The Fourth Dimension

GA 324a — 24 March 1905, Berlin

First Lecture

If you are disappointed about what you are about to hear, I would like to say in advance that today I want to discuss very elementary things [about the fourth dimension]. Those who want to delve deeper into this question should be very familiar with the higher concepts of mathematics. I would like to give you some very elementary and general concepts. One must distinguish between the possibility of thinking in a four-dimensional space and reality. Whoever is able to make observations there is dealing with a reality that extends far beyond what we know as the sensual-real. You have to do thought transformations when you go there. You have to let things play into mathematics a little, find your way into the way of thinking of the mathematician.

You have to realize that the mathematician does not take a step without accounting for what arrives at his conclusions. But we must also realize when we deal with mathematics that even the mathematician cannot penetrate a single step [into reality], that he cannot draw any conclusions [that go beyond what is merely possible in thought]. First of all, it is about simple things, but they become more complicated when one wants to arrive at the concept of the fourth dimension. We must be clear about what we mean by dimensions. It is best to examine the various spatial structures in terms of their dimensionality. They lead to considerations that were only tackled in the 19th century by great mathematicians such as Bolyai, Gauss and Riemann.

The simplest spatial size is the point. It has no extension at all; it must be conceived. It is the fixation of an extension in space. It has no dimension. The first dimension is the line. The straight line has one dimension, length. If we move the line, which has no thickness, ourselves, we step out of the one dimension, and the line becomes a surface. This has two dimensions, a length and a width. If we move the surface, we step out of these two dimensions and we get the body. It has three dimensions: height, width, depth (Figure 1).

If you move the body itself, if you move a cube around in space, you will again only get a spatial body. You cannot move space out of itself.

We need to turn to a few other concepts. If you look at a straight line, it has two boundaries, two endpoints A and B (Figure 2).

Let's imagine that we want A and B to touch. But if they are to touch, we have to curve the line. What happens? You cannot possibly remain within the [one-dimensional] line if you want to make A and B coincide. To connect points A and B, we have to step out of the straight line itself, we have to step out of the first dimension and into the second dimension, the plane. In this way, the straight line becomes a closed curve (that is, in the simplest case, a circle) by bringing its endpoints into alignment (Figure 3).

It is therefore necessary to go beyond the first dimension; you cannot remain within it. Only in this way can the circle be created. You can perform the same operation with a surface. However, this only works if you do not remain within the two dimensions. You have to enter the third dimension and then you can turn the surface into a tube, a cylinder. This operation is done in a very similar way to the way we brought two points into coincidence earlier, thereby moving out of the first dimension. Here, in order to bring two boundaries of the surface into coincidence, we have to move into the third dimension (Figure 4).

Is it conceivable that a similar operation could be carried out with a spatial structure that already has three dimensions itself? If you have two congruent cubes, you can slide one into the other. [Now imagine two congruent cubes as the boundaries of a three-dimensional prismatic body.] If you try to make one cube, which is colored red on one side [and blue on the opposite side], fit exactly over the other cube, which is otherwise [geometrically] identical but with the red and blue colors swapped, then you cannot make them fit except by rotating the cube (Figure 5).

Let us consider another spatial structure. If you take the left-hand glove, it is impossible for you to pull the left-hand glove over the right hand. But if you look at the two [mirror-symmetrical] gloves together, like the straight line with the end points A and B, you have something that belongs together. It is then a single entity, with a boundary [that is, with a mirror plane] in the middle. It is very similar with the two symmetrical halves of the human outer skin. 2

How can we now make two [mirror] symmetrical three-dimensional structures coincide? Only if we go beyond the third dimension, as we did with the first and second. We can also put the right or left glove over the left or right hand, respectively, when we walk through four-dimensional space.

[When constructing the third dimension (depth dimension) of the visualization space, we align the image of the right eye with that of the left eye and place it over it.

We now look at an example from Zöllner. We have a circle and a point P outside of it. How can we bring the point P into the circle without crossing the circle? This is not possible if we remain within the plane. Just as one has to go from the second dimension into the third when moving from a square to a cube, we also have to go out of the second dimension here. With a sphere, there is also no possibility of entering [into the interior] without [piercing the surface of the sphere or] going beyond the third dimension. Figure 6 Figure 7 Figure 8 Now, however, we can imagine that if we go through [the straight line and] remain within the line, we will come back from the other side of infinity. But in doing so, we have to go through infinity.

These are possibilities for thought, but they have a practical significance for the theory of knowledge, [in particular for the problem of the objectivity of the content of perception]. If we realize how we actually perceive, we will come to the following view. Let us first ask ourselves: How do we gain knowledge of bodies through our senses? We see a color. Without eyes, we would not perceive it. The physicist then says: Out there in space is not what we call color, but purely spatial forms of movement; they penetrate through our eye, are captured by the optic nerve, transmitted to the brain, and there, for example, the red arises. One may now ask: Is the red also present when there is no sensation?

Red could not be perceived without the eye. The ringing of a bell could not be perceived without the ear. All our sensations depend on the transformation of forms of motion by our physical and mental apparatus. However, the matter becomes even more complicated when we ask ourselves: Where is the red, this peculiar quality, actually located? Is it in the body? Is it a process of vibration? Outside there is a process of movement, and this continues right into the eye and into the brain itself. There are vibrational [and nervous] processes everywhere, but red is nowhere to be found. Even if you examine the eye, you would not find red anywhere. It is not outside, but it is also not in the brain. We only have red when we, as a subject, confront these processes of movement. So do we have no possibility at all to talk about how the red meets the eye, how a c sharp meets the ear?

The question is, what is this inner [representation], where does it arise? In the philosophical literature of the 19th century, you will find that this question runs through everything. Schopenhauer, in particular, has provided the following definition: The world is our representation. But what then remains for the external body? [Just as a color representation can be “created” by movements, so can] movement can arise in our inner self through something that is basically not moved. Let us consider twelve snapshots of a [moving] horse figure on [the inside of] a [cylinder] surface, [which is provided with twelve fine slits in the spaces between. If we look at the rotating cylinder from the side,] we will have the impression that it is always the same horse and that only its feet are moving. So [the impression of] movement can also arise through our [physical organization] when something is not moving at all [in reality]. This is how we arrive at a complete dissolution of what we call movement.

But what then is matter? If you subtract color, movement [shape, etc., i.e. what is conveyed by sensory perception] from matter, then nothing remains. If we already have the [secondary, i.e. “subjective” sensations [color, sound, warmth, taste, smell] within us, we must also place [the primary sensations, that is, shape and movement,] within us, and with that the external world completely disappears. However, this results in major difficulties [for the theory of knowledge].

Let us assume that everything is outside, how then do the properties of the object outside come into us? Where is the point [where the outside merges into the inside]? If we subtract all [sensory perceptions], there is no outside anymore. In this way, epistemology puts itself in the position of Münchhausen, who wants to pull himself up by his own hair. But only if we assume that there is an outside, only then can we come to [an explanation of] the sensations inside. How can something from the outside enter our inside and appear as our imagination?

We need to pose the question differently. Let us look at some analogies first. You will not be able to find a relationship [between the outside world and the sensation inside] unless you resort to the following. We return to the consideration of the straight line with endpoints A and B. We have to go beyond the first dimension, curve the line, to make the endpoints coincide (Figure 7).

Now imagine the left endpoint A [of this straight line] brought together with the right endpoint B so that they touch at the bottom, so that we are able to return to the starting point [via the coinciding endpoints]. If the line is small, the corresponding circle is also small. If I turn the [initially given] line into a circle and then turn larger and larger lines into circles, the point at which the endpoints meet moves further and further away from the [original] line and goes to infinity.

of the [original] line and goes to infinity. Only at infinity do the [increasingly large] circle lines have their endpoint. The curvature becomes weaker and weaker, and eventually we will not be able to distinguish the circle line from the straight line with the naked eye (Figure 8).

In the same way, when we walk on the Earth, it appears to us as a straight piece, although it is round. If we imagine that the two halves of the straight line extend to infinity, the circle actually coincides with the straight line.

The straight line can be conceived as a circle whose diameter is infinite.

Now, instead of a [geometric] line, imagine something that is real and that connects to a reality. Let us imagine that as the point C [on the circumference of the circle] progresses, cooling occurs, that the point becomes colder and colder the further it moves away [from its starting point] (Figure 9). Let us leave the point within the circle for the time being, and, as it becomes colder and colder, let it reach the lower limit A, B. When it returns on the other side, the temperature increases again. So on the way back, the opposite condition to the one on the way there occurs. The warming increases until the temperature at C is reached again, from which we started. No matter how extended the circle is, it is always the same process: a flow of heat out and a flow of heat in. Let us also imagine this with the [infinitely extended straight] line: as the temperature [on one side increasingly] dissipates, it can rise on the other side. We have here a state that dissipates on one side while it rebuilds on the other.

In this way, we bring life and movement into the world and approach what, in a higher sense, we can call an understanding of the world. We have here two states that are interdependent and interrelated. However, for everything you can observe [sensually], the process that goes, say, to the right has nothing to do with the one that comes back from the left, and yet they are mutually dependent.

We now compare the body of the external world with the state of cooling and, in contrast, our inner sensation with the state of warming. [Although the external world and inner sensation have nothing directly perceptible in common,] they are related to each other, mutually dependent [in an analogous way to the processes described above]. This results in a connection between the external world [and our internal world] that we can support with an image: [through the relationship between] the seal and the sealing wax. The seal leaves behind an exact imprint, an exact reproduction of the seal in the sealing wax, without the seal remaining in the sealing wax [and without any material from the seal being transferred to the sealing wax]. So in the sealing wax there remains a faithful reproduction of the seal. It is quite the same with the connection between the outside world and inner sensations. Only the essential is transferred. One state determines the other, but nothing (material) is transferred.

If we imagine that this is the case with [the connection between the] outside world and our impressions, we come to the following. [Geometric] mirror images in space behave like gloves from the left and right hand. [In order to relate these directly and continuously to each other,] we have to use a new dimension of space to help us. [Now the outside world and the inner impression behave analogously to geometric mirror images and can therefore only be directly related to each other through an additional dimension.] In order to establish a relationship between the outside world and inner impressions, we must therefore go through a fourth dimension and be in a third element. We can only seek the common ground [of the outside world and inner impressions] where we [are one] with them. [One can imagine these mirror images as] floating in a sea, within which we can align the mirror images. And so we come [initially in thought] to something that transcends three-dimensional space and yet has a reality. We must therefore bring our spatial ideas to life.

Oskar Simony has tried to represent these animated spatial structures with models. [As we have seen, one comes] from the consideration of the zero-dimensional [step by step] to the possibility of imagining four-dimensional space. [On the basis of the consideration of mirror-symmetrical bodies, that is, with the help of] symmetries, we can first [most easily] recognize this space. [Another way to study the peculiarities of empirical three-dimensional space in relation to four-dimensional space is to study the knotting of curves and ribbons.] What are symmetry conditions? By intertwining spatial structures, we cause certain complications. [These complications are peculiar to three-dimensional space; they do not occur in four-dimensional spaces.]

Let's do some practical thinking exercises. If we cut a band ring in the middle, we get two such rings. If we now cut a band whose ends have been twisted by 180° and then glued, we get a single twisted ring that does not disintegrate. If we twist the ends of the tape 360° before gluing them together, then when we cut it, we get two intertwined rings. Finally, if we twist the tape ends 720°, the same process results in a knot.

Anyone who reflects on natural processes knows that such convolutions occur in nature; [in reality,] such intertwined spatial structures are endowed with forces. Take, for example, the movement of the Earth around the Sun, and then the movement of the Moon around the Earth. It is said that the Moon describes a circle around the Earth, but [if you look more closely] it is a line that is wrapped around [a circle, the orbit of the Earth], thus a helix around a circular line. And then we have the sun, which rushes through space so fast that the moon makes an additional spiral movement around it. So there are very complicated lines of force extending in space. We have to realize that we are dealing with complicated concepts of space that we can only grasp if we do not let them become rigid, if we have them in a fluid state.

Let us recall what has been said: the zero-dimensional is the point, the one-dimensional is the line, the two-dimensional the surface and the three-dimensional the body. How do these concepts of space relate to each other?

Imagine you are a creature that can only move along a straight line. What would the spatial perceptions of such a being, which itself is only one-dimensional, be like? It would not perceive its own one-dimensionality, but would only imagine points. This is because, if we want to draw something on a straight line, there are only points on the straight line. A two-dimensional being could encounter lines, and thus distinguish one-dimensional beings. A three-dimensional being, such as a cube, would perceive the two-dimensional beings. Man, then, can perceive three dimensions. If we reason correctly, we must say to ourselves: Just as a one-dimensional being can only perceive points, as a two-dimensional being can only perceive one dimension, and a three-dimensional being can only perceive two dimensions, so a being that perceives three dimensions can only be a four-dimensional being. The fact that a human being can define external beings in three dimensions, can [deal with] spaces of three dimensions, means that he must be four-dimensional. And just as a cube can only perceive two dimensions and not its third, it is true that the human being cannot perceive the fourth dimension in which he lives.