project the 3-sphere from the North Pole onto the tangent space at the South Pole

which is our three-dimensional space. Here is the projection of one of the circles which as we have seen is the intersection of one complex line and the 3-sphere. But there are many such circles one for each complex line going through the origin. For each complex number a

we can consider the line z2 equal to a times z1 and its associated circle. Let us vary this number a or what will amount to the same thing. Let us rotate this line in order to see how the circle changes. Notice that sometimes the circle appears to be a straight line but this is simply because it passes through

the North Pole of our 3-sphere. Let us look at two of these circles simultaneously. In the lower left hand corner there are the two moving complex points one red, the other green.

You can see the circles associated to the red and green points. Notice that these two circles are linked together like two links of a chain. It is impossible to separate them without breaking them.

For the fun of it let us consider three circles. Look at the dance of these three linked circles.

Now let us take many more complex lines chosen randomly and let us look at them all at once.

The circles fill up the space and no two of them intersect. This is an example of a fibration.

Let us try to understand this better by returning to the board for a moment. Look, we have a Hopf circle for each line. Each one of these lines has an equation of the form Z2 equal to A times Z1 where A is a complex number

the slope of the line and is indicated by the red point moving on the green line. Actually, the vertical axis does not have such an equation but in this case we may see that A is infinite. Do not forget that A is a complex number. The green line is also a complex line.

So it is a real plane of course. Summing up, the complex lines that we are interested in are completely described by a point on the green line and an additional point at infinity.

But we already saw that if one adds a point at infinity to the complex line

we get the usual two sphere. Once more, this is stereographic projection.

So the complex lines that interest us are described by points on the yellow sphere the two dimensional sphere S2. So we have a circle for each point on the two sphere.

But a circle is a sphere of dimension one, isn't it? All these circles fill up the three sphere. Each point on the three sphere belongs to a single circle and therefore defines a point on the two sphere.

In this way we get a projection from the three sphere to the two sphere. Complicated, isn't it? Mathematicians say that above any point of the base S2 there is a fibre which is a circle S1

and that the total space of this vibration is the sphere S3. I am very proud of my vibration all the more so because it has become a fundamental object in topology.