# The Fourth Dimension

## GA 324a — 22 October 1908, Berlin

### On Higher-Dimensional Space

The subject we are to discuss today will present us with a number of difficulties. Consider the lecture as an episode; it is being held at your request. If you only want to grasp the subject formally in its depth, some mathematical knowledge is necessary. But if you want to grasp it in its reality, you have to penetrate very deeply into occultism. So today we can only talk about it very superficially, only give a suggestion for this or that.

It is very difficult to talk about multidimensionality at all, because if you want to get an idea of what more than three dimensions are, you have to delve into abstract areas, and there the concepts must be very precisely and strictly defined, otherwise you end up in a bottomless pit. And that's where many friends and enemies have ended up.

The concept of multidimensional space is not as foreign to the world of mathematicians as one might think.® In mathematical circles, there is already a way of calculating with a multidimensional type of calculation. Of course, the mathematician can only speak of this space in a very limited sense; he can only discuss the possibility. Whether it really is can only be determined by someone who can see into a multidimensional space. Here we are already dealing with a lot of concepts that, if we grasp them precisely, really provide us with clarity about the concept of space.

What is space? We usually say: there is space around me, I walk around in space — and so on. If you want a clearer idea, you have to go into some abstractions. We call the space in which we move three-dimensional. It has an extension in height and depth, to the right and left, to the front and back, it has length, width and height. When we look at bodies, these bodies are extended for us in this three-dimensional space; they have a certain length, a certain width and height for us.

However, we have to deal with the details of the concept of space if we want to arrive at a more precise concept. Let us look at the simplest body, the cube. It shows us most clearly what length, width and height are. We find a base of the cube that is the same in length and width. If we move the base up, just as far as the base is wide and long, we get the cube, which is therefore a three-dimensional object. The cube is the clearest way for us to learn about the details of a three-dimensional object. We examine the boundaries of the cube. These are formed everywhere by surfaces bounded by sides of equal length. There are six such surfaces.

What is a surface? Those who are not capable of very sharp abstractions will already falter here. For example, you cannot cut the boundaries of a wax cube as a fine layer of wax. You would still get a layer of a certain thickness, so you would get a body. We will never get to the boundary of the cube this way. The real boundary has only length and width, no height. Thickness is eliminated. We thus arrive at the formulaic sentence: The area is the boundary [of a three-dimensional object] in which one dimension is eliminated.

What then is the boundary of a surface, for example of a square? Here we must again take the most extreme abstraction. [The boundary of a surface] is a line that has only one dimension, length. The width is canceled. What is the boundary of a line? It is the point, which has no dimension at all. So you always get the boundary of a thing by leaving out a dimension.

So you could say to yourself, and this is also the line of thought that many mathematicians have followed, especially Riemann,* who has achieved the most solid work here: We take the point, which has none, the line, which has one, the plane, which has two, the solid, which has three dimensions. Now mathematicians asked themselves: Could it not be that formally one could say that one could add a fourth dimension? Then the [three-dimensional] body would have to be the boundary of the four-dimensional object, just as the surface is the boundary of the body, the line is the boundary of the surface, and the point is the boundary of the line. Of course, the mathematician then goes even further to five-, six- and seven-dimensional objects and so on. We have [even arbitrary] “-dimensional objects [where ” is a positive integer].

Now, there is already some ambiguity in the matter when we say: the point has none, the line has one, the plane two, the solid three dimensions. We can now make such a solid, for example a cube, out of wax, silver, gold and so on. They are different in terms of matter. We make them the same size, then they all occupy the same space. If we now eliminate all material, only a certain part of space remains, which is the spatial image of the body. These parts of space are the same [among themselves], regardless of what material the cube was made of. These parts of space also have length, width and height. We can now imagine these cubes as infinitely extended and thus arrive at an infinitely extended three-dimensional space. The (material) body is, after all, only a part of it.

The question now is whether we can simply extend such conceptual considerations, which we make starting from space, to higher realities. In these considerations, the mathematician actually only calculates, and does so with numbers. Now the question is whether one can do that at all. I will show you how much confusion can arise when calculating with spatial quantities. Why? I only need to tell you one thing: Imagine you have a square figure here. I can make this figure, this area, wider and wider on both sides and thus arrive at an area that extends indefinitely between two lines (Figure 56).

This area is infinitely large, so it is >. Now imagine someone who hears that the area between these two lines is infinite. Of course, he thinks of infinity. If you now talk to him about infinity, he may have very wrong ideas about it. Imagine that I now add below [each square one more, so another row of] an infinite number of squares, and I get a [different] infinity that is exactly twice as large as the first (Figure 57). So we have > = 2 + 0,

In the same way I could get: “ = 3 +,

In calculating with numbers, you can just as well use infinity as finiteness. Just as it is true that space was already infinite in the first case, it is just as true that it is 2 + c, 3 - c, and so on. So we are calculating numerically here.

We see that the concept of the infinity of space [which follows from the numerical representation] does not give us any possibility of penetrating deeper [into the higher realities]. Numbers actually have no relation to space at all, they relate to it quite neutrally, like peas or any other objects. You now know that nothing changes in reality as a result of calculation. If someone has three peas, multiplication does not change that, even if the calculation is done correctly. The calculation 3 + 3 = 9 does not give nine peas. A mere consideration does not change anything here, and calculation is a mere consideration. Just as three peas are left behind, [you do not actually create nine peas,] even if you multiply correctly, three-dimensional space must also be left behind if the mathematician also calculates: two-, three-, four-, five-dimensional space. You will feel that there is something very convincing about such a mathematical consideration. But this consideration only proves that the mathematician could indeed calculate with such a multidimensional space; [but whether a multidimensional space actually exists, that is,] he cannot determine anything about the validity of such a concept [for reality]. Let us be clear about that here in all strictness.

Now we want to consider some other considerations that have been made very astutely by mathematicians, one might say. We humans think, hear, feel and so on in three-dimensional space. Let us imagine that there are beings that could only perceive in two-dimensional space, that would be organized so that they always have to remain in the plane, that they could not get out of the second dimension. Such beings are quite conceivable: they can only move [and perceive] to the right and left [and backwards and forwards] and have no idea of what is above and below.

Now it could be the same for man in his three-dimensional space. He could only be organized for the three dimensions, so that he could not perceive the fourth dimension, but for him it arises just as the third arises for the others. Now mathematicians say that it is quite possible to think of man as such a being. But now one could say that this is also only one interpretation. One could certainly say that. But here one must again proceed somewhat more precisely. The matter is not as simple as in the first case [with the numerical determination of the infinity of space]. I am intentionally only giving very simple discussions today.

This conclusion is not the same as the first purely formal [calculative] consideration. Here we come to a point where we can take hold. It is true that there can be a being that can only perceive what moves in the plane, that has no idea that there is anything above or below. Now imagine the following: Imagine that a point becomes visible to the being within the surface, which is of course perceptible because it is located in the surface. If the point only moves within the surface, it remains visible; but if it moves out of the surface, it becomes invisible. It would have disappeared for the surface being. Now let us assume that the point reappears, thus becoming visible again, only to disappear again, and so on. The being cannot follow the point [as it moves out of the surface], but the being can say to itself: the point has now gone somewhere I cannot see. The being with the surface vision could now do one of two things. Let us put ourselves in the place of the soul of this flat creature. It could say: There is a third dimension into which the object has disappeared, and then it has reappeared afterwards. Or it could also say: These are very foolish creatures who speak of a third dimension; the object has always disappeared, perished and been reborn [in every case]. One would have to say: the being sins against reason. If it does not want to assume a continuous disappearance and re-emergence, the being must say to itself: the object has submerged somewhere, disappeared, where I cannot see.

A comet, when it disappears, passes through four-dimensional space.

We see here what we have to add to the mathematical consideration. There should be something in the field of our observations that always emerges and disappears again. You don't need to be clairvoyant for that. If the surface being were clairvoyant, it wouldn't need to conclude, because it would know from experience that there is a third dimension. It is the same for humans. Unless they are clairvoyant, they would have to say: I remain in the three dimensions; but as soon as I observe something that disappears from time to time and reappears, I am justified in saying: there is a fourth dimension here.

Everything that has been said so far is as unassailable as it can possibly be. And the confirmation is so simple that it will not even occur to man in his present deluded state to admit it. The answer to the question: Is there something that always disappears and reappears? — is so easy. Just imagine, a feeling of joy arises in you and then it disappears again. It is impossible that anyone who is not clairvoyant will perceive it. Now the same sensation reappears through some event. Now you, just like the surface creature, could behave in different ways. Either you say to yourself that the sensation has disappeared somewhere where I cannot follow it, or you take the view that the sensation passes away and arises again and again.

But it is true: every thought that has vanished into the unconscious is proof that something disappears and then reappears. At most, the following can be objected to: if you endeavor to object to such a thought, which is already plausible to you, with everything that could be objected to from a materialistic point of view, you are quite right. I will make the most subtle objection here, all the others are very easy to refute. For example, one says to oneself: everything is explained in a purely materialistic way. Now I will show you that something can quite well disappear within material processes, only to reappear later. Imagine that some kind of vapor piston is always acting in the same direction. It can be perceived as a progressive piston as long as the force is acting. Now suppose I set a piston that is exactly the same but acting in the opposite direction. Then the movement is canceled out and a state of rest sets in. So here the movement actually disappears.

In the same way, one could say here: For me, the sensation of joy is nothing more than molecules moving in the brain. As long as this movement takes place, I feel this joy. Now, let us assume that something else causes an opposite movement of the molecules in the brain, and the joy disappears. Wouldn't someone who doesn't go very far with their considerations find a very meaningful objection here? But let's take a look at what this objection is actually about. Just as one [piston] movement disappears when the opposite [piston movement] occurs, so the [molecular movement underlying the sensation] is extinguished by the opposite [molecular movement]. What happens when one piston movement extinguishes the other? Then both movements disappear. The second movement also disappears immediately. The second movement cannot extinguish the first without itself being extinguished. [A total standstill results, no movement whatsoever remains.] Yes, but then a [new] sensation can never extinguish the [already existing] sensation [without perishing itself]. So no sensation that is in my consciousness could ever extinguish another [without extinguishing itself in the process]. It is therefore a completely false assumption that one sensation could extinguish another [at all]. [If that were the case, no sensation would remain, and a totally sensationless state would arise.]

Now, at most, it could be said that the first sensation is pushed into the subconscious by the second. But then one admits that something exists that eludes our [immediate] observation.

We have not considered any clairvoyant observations today, but have only spoken of purely mathematical ideas. Now that we have admitted the possibility of such a four-dimensional world, we ask ourselves: Is there a way to observe something [four-dimensional] without being clairvoyant? — Yes, but we have to use a kind of projection to help us. If you have a piece of a surface, you can rotate it so that the shadow becomes a line. Similarly, you can get a point from a line as a shadow. For a [three-dimensional] body, the silhouette is a [two-dimensional] surface. Likewise, one can say: So it is quite natural, if we are aware that there is a fourth dimension, that we say: [Three-dimensional] bodies are silhouettes of four-dimensional entities.

Here we have arrived at the idea of [four-dimensional space] in a purely geometrical way. But [with the help of geometry] this is also possible in another way. Imagine a square, which has two dimensions. If you imagine the four [bounding] lines laid down next to each other [i.e., developed], you have laid out the [boundary figures] of a two-dimensional figure in one dimension (Figure 58). Let's move on. Imagine we have a line. If we proceed in the same way as with the square, we can also decompose it into two points [and thus decompose the boundaries of a one-dimensional structure into

zero dimensions]. You can also decompose a cube into six squares (Figure 59). So there we have the cube in terms of its boundaries decomposed into surfaces, so that we can say: a line is decomposed into two points, a surface into four lines, a cube into six surfaces. We have the numerical sequence two, four, six here.

Now we take eight cubes. Just as [the above developments each consist of] unfolded boundaries, here the eight cubes form the boundary of the four-dimensional body (Figure 60). The [development of these] boundaries forms a double cross, which, we can say, indicates the boundaries of the regular [four-dimensional] body. [This body, a four-dimensional cube, is named the Hinton Tessaract after Hinton.]

We can therefore form an idea of the boundaries of this body, the tessaract. We have here the same idea of the four-dimensional body as a two-dimensional being could have of a cube, for example by unfolding the boundaries.