The Fourth Dimension

GA 324a — 24 May 1905, Berlin

Fourth Lecture

I recently tried to give you a schematic idea of four-dimensional space. But it would be very difficult if we were not able to form a picture of four-dimensional space in some kind of analogy. If it were a matter of characterizing our task, then it would be this: to show a four-dimensional structure here in three-dimensional space. Initially, we only have three-dimensional space at our disposal. If we want to link something unknown to us with something known, then, just as we have mapped a three-dimensional object into two dimensions, we have to bring a four-dimensional object into the third dimension. Now I would like to show, in the most popular way possible, using Mr. Hinton's method, how four-dimensional space can be mapped within three dimensions. So I would like to show how this task can be solved.

First, let me assume how to bring three-dimensional space into two-dimensional space. Our blackboard here is a two-dimensional space. If we were to add depth to height and width, we would have three-dimensional space. Now let's try to visualize a three-dimensional object on the blackboard.

A cube is a three-dimensional object because it has height, width and depth. Let's try to bring it into two-dimensional space, or onto a plane. If you take the whole cube and roll it up, or rather unroll it, you can do it like this. The sides, the six squares that we have in three-dimensional space, can be spread out once in a plane (Figure 25). So I could imagine the boundary surfaces of the cube spread out on a plane in a cross shape.

There are six squares that can be rearranged to form a cube again if I fold them back, so that squares 1 and 3, 2 and 4, and 5 and 6 are opposite each other. Thus we have a three-dimensional structure simply laid in the plane.

This is not a method that we can use directly to draw the fourth dimension in three-dimensional space. For that, we have to look for a different analogy. We have to use colors to help us. To do that, I will label the six squares along their sides with different colors. The squares facing each other [in the cube] should have the same colors when they are unfolded. I will draw the squares 1 and 3 so that one side is red [dotted lines] and the other is blue [solid lines]. Now I will complete these squares so that I keep blue for the whole horizontal direction (Figure 26). So I will draw all the vertical sides of these squares in red and all the horizontal sides in blue.

If you look at these two squares, 1 and 3, you have the two dimensions that the squares have, expressed in two colors, red and blue. So here for us [at the vertical blackboard, where square 2 is “stuck” to the blackboard], red would mean height and blue depth.

Let us now keep in mind that we apply red wherever height occurs and blue wherever depth occurs; and then we want to take green [dashed line] for the third dimension, width. Now we want to complete the unfolded cube in this way. The square 5 has sides that are blue and green, so the square 6 must look the same. Now only the squares 2 and 4 remain, and if you imagine them unfolded, it follows that the sides will be red and green.

Now, if you imagine it, you will see that we have transformed the three dimensions into three colors. We now say red [dotted], green [dashed], and blue [(solid line)] for height, width, and depth. We name the three colors that are to be images for us instead of the three spatial dimensions. If you imagine the whole cube opened up, you can explain the third dimension in two dimensions in such a way as if, for example, you had let the blue-red square [from left to right in Figure 26] march through green. We want to say that red and blue passed through green. We will describe the marching through green, the disappearance into the third color dimension, as the passage through the third dimension. So, if you imagine that the green fog colors the red-blue square, both sides – red and blue – will appear colored. Blue will take on a blue-green hue and red a cloudy shade, and only where the green stops will both appear in their own color again. I could do the same with squares 2 and 4. So I let the red-green square move through a space that is blue, and then you can do the same with the other two squares, 5 and 6, where the blue-green square would have to pass through the red. In this way, you let each square disappear on one side, submerging it in a different color. It takes on a different color itself through this third color, until it emerges on the other side in its original state.

We thus have an allegorical representation of our cube using three perpendicular colors. We have simply used three colors to represent the three directions we are dealing with here. If we want to imagine the changes that the three pairs of squares have undergone, we can do so by imagining that the squares pass through green the first time, red the second time, and blue the third time.

Now imagine squares instead of these [colored] lines, and squares everywhere for the bare space. Then I can draw the whole figure differently (Figure 27). We draw the transit square blue, and the two that pass through it – before and after the transit – we draw them above and below, here in red-green. [In a second step] I take the red square as the one that allows the blue-green squares to pass through it. And [in a third step] we have the green square here. The two corresponding other colors, red and blue, pass through the green square.

You see, now I have shown you another form of propagation with nine adjacent squares, but only six of which are on the cube itself, namely the squares drawn at the top and bottom of the figure (Figure 27). The other three [middle] squares are transition squares that denote nothing more than the disappearance of the individual colors into a third [color]. [For the transition movement, we] therefore always have to take two dimensions together, because each of these squares [in the upper and lower rows] is composed of two colors and disappears into the color that it does not contain itself. To make these squares reappear on the other side, we let them disappear into the third color. Red and blue disappear into green, red and green have no blue, so they disappear into blue [and green and blue disappear into red].

So, you see, we have the option here of assembling our cube using squares from two color dimensions that pass through the third color dimension.

Now it stands to reason that we imagine cubes instead of squares, and in doing so we put the cubes together out of three color dimensions – just as we put the square together out of two lines of different colors – so that we have three colors, according to the three dimensions of space. If we now want to do the same as we did with the square, we have to add a fourth color. This will allow us to make the cube disappear as well, of course only through a color that it does not have itself. Instead of the three pass squares, we now have four pass cubes in four colors: blue, white, green, and red. So instead of the pass square, we have the pass cube. Mr. Schouten has now produced these colored cubes in his models.

Now, just as we have a square pass through another that is not its color, we must now let a cube pass through another that is not its color. So we let the white-red-green cube pass through a blue one. It will submerge into the fourth color on one side and reappear in its [original] colors on the other side (Figure 28.1).

So here we have a [color] dimension bounded by two cubes that have three colored faces. In the same way, we now have to let the green-blue-red cube pass through the white cube (Figure 28.2), and then let the blue-white-red cube pass through the green (Figure 28.3). In the last figure (Figure 28.4), we have a blue-green-white cube that has to pass through a red dimension, that is, it has to disappear into a color that it does not itself have, in order to reappear on the other side in its very own colors.

These four cubes behave exactly like our three squares did before. If you now realize that we need six squares to bound a cube, we need eight cubes to bound a four-dimensional object, the tessaract.

Just as we obtained three auxiliary squares there, which only signify their disappearance through the other dimension, so here we obtain twelve cubes in all, which are related to each other in the same way that these nine figures are related in the plane. Then we did the same with the cube as we did earlier with the squares, and by choosing a new color each time, a new dimension was added to the others. So we think, we represent a body that has four dimensions in color, in that we have four different colors in four directions, with each [single] cube having three colors and passing through the fourth [color].

The purpose of this substitution of dimensions with colors is that, as long as we stick with the [three] dimensions, we cannot bring the three dimensions into the [two-dimensional] plane. But if we use three colors instead, we can do it. We do the same with four dimensions if we want to visualize them using [four] colors in three-dimensional space. This is one way in which I would like to introduce you to these otherwise complicated things, and how Hinton used them in his problem [of the three-dimensional representation of four-dimensional structures].

I would now like to spread out the cube in the plane again, to turn it over into the plane once more. I will draw this on the board. First, disregard the bottom square [of Figure 25] and imagine that you can only see two-dimensionally, so you can only see what is spread out on the surface of the board. If we put five squares together as in this case, so that they are arranged in such a way that the one square comes into the middle, this inner area remains invisible (Figure 29). You can go around it from all sides. You cannot see square 5 because you can only see in two dimensions.

Now let us do the same thing that we have done here with five of the six side squares of the cube with seven of the eight boundary cubes that form the tessaract when we spread our four-dimensional structure into space. I will lay out the seven cubes in the same way as I did with the faces of the cube on the board; only now we have cubes where we previously had squares. Now we have here the corresponding spatial figure, formed entirely analogously. Thus we have the same for three-dimensional space as we previously had for two-dimensional surface. Just as a square is completely hidden from all sides, so is the seventh cube, which a being that has [only] the ability to see three-dimensionally will never be able to see (Figure 30). If we could fold up these figures in the same way as the six unfolded squares of the cube, we could pass from the third into the fourth dimension. We have shown how one can form an idea of this by means of color transitions."

With this, we have at least shown how, despite the fact that humans can only perceive three dimensions, we can still imagine four-dimensional space. Now you might still wonder how one can gain a possible conception of the real four-dimensional space. And here I would like to point you to something that is called the actual “alchemical secret.” For the real insight into four-dimensional space is in some way connected with what the alchemists called “transformation”.

[First variant:] He who wishes to acquire a true intuitive grasp of four-dimensional space must perform very definite exercises in intuitive grasp. These consist in his first forming a very clear intuitive perception, a deepened intuitive perception, not an imagination, of what is called water. Such an intuitive perception of water is not so easy to come by. One must meditate for a long time and delve very deeply into the nature of water; one must, so to speak, creep into the nature of water. The second thing is to gain an insight into the nature of light. Man is familiar with light, but only in the sense that he receives it from outside. Now, through meditation, man comes to receive the inner counter-image of outer light, to know where and from what light arises, so that he can himself bring forth and generate something like light. The yogi acquires this ability to produce and generate light through meditation. This is possible for the person who is able to have pure concepts truly meditatively present in his soul, who truly allows pure concepts to have a meditative effect on his soul, who is able to think free of sensuality. Then the light arises from the concept. Then the whole environment opens up to him as flooding light. The secret disciple must now, as it were, chemically combine the conception he has formed of water with the conception of light. The water, completely permeated by light, is a body called by the alchemists Mercury. Water plus light is called Mercury in the language of the alchemists. But this alchemical Mercury is not ordinary mercury. You will not have received the matter in this form. One must first awaken within oneself the ability to generate the light from the [dealing with the pure] concepts. Mercury is this mixture [of light] with the contemplation of water, this light-imbued water power, in whose possession one then puts oneself. That is one element of the astral world.

The second [element] arises from the fact that, just as one has formed an idea of water, one forms an idea of air, that we therefore suck out the power of the air through a mental process. If you concentrate your feeling in a certain way, you create a fire through feeling. If you combine the power of the air chemically with the fire created by feeling, you get “fire air.” You know that Goethe's Faust speaks of fire air.” This is something in which the inner being of the person must participate. So one element is sucked out of a given element, the air, and the other [fire or warmth] is generated by yourself. This air plus fire was called sulfur, sulphur, luminous fire-air by the alchemists.

If you now have this luminous fire air in an aqueous element, then you truly have that [astral] matter of which it says in the Bible: “And the Spirit of God hovered, or brooded, over the ‘waters’.”

[The third element arises when] you draw the power from the earth and then connect it with the [spiritual forces in the] “sound”; then you have what is called the Spirit of God [here]. Therefore, it is also called “thunder”. [The acting] Spirit of God is thunder, is earth plus sound. The Spirit of God [thus hovers over the] astral matter.

Those “waters” are not ordinary water, but what is actually called astral matter. This consists of four types of forces: water, air, light and fire. The arrangement of these four forces presents itself to the astral view as the four dimensions of astral space. That is how they are in reality. It looks quite different in the astral than in our world, some things that are perceived as astral are only a projection of the astral into physical space.

You see, that which is astral is half subjective [that is, passively given to the subject], half water and air, because light and feeling [fire] are objective, [that is, actively brought to appearance by the subject]. Only part of what is astral can be found outside [given to the subject] and obtained from the environment. The other part must be brought about subjectively [through one's own activity]. Through conceptual and emotional powers, one gains the other [from the given] through [active] objectification. In the astral, we thus have subjective-objective elements.

In devachan, there is no longer any objectivity [that is merely given to the subject]. One would have a completely subjective element there.

When we speak of the astral realm, we have something that the human being must first create [out of himself]. So everything we do here is symbolic, an allegorical representation of the higher worlds, of the devachanic world, which are real in the way I have explained to you in these suggestions. What lies in these higher worlds can only be attained by developing new possibilities of perception within oneself. Man must do something himself for this.

[Second text variant (Vegelahn):] Those who want to acquire a real view of four-dimensional space must do very specific visual exercises. First of all, they form a very clear, in-depth view of water. Such a view is not easy to come by; one has to delve very deeply into the nature of water; one has to, so to speak, get into the water. The second thing is to gain an insight into the nature of light. Light is something that man knows, but only in the sense that he receives it from outside. Through meditation, he can gain an inner image of light, know where light comes from and therefore produce light himself. This can be done by someone who allows pure concepts to have a real meditative effect on his soul, who has a thinking free of sensuality. Then the whole of his environment will reveal itself to him as flooding light, and now he must, as it were chemically, combine the idea he has formed of water with that of light. This water, completely permeated by light, is a body that was called “Mercury” by the alchemists. But the alchemical Mercury is not the ordinary mercury. First you have to awaken within yourself the ability to generate Merkurius from the concept of light. Merkurius, light-imbued water power, is what you then place yourself in possession of. That is the one element of the astral world.

The second is created by you also forming a vivid mental image of air, then sucking out the power of the air through a spiritual process, connecting it with feeling, and you ignite the concept of “warmth”, “fire”, then you get “fire air”. So one element is sucked out, the other is produced by yourself. This - air and fire - the alchemists called “sulfur”, sulfur, luminous fire air. In the aqueous element, there you have in truth that matter of which it is said: “and the Spirit of God hovered over the waters”.

The third element is “spirit-God”, which is connected to “earth” and “sound”. This is what happens when you extract the earth's forces and combine them with sound. These “waters” are not ordinary water, but what is actually called astral matter. This consists of four types of forces: water, air, light and fire. And this manifests itself as the four dimensions of astral space.

You see, that which is astral is half subjective; only part of what is astral can be gained from the environment; from conceptual and emotional powers, one gains the other through objectification. In devachan, you would have a completely subjective element; there is no objectivity there. So everything we do here, the symbolic, is an allegorical representation of the devachanic world. Everything that lies in the higher worlds can only be attained by developing new views within yourself. Man must do something about it himself.