How can we draw the 4th dimension

Chapters 3 and 4: The Fourth Dimension

The mathematician Ludwig Schläfli speaks about objects in the fourth dimension and shows us a series of regular polyhedra in the fourth dimension; strange objects with 24, 120 and even 600 pages!


1. Ludwig Schläfli and the other

We have doubted much to choose an announcer for this chapter. The idea of ​​the fourth dimension is not the result of a single man and it took many creative minds to understand and justify it in mathematics. We can mention Riemann among the forerunners. He will be the announcer of the ninth chapter and no doubt had a very precise idea of ​​the fourth dimension since the mid-19th century.

We gave the floor to Ludwig Schläfli (1814-1895), partly to remember this original spirit, which is now almost forgotten even among mathematicians. He was one of the first to realize that we can imagine four-dimensional space, and that we can prove geometry theorems regarding four-dimensional mathematical objects, even if our physical space looks three-dimensional. For him, the fourth dimension was a pure abstraction, but there is no doubt that after years of investigation he felt better in the fourth dimension than in the third dimension. His main work is entitled "Theory of Multiple Continuity" and was published in 1852. It must be admitted that few readers rated the importance of this book in its time. It has been necessary to wait until the beginning of the twentieth century for mathematicians to understand the usefulness of such a great work. For more information about Schläfli, see here and there.

Even among mathematicians, the fourth dimension has long been kept mysterious and impossible. For the general public, the fourth dimension often evokes science fiction novels in which parapsychological phenomena occur, or Einstein's theory of relativity: "the fourth dimension is time, isn't it?" That means confusing mathematical and physical questions. We will take up the topic again recently. First we try to understand the idea of ​​the fourth dimension, as Schläfli makes, as the work of the spirit.

2. The idea of ​​dimension

Schläfli reminds us of things we saw in the previous chapter, and for that he uses a blackboard. A straight line has 1st dimension because the position of a point on a straight line is only described by a number. It is the abscissa of a point with a minus if the point is to the left of the origin or a plus if it is to the right of it.

Schläfli reminds us of things we saw in the previous chapter, and for that he uses a blackboard. A straight line has 1st dimension because the position of a point on a straight line is only described by a number. It is the abscissa of a point with a minus if the point is to the left of the origin or a plus if it is to the right of it.

The board plane is two-dimensional because you can draw two vertical lines so that each point can be described by two numbers: they are the abscissa and the ordinate. In the room where we live, we can draw a third straight line that is perpendicular to the board to complement the other two axes. Well, it's a bit strange to have a piece of chalk that writes straight lines from the table, but, because we want to leave for the fourth dimension, we need magical chalk!

A point in space is therefore described by three numbers, which are almost always called x, y and z, and that is why our space has three dimensions. Of course it would be great to be able to continue like this, but it is not possible to draw a fourth axis perpendicular to the previous three axes. This is not a surprise because our physical space has 3rd dimension, and we are not looking for the fourth dimension there, but in our imagination ...

Schläfli suggests a few solutions so that we can form an idea about the fourth dimension. There are different methods, as there were also many methods, to explain the plane lizards the third dimension. These methods will allow us to glimpse the fourth dimension.

The first method is the most pragmatic. We can simply say that a point in four-dimensional space is given exactly by the four numbers x, y, z and t. This is not particularly illuminating, and it is a disadvantage, but it is quite a logical process and the majority of mathematicians are happy with it.
We can also copy ordinary definitions in 2 and 3 dimensions in order to be able to describe four-dimensional objects. For example, we can copy the definition of plane, and then we call the set of all points x, y, z, t that correspond to the linear equation ax + by + cz + dt = e a (hyper) plane. With these definitions one can develop a fixed geometry, prove theorems, etc. It is actually the only way to handle high-dimensional spaces seriously.
The aim of this film is not to be “too serious”, but to employ the fourth dimension, as some mathematicians suspect.

Schläfli shows us a method that uses the analogy. The idea is to examine dimensions 1, 2 and 3 in order to note certain phenomena, and then one has to suspect that these phenomena also occur in the fourth dimension. It's a difficult game that doesn't always work. A lizard that escapes its plane and enters three-dimensional space will be surprised and will need time to get used to it. The same thing happens with a mathematician who enters with analogies in four-dimensional space ... Schläfli's example is the result: “Straight line segment, isosceles triangle, tetrahedron”. There is an analogy between these objects; it is clear that the tetrahedron is the equivalent of the isosceles triangle in the 3rd dimension.

And then, which object corresponds to the tetrahedron in the 4th dimension?

The straight line segment has two corner points and is in the 1st dimension. The triangle has three corner points and it has a 2nd dimension. The tetrahedron has four corner points and it has a third dimension. One tends to think that there is an object in the 4th dimension that has five vertices and that the series continues. We also see that with the edges, the triangle and the tetrahedron, two corner points are each connected by an edge. We would have to connect the 5 corner points in pairs, but let's not worry about which room we make the drawing in, and then we pay 10 edges. Now the sides: in our object, each triple of corners borders a triangular side. We also see ten. Now we also need to insert a tetrahedron for every quadruple of corners. The finished object is not very clear ... We know its corners, edges, sides, three-dimensional sides, but we cannot see it. The mathematician talks about combinatorics, with which he describes what we have: we know which edges connect which corners, but we have no geometrical representation of the object. The object whose existence we have predicted and which continues the sequence “straight line segment, triangle, tetrahedron” is called “simplex”.

Clicking on the picture starts the film.

3. Schläfli's polyhedra

The polygons are drawn in the plane and so are the polyhedra in space. The equivalents in the fourth (or more!) Dimension are generally called polytopes, although they are often called polyhedra too, without further ado.
  

As soon as Plato was talking about the regular polyhedra in space, Schläfli described all the regular polyhedra in the 4th dimension. Some have an unimaginable richness that the film tries to show the three-dimensional audience (all of us). When he showed the Platonic polyhedra to the lizards, he was more likely to present a bouquet of flowers or a book (frankly, the authors of the film would like to show us a bouquet of flowers in dimension 4, but unfortunately that is not possible!) Here is one of Schläfli's most beautiful contributions: the precise description of the six regular polyhedra in the fourth dimension. Since they have 4th dimension, they have corners, edges, sides and three-dimensional sides. In the table below you can read the names of each polyhedron, as well as its corners, edges, planes and three-dimensional sides.
  

Simple nameSurnameCornersedge2D sides3D pages
simplexPentachoron51010 triangles5 tetrahedra
HypercubeTesseract163224 squares8 dice
 16Hexadecachoron82432 triangles16 tetrahedra
24Icositetrachoron249696 triangles24 octahedron
120Hecatonicosachoron6001200720 pentagons120 dodecahedron
600Hexacosichoron1207201200 triangles600 tetrahedra

This will be very useful to illustrate to you, see here or here, or here too.

4. "See" in the 4th dimension

How can we see in the 4th dimension? Unfortunately we don't have “4D glasses”, but there are other ways.

The method of cuts:

At first we can pretend like the lizards. We are in our three-dimensional space and we imagine that an object gradually penetrates our space.

Now the section is not a polygon, but a polyhedron that deforms. We can make an intuitive estimate of the shape of the polymer. To do this, we have to observe the cuts that gradually deform and finally disappear. It is not easy to grasp the object: it is more difficult for us than for the lizards ...

In the film we familiarize ourselves with three polyhedra: namely the hypercube and the so-called “120” and “600”. They cut out our space and put out the cuts which are deforming three-dimensional polyhedra. Impressive! Still, it's not easy to understand.

On the right you can see the "600" that penetrates our three-dimensional space.

Clicking on the picture starts the film.

Since it is not easy to understand the fourth dimension, it is not useless to use complementary methods.

The method of shadows :

The other method shown in the film is more understandable than that of the cuts. We could have used them with the lizards too. This is done by the artist who wants to present a three-dimensional landscape with his two-dimensional canvas. He projects the image onto his screen. For example, he can put a bundle of rays behind the object and observe the shadow on his canvas. The shadow provides only incomplete information about the object, but if we rotate the object in the light, then we see the deforming shadow and we can build up a very precise picture of the shape of the object. That is the art of perspective.

It's the same here: we can consider that the four-dimensional object that we want to depict has a spotlight behind it, which projects its shadow onto our three-dimensional space. If the object rotates, the shadow deforms and we imagine the shape of the object, even if we cannot see it.

First we see the hypercube, obviously more clearly than the cuts.

Clicking on the picture starts the film.

Then the "24", which we think Schläfli was most proud of. The reason is that the newcomer is really new, it is no equivalent to a three-dimensional polyhedron like the others. It also has the wonderful property of being self-dual: it has, for example, as many flat sides as edges and as many three-dimensional sides as corners.

This new view introduces us to other aspects of the four-dimensional polyhedra that are obviously complex. Both methods, the cuts and the shadows, have many advantages, but it must be admitted that they do not show all the uniformity that all these beautiful objects offer.

In the following chapter we will use another method, namely stereographic projection. Maybe it's more illuminating.

5. “To be seen” in dimension 4: the stereographic projection.

(see the fourth chapter of the film: the fourth dimension, sequel)

Schläfli offers a final method of imagining four-dimensional objects. It is simply the stereographic projection. Nevertheless, it is of course not the projection that Hipparchus showed us in Chapter 1.

Let us imagine that we are in four-dimensional space and that we are looking at a sphere. To define it, we use the usual definition: it is the set of all those points that are equidistant from a center called the center. We know that the sphere that three-dimensional space contains is two-dimensional because every point on the sphere is determined by its latitude and longitude. To a certain extent we can say that the sphere in three-dimensional space has only 2nd dimension because it “lacks” one dimension, namely the height above the sphere. Likewise, the sphere in four-dimensional space has the 3rd dimension and it “lacks” a dimension that is again the height above the sphere.

What is the sphere in a plane, that is, in a two-dimensional space? It is the set of all those points that are equidistant from a center, namely a circle. Such a circle is a sphere in a two-dimensional space, and it has 1st dimension, since only one number is necessary to determine a point of the circle.

Even more surprising: what is a sphere in a one-dimensional space, that is, in a straight line? It is the set of all those points that are equidistant from a center. There are only two points, one on the left and the other on the right. So the sphere in one-dimensional space contains only two points. No wonder it has zero dimension.

In short: In n-dimensional space the sphere has n-1 dimension. That's why they call them mathematicians S.n-1.

The beginning of the chapter tells what theS.3 Sphere is, but well, not even Schläfli can show it to us. At most he can show us an S2 sphere and encourage us to look at theS.3 Imagine the sphere as if we were in four-dimensional space. The stereographic projection presented by Hipparch projects theS.2 Sphere on the tangential plane at the south pole. We can do that with theS.3 Move sphere. We'll take the one at the South PoleS.3 Sphere tangent space, which is three-dimensional, and now we can each point of theS.3 Map the sphere to our space, with the exception of the North Pole. It is sufficient to follow the straight line connecting the North Pole with the point until it meets the tangent space at the South Pole. The constellation is completely analogous to the one seen earlier, even if this is a four-dimensional process.

Let us assume that Schläfli wants to show us a four-dimensional polyhedron. He will continue as we did with the reptiles. He will inflate it into a sphere until it is mapped onto the sphere. Now it can stereographically project onto the tangential space at the south pole, which is “our” space, and so we can see the projection.

We can do that tooS.3 Turn the sphere and then project it down and watch the dance of the polyhedron. We notice that when we turn the sphere, from time to time one side hits the projection pole and the projection becomes infinitely large. It seems like it is about to explode onto the screen. We have the same impression as in the 2nd chapter when the polyhedra were projected onto the plane.

This is suggested in Chapter 4: projecting Schläfli's polyhedra stereographically and rotating them at the same time ...

Clicking on the picture starts the film.

The geometry of four-dimensional space is only the beginning because there is the fifth, the sixth, and even the infinite dimension! They, which were initially regarded as pure abstractions, are now widely used in modern physics. Einstein's theory of relativity postulates that space and time are connected to one another to form a four-dimensional space-time. A point in this spacetime is an event that is characterized by its spatial position x, y, z and by the moment t at which it takes place.

The power of the theory of relativity is to be able to mix these four coordinates to a certain extent without qualitatively distinguishing between time and space, and so they lose their peculiarity. We will not explain this theory here because, among other things, Schläfli did not know it. Einstein's theory dates back to 1905, much later than the discovery of the fourth dimension. It is not the first time, not even the last, when physics and mathematics interact, each with its own method, with different goals and reasons, but still so similar ...

On the other hand, current physics is talking about spaces that are 10th dimension or more, and quantum physics works in an infinite dimension, doesn't it? One will have to wait until we make a film about ten-dimensional spaces ...