Chapters 3 and 4 : The fourth dimension  

Mathematician Ludwig Schläfli talks to us about objects in the fourth dimension and shows us a procession of regular polyhedra in dimension 4, strange objects with 24, 120 and even 600 faces!

1. Ludwig Schläfli and the others

We hesitated for quite some time before choosing the presenter of this chapter. The idea of the fourth dimension does not come from just one person and many creative spirits were needed before it was definitively established and assimilated into mathematics. Among the precursors, one can cite the great Riemann who will present the final chapter and who had, without doubt, a very clear idea of the fourth dimension as of the middle of the nineteenth century.

But we called on  Ludwig Schläfli (1814-1895), mainly because this original mind is almost forgotten today, even amongst mathematicians. He is one of the first to have grasped the idea that even if our physical space seems to be of dimension 3, there is nothing to stop us imagining a space of dimension 4, or even proving geometrical theorems about 4 dimensional mathematical objects. For him, the fourth dimension was a pure abstraction, but after years of work, he must have felt more at ease in four  dimensions than in three ! His major work is the "Theorie der vielfachen Kontinuität", published in 1852. It must be said that few at the time realised the importance of this treatise. It was not until the beginning of the twentieth century that mathematicians understood the point of this monumental work. For more information on Schläfli, see here or here.

Even within the mathematical community, the fourth dimension maintained an aspect of mystery and impossibility for many years. To the general public, the fourth dimension often suggests science fiction full of paranormal phenomena, or sometimes,  Einstein's theory of relativity: "the fourth dimension is time,  isn't it?" However, this is confusing mathematical questions with those of physics. We will return to this briefly later. Let us first try to grasp the fourth dimension as Schläfli did, as a pure creation of the mind! 

2. The idea of dimension

Schläfli uses the blackboard to remind us of some of the things we have seen in the preceding chapters.  A line is of dimension 1 because to locate a point on it, one needs only one number. This is the  abscissa, or x-coordinate, of the point, negative on the left of the  origin, and positive on the right.

The plane of the blackboard  is of dimension 2  because to locate a point in this plane, one can plot two perpendicular straight lines on the blackboard and describe the position of points with respect to these two axes: they are the abscissa  and the ordinate (the x and y coordinates). For the space in which we live, one can supplement the two axes of the blackboard by tracing a third axis, perpendicular to the blackboard. Of course, it is rather rare to have chalk that plots straight lines which leave the blackboard, but as we are ready to go off into the fourth dimension, we shall need magic chalk!

Any point in space can then be described by three numbers noted traditionally x, y and z,  and that's why we say that space has three dimensions. One would of course like to be able to continue, but it is not possible to trace a fourth axis perpendicular to the previous three; this is not a surprise, because the physical space in which we live is of dimension 3 and we shouldn't look for the fourth dimension here, but rather in our imagination…

Schläfli suggests several ways in which we can get an idea of the fourth dimension. There is not just one single method, just as there are several ways to explain the third dimension to flat lizards. It's the combination of different methods which allows us to get a look into the fourth dimension.

The first method method is the most pragmatic. We can simply decree that a point in 4 dimensional space is nothing more than the set of data consisting of four numbers: x, y, z, t. The disadvantage of this approach it is that it is hard to visualise anything. But it is completely logical and the majority of mathematicians are happy with it. One can then try to copy the usual definitions in dimension 2 and 3 and try to define objects in the fourth dimension. For example, one can call (hyper-)plane the collection of  points (x, y, z, t) satisfying a linear equation of the form ax+by+cz+dt = e, copying the similar definition of a plane in space. With this kind of definition, one can develop a consistent geometry, prove theorems and so on. In fact, this is the only way in which to treat spaces of higher dimension seriously. But the aim of this film is not to be "too serious" but rather  to "show" the fourth dimension and  to explain the intuition that certain mathematicians have of it. 

Schläfli then gives us an explanation "by analogy". The idea is to carefully observe dimensions 1, 2 and 3, to notice certain phenomena, and then to suppose that these phenomena continue to exist in the fourth dimension. It is a difficult game and does not work all of the time. A lizard which leaves its world and enters the third dimension must expect some surprises and needs time to adapt. The same is true for the mathematician who climbs into the fourth dimension "by analogy "… Schläfli takes the example of the sequence "line segment, equilateral triangle, regular tetrahedron". One senses that there is an analogy between these objects, and there is no doubt that the tetrahedron in some way generalizes the equilateral triangle to dimension 3.

What then is the object that  generalizes the tetrahedron to the fourth dimension?

The segment has two vertices and it lies in dimension 1. The triangle has three vertices and it lies in dimension 2. The tetrahedron has four vertices and it lies in dimension 3. It is tempting to think that the sequence continues and that there is an object in 4 dimensional space that has five vertices and that continues the sequence. We can see that in the triangle and in the tetrahedron, there is an edge joining each pair of vertices. If one tries to join  five vertices to each other in pairs,  without thinking too hard about the space  in which one makes the drawing, one sees that ten edges are needed. Then, it is very natural to try to place triangular faces on each triplet of vertices. Again, one finds ten of them. And then, one continues by placing a tetrahedron on each quadruplet of edges. The object which we have just built does not yet have  a very clear status… we know the vertices, the edges, the faces and the 3 dimensional faces, but we do not yet see it very clearly. The mathematician speaks about combinatorics to describe what we know: we know which edges connect which vertices, but we still don't have a geometrical view of the object. This object, whose existence we have just guessed, and that continues the sequence segment, triangle, tetrahedron, is called a simplex!

3.  Schläfli's polyhedra

Polygons are drawn in the plane and polyhedra in ordinary 3 dimensional space. Similar objects in dimension 4 (or more!) are generally called polytopes  though they are often simply called polyhedra. 

Whereas Plato discussed the regular polyhedra in ordinary 3 dimensional space, Schäfli described the regular polyhedrons in dimension 4. Some are amazingly rich and the film proposes to show them to 3 dimensional  spectators  (you and me!) in the same way that the film showed the polyhedra of Plato to the lizards, rather than a pot of flowers or a book (admittedly, it would be very hard for the authors of the film to show you flowers in dimension 4, which is a pity!). Here we have one of  Schläfli's most beautiful contributions:  the complete and precise description of the six regular polyhedra in dimension 4. As they lie in dimension 4, they have vertices, edges, faces of dimension 2 and faces of dimension 3. Here is a table with the names of these polyhedra, the numbers of edges, of faces etc

This will be useful to help  appreciate their visualisations. For more information about polyhedra in dimension 4, see here or here, or also here.

4. "Seeing" in 4 dimensions

How does one "see" in 4 dimensions? Unfortunately we can't give you 4D glasses, but there are other ways.

The method of sections :

We begin as we did with the lizards. We are in our 3 dimensional space and we imagine that an obect moves through 4 dimensional space and progressively cuts our 3 dimension space.

The section is now in our space and instead of being a polygon which is deformed, it is now a polyhedron which is deformed. We can get an intuition of the shape of the 4 dimensional polyhedron by observing the sections as they slowly deform, ending up by disappearing. Recognising the object in this way is not easy, and even harder than for the lizards.

In the film, we get to know three of these polyhedra : the hypercube and those called the 120 and the 600. You see them cut through our space and show their sections which are 3 dimensional polyhedra which deform. Impressive ! But not easy to understand.

The image on the right shows the 600 going through our 3 dimensional space. 

As the fourth dimension is not easy to understand, it is not a bad idea to use several complementary methods. 

The method of shadows :

The other method we give in this chapter is almost more evident than the sections method. We could also have used it with the lizards. It is the technique of a painter who wishes to represent a landscape containing 3 dimensional objects on a 2 dimensional canvas. He projects the image onto the canvas. For instance, he can place a light source behind the object and observe the shadow of the object on the canvas. The shadow of the object only gives partial information, but if one turns the object in front of the light and if one observes the way in which the shadow changes, one can often get a very precise idea of the object. All this is the art of perspective.

Here it's the same thing : think of the 4 dimensional object that we want to  represent as lying in 4 dimensional space and that a lamp projects its shadow onto a canvas which is now our 3 dimensional space. If the object turns in 4 dimensional space, the shadow is modified, and we get an idea of the object even if we don't see it!

First we see the hypercube, much more clearly than we did with the sections.

Then  the 24, the object of which we think Schläfli was the proudest ! The reason is that this new arrival is really new ; it does not generalise any 3 dimensional polyhedron, as is the case for the other polyhedra. Moreover it has the wonderful property of being self-dual : for example, it has as many 2 dimensional faces as 1 dimensional faces (edges), and as many 3 dimensional faces as 0 dimensional faces (vertices). 

Finally we see the polyhedra 120 and 600 whose sections we have already examined. This new point of view shows us other aspects of these 4 dimensional polyhedra, which are indeed complicated. These two methods, sections and shadows, have advantages, but it has to be admitted that they do not do justice to all of the symmetries of these magnificent objects.

In the next chapter, we shall use another method, called stereographic projection ! Maybe that will help us to see more clearly ? 

5. "Seeing" in 4 dimensions :  stereographic projection

(cfr. the film,  Chapter 4 : the fourth dimension, continued)

Schläfli gives us one last method to represent polyhedra in 4 dimensions.  It consists of using quite simply stereographic projection. But of course this is not the same projection that Hipparchus showed us in chapter 1 !

Consider a sphere in  4 dimensional space. To define such a sphere, we use the usual definition : it consists of the set of points in this space which are all at the same distance from a point called the centre. We have seen that the sphere in 3 dimensional space is 2 dimensional, as its points are described by a longitude and a latitude. In some sense, the sphere in 3 dimensional space is only 2 dimensional because "it is missing a dimension" : the height above the sphere. In the same way, the sphere in 4 dimensional space is 3 dimensional, and it also is "missing" a dimension which is again the height above the sphere.

What is the sphere in the plane, i.e. in 2 dimensional space ? It's the set of all points at the same distance from a centre, otherwise known as a circle. A circle is thus a sphere in 2 dimensional space ! And it is clearly 1 dimensional as one number suffices to describe a position on a circle.

More surprising : what is a sphere in 1 dimensional space, that is, on a line ? The set of points at the same distance from a given point on a line. There are only two, one on the left and the other on the right ... The sphere in 1 dimension then contains only two points... Hardly surprising that we say that it is of dimension 0. 

To sum up : in n dimensional space,  the sphere is of dimension n-1 and it's for this reason that mathematicians use the symbol  S^n-1.

The beginning of the chapter explains what the sphere S³ is, but of course not even Schläfli can show it to us. The best he can do is to show you a sphere S² and encourage you to proceed as if you were in 4 dimensions and imagine the sphere S³ ... Stereographic projection as presented by Hipparchus projects the sphere S² onto its tangent plane at the south pole. One can proceed in exactly the same way with S³. One takes the tangent plane at the south pole of the sphere S³ which is a 3 dimensional space and one can then project any point of S³ (except the north pole) onto this space.  It suffices to extend the straight line from the north pole passing throught the point until it meets the tangent space to the south pole... Even if this takes place in 4 dimensions, the figure is analogous to what we have already seen.

Suppose then that Schläfli wants to show us one of these 4 dimensional polyhedra. He does what we have already done with the reptiles. He inflates the polyhedron until it is traced out on the sphere S³. Then he can project stereographically onto the tangent plane at the south pole, which is our 3 dimensional space, and we can observe this projection.

We can also roll the sphere S³ on its tangent plane and then project so that we can watch the polyhedron dance. Notice that when the rotation of the sphere takes a face of the polyhedron through the projection pole, the corresponding face has an infinite projection and we get the impression that the face explodes on the screen. We had the same impression in chapter 1 when polyhedra were projected on the plane.

This is the display proposed in chapter 4 : projecting Schläfli polyhedra stereographically and  turning them at the same time.

The geometry of 4 dimensional spaces is just the beginning, as there are also spaces of dimension 5, 6... and inifinite dimensional spaces ! Initially thought of as pure abstractions, modern physics makes great use of them. Einstein's theory of relativity uses a 4 dimensional  space-time. A point of this space-time is described by three numbers describing a position and by a fourth describing an instant in time.

But the force of the theory of relativity lies precisely in mixing these four coordinates in some way without trying to privilege time or space, and these thus lose their individual specifics. We shall not explain this theory here because Schläfli didn't know it ! Einstein's theory dates from 1905, well after the birth of the mathematical idea of 4 dimensions. This is not the first, nor the last, time that physics and mathematics interact fruitfully, each contributing its methods, with differing goals and motivations, yet so close... 

Besides, doesn't today's physics postulate the existence of spaces of dimension 10 or more, and doesn't quantum physics work with infinite dimensional spaces ? There will be a bit of a wait before we produce a film on spaces of dimension 10....