The Fourth Dimension
24 May 1905, Berlin
In a recent lecture I attempted to develop a schematic idea of four-dimensional space, which would be very difficult to do without using an analogy of sorts. The problem that confronts us is how to indicate a four-dimensional figure here in three-dimensional space, which is the only type of space initially accessible to us. To link the unfamiliar element of four-dimensional space to something we know about, we must find ways to bring a four-dimensional object into three dimensions, just as we brought a three-dimensional object into two dimensions. I would like to use the method popularized by Mr. Hinton to demonstrate a solution to the problem of how to represent four-dimensional space in three dimensions. [Note 27]
Let me begin by showing how three-dimensional space can be depicted in two dimensions. Our chalkboard here is a two-dimensional surface. Adding depth to its dimensions of height and width would give us a three-dimensional space. Now let's attempt to depict a three-dimensional figure here on the chalkboard.
A cube is a three-dimensional figure because it has height, width, and depth. Let's try to bring a cube into two-dimensional space — that is, into a plane. We can take a cube and unfold it so that its six square sides are spread out in a plane (Figure 25). In two dimensions, therefore, the surfaces defining a cube can be imagined as forming a cross.
These six squares form a cube again when I fold them up so that squares 1 and 3 are opposite each other. Squares 2 and 4 are also opposite each other, as are 5 and 6. This is a simple way of transferring a three-dimensional figure to a plane.
We cannot use this method directly when we want to draw the fourth dimension in three-dimensional space. For that, we need a different analogy. We will need to use colors. I will color the edges of the six squares differently, so that opposite sets of squares are of the same colors. For squares 1 and 3, I will make one pair of edges red (dotted lines) and another blue (solid lines). I also will color all the horizontal edges of the other squares blue and all the verticals red (Figure 26).
Look at these two squares, 1 and 3. Their two dimensions are represented by two colors, red and blue. For us, then, on the vertical board, where square 2 is flat against the board, red means height, and blue means depth.
Having consistently used red for height and blue for depth, let's add green (dashed lines) for width, the third dimension, and complete our unfolded cube. Square 5 has blue and green sides, so square 6 must look the same. Now only squares 2 and 4 are left. When you imagine them unfolded, you find that their sides are red and green.
Having visualized these colored edges, you realize that we have transformed the three dimensions into three colors. Instead of height, width, and depth, we now call them red (dotted), green (dashed), and blue (solid). These three colors replace and represent the three dimensions of space. Now imagine the whole cube folded up again. You can explain the addition of the third dimension by saying that the blue and red square has moved through green i.e., from left to right in Figure 26. Moving through green, or disappearing into the dimension of the third color, represents the transition through the third dimension. Imagine that a green fog tints the red-and-blue squares, so that both edges (red and blue) appear colored. The blue edge becomes blue-green and the red acquires a murky tint. Both edges reappear in their own color only where the green stops. I could do the same thing with squares 2 and 4 by allowing a red-and- green square to move through a blue space. You could do the same with the two blue-and-green squares, 5 and 6, moving one of them through red. In each case, the square disappears on one side, submerging into a different color that tints it until it emerges on the other side in its original coloration. Thus, the three colors positioned at right angles to each other are a symbolic representation of our cube. We simply have used colors for the three directions. To visualize the changes the cube's three pairs of surfaces undergo, we imagine them passing through green, red, and blue, respectively.
Instead of these colored lines, imagine squares, and instead of empty space, picture squares everywhere. Then I can draw the entire figure in a still different way (Figure 27). The square through which the others pass is colored blue, and the two that pass through it — before and after they make the transition — are drawn flanking it. Here they are in red and green. In a second step, the blue-and-green squares pass through the red square, and, in a third step, the two red-and-blue squares pass through the green.
Here you see a different way of flattening out a cube. Of the nine squares arranged here, only six — the upper and lower rows — form the boundaries of the cube itself (Figure 27). The other three squares in the middle row represent transitions, — they simply signify that the other two colors disappear into a third. Thus, with regard to the movement of transition, we must always take two dimensions at once, because each of these squares in the upper and lower rows is made up of two colors and disappears into the color that does not contain it. We make these squares disappear into the third color in order to reappear on the other side. The red-and-blue squares pass through green. The red-and-green squares have no blue sides, so they disappear into blue, while the green-and-blue squares pass through red. As you see, we can thus construct our cube out of two-dimensional — that is, bi-colored — squares that pass through a third dimension or color. [Note 28]
The next obvious step is to imagine cubes in the place of squares and to visualize these cubes as being composed of squares of three colors (dimensions), just as we constructed our squares out of lines of two colors. The three colors correspond to the three dimensions of space. If we want to proceed just as we did with the squares, we must add a fourth color so that we can make each cube disappear through the color it lacks. We simply have four differently colored transition cubes — blue, white, green, and red — instead of three transition squares. Instead of squares passing through squares, we now have cubes passing through cubes. Mr. Schouten's models use such colored cubes. [Note 29]
Just as we made one square pass through a second square, we must now make one cube pass through a second cube of the remaining color. Thus the white-red-and-green cube passes through a blue one. On one side, it submerges in the fourth color, — on the other side, it reappears in its original colors (Figure 28.1). Thus we have here one color or dimension that is bounded by two cubes whose surfaces are three different colors.
Similarly, we must now make the green-blue-and-red cube pass through the white cube (Figure 28.2). The blue-red-and-white cube passes through the green one (Figure 28.3), and, in the last figure (Figure 28.4), the blue-green-and-white cube has to pass through a red dimension, — that is, each cube must disappear into the color it lacks and reappear on the other side in its original colors.
These four cubes relate to each other in the same way as the three squares in our previous example. We needed six squares to delineate the boundaries of a cube. [Note 30] Similarly, we need eight cubes to form the boundaries of the analogous four-dimensional figure, the tessaract. [Note 31] In the case of a cube, we needed three accessory squares that simply signified disappearance through the remaining dimension. A tessaract requires a total of twelve cubes, which relate to one another in the same way as the nine squares in a plane. We have now done to a cube what we did with squares in the earlier example. Each time we chose a new color, we added a new dimension. We used colors to represent the four directions incorporated by a four-dimensional figure. Each of the cubes in this figure has three colors and passes through a fourth. The point in replacing dimensions with colors is that three dimensions as such cannot be incorporated into a two-dimensional plane. Using three colors makes this possible. We do the same thing with four dimensions when we use four colors to create an image in three-dimensional space. This is one way of introducing this otherwise complicated subject. Hinton used this method to solve the problem of how to represent four-dimensional figures in three dimensions.
Next I would like to unfold the cube again and lay it down in the plane. I'll draw it on the board. For the moment, disregard the bottom square in Figure 25 and imagine that you can see in two dimensions only — that is, you can see only what you can encounter on the surface of the board. In this instance, we have placed five squares so that one square is in the middle. The interior area remains invisible (Figure 29). You can go all the way around the outside, but since you can see only in two dimensions, you will never see square 5.
Now instead of taking five of the six square sides of a cube, let's do the same thing with seven of the eight cubes that form the boundaries of a tessaract, spreading our four-dimensional figure out in space. The placement of the seven cubes is analogous to that of the cube's surfaces laid down in a plane on the board, but now we have cubes instead of squares. The resulting three-dimensional figure is analogous in structure to the two-dimensional cross made of squares and is its equivalent in three-dimensional space. The seventh cube, like one of the squares, is invisible from all sides. It cannot be seen by any being capable only of three-dimensional sight (Figure 30). If we could fold up this figure, as we can do with the six unfolded squares in a cube, we could move from the third into the fourth dimension. Transitions indicated by colors show us how this process can be visualized. [Note 32]
We have demonstrated at least how we humans can visualize four-dimensional space in spite of being able to perceive only three dimensions. At this point, since you also may wonder how we can gain an idea of real four-dimensional space, I would like to make you aware of the so-called alchemical mystery, because a true view of four-dimensional space is related to what the alchemists called transformation.
[First text variant:] If we want to acquire a true view of four-dimensional space, we must do very specific exercises. First we must cultivate a very clear and profound vision — not a mental image — of what we call water. Such vision is difficult to achieve and requires lengthy meditation. We must immerse ourselves in the nature of water with great precision. We must creep inside the nature of water, so to speak. As a second exercise, we must create a vision of the nature of light. Although light is familiar to us, we know it only in the form in which we receive it from outside. By meditating, we acquire the inner counterpart of outer light. We know where and how light arises, — we ourselves become able to produce something like light. Through meditation, yogis or students of esotericism acquire the ability to produce light. When we truly meditate on pure concepts, when we allow these concepts to work on our souls during meditation or sense-free thinking, light arises out of the concepts. Our entire surroundings are revealed to us as streaming light. Esoteric students must "chemically combine" the vision of water that they have cultivated with their vision of light. Water fully imbued with light is what the alchemists called mercury. In the language of alchemy, water plus light equals mercury. In the alchemical tradition, however, mercury is not simply ordinary quicksilver. After we awaken our own ability to create light out of our own work with pure concepts, mercury comes about as the mingling of this light with our vision of water. We take possession of this light-imbued power of water, which is one of the elements of the astral world.
The second element arises when we cultivate a vision of air, just as we previously cultivated a vision of water. Through a spiritual process, we extract the power of air. Then, by concentrating the power of feeling in certain ways, feeling kindles fire. When you chemically combine, as it were, the power of air with the fire kindled by feeling, the result is "fire-air." As you may know, this fireair is mentioned in Goethe's Faust. [Note 33] It requires the inner participation of the human being. One component is extracted from an existing element, the air, while we ourselves produce the other fire or warmth. Air plus fire yields what the alchemists called sulfur, or shining fire-air. The presence of this fire-air in a watery element is truly what is meant when the Bible says, "And the Spirit of God brooded upon the face of the waters." [Note 34]
The third element comes about when we extract the power of the earth and combine it with the spiritual forces in sound. The result is what is called the Spirit of God. It also is called thunder. The active Spirit of God is thunder, earth plus sound. Thus, the Spirit of God hovered over astral substance. The biblical "waters' are not ordinary water but what we know as astral substance, which consists of four types of forces: — water, air, light, and fire. The sequence of these four forces is revealed to astral vision as the four dimensions of astral space. That is what they really are. Astral space looks very different from our world. Many supposedly astral phenomena are simply projections of aspects of the astral world into physical space.
As you can see, astral substance is half-subjective, that is, passively given to the subject, and half water and air. Light and feeling (fire), on the other hand, are objective, that is, made to appear by the activity of the subject. Only one part of astral substance can be found outside, given to the subject in the environment. The other part must be added by subjective means, through personal activity. Conceptual and emotional forces allow us to extract the other aspect from what is given through active objectification. In the astral realm, therefore, we find subjective- objective substance. In devachan, we would find only a completely subjective element; there is no longer any objectivity at all that is simply given to the subject.
In the astral realm, therefore, we find an element that must be created by human beings. Everything we do here is simply a symbolic representation of the higher worlds, or devachan. These worlds are real, as I have explained to you in these lectures. What lies within these higher worlds can be attained only by developing new possibilities for vision. Human beings must be active in order to reach these worlds.
[Second text variant: (Vetfelahn):] If we want to acquire a true view of four-dimensional space, we must do very specific exercises. First, we must cultivate a clear and profound vision of water. Such vision cannot be achieved as a matter of course. We must immerse ourselves in the nature of water with great precision. We must creep inside water, so to speak. Second, we must create a vision of the nature of light. Although light is familiar to us, we know it only in the form in which we receive it from outside. By meditating, we acquire the inner counterpart of outer light. We learn where light comes from, so we ourselves become able to produce light. We can do this by truly allowing these concepts to work on our souls during meditation or sense-free thinking. Our entire surroundings are revealed to us as streaming light. Then we must "chemically combine" the mental image of water that we have cultivated with that of light. Water fully imbued with light is what the alchemists called mercury. In the language of alchemy, water plus light equals mercury. This alchemical mercury, however, is not simply ordinary quicksilver. We must first awaken our own ability to create mercury out of the concept of light. We then take possession of mercury, the light-imbued power of water, which is one element of the astral world.
The second element arises when we cultivate a vivid mental image of air and then extract the power of air through a spiritual process, combining it with feeling inside us to kindle the concept of warmth, or fire. One element is extracted, while we ourselves produce the other. These two — air plus fire — yield what the alchemists called sulfur, or shining fire-air. The watery element is truly the substance referred to in the biblical statement 'The Spirit of God brooded upon the face of the waters." [Note 35]
The third element is "Spirit-God," or earth combined with sound. It comes about when we extract the power of the earth and combine it with sound. The biblical "waters" are not ordinary water but what we know as astral substance, which consists of four types of forces: water, air, light, and fire. These four forces constitute the four dimensions of astral space.
As you can see, astral substance is half subjective, — only one part of astral substance can be acquired from the environment. The other part is acquired through objectification from conceptual and emotional forces. In devachan, we would find only a completely subjective element, — there is no objectivity there. Everything we do here is simply a symbolic representation of the world of devachan. What lies within the higher worlds can be reached only by developing in ourselves new ways of perceiving. Human beings must be active in order to reach these worlds.
In the course of his life, Hinton developed and popularized not one but many methods of representing four-dimensional space in three-dimensional perceived space. He was noted more for his popularization of the subject than for his mathematical originality. See the list of Hinton's works in the bibliography.
Hinton employed several different color systems and distributions of color. He saw the two-dimensional representation of three-dimensional figures as preparation for the three-dimensional representation of four-dimensional figures (see A New Era of Thought , part II, chapters I-IV and VII, and The Fourth Dimension , chapters XI-XIII). Steiner seems to be referring to a very simplified version of one of Hinton's systems.
It is not evident from the context of the lecture whether Steiner intended the colors to suggest specific attributes of the corresponding dimensions, but it seems unlikely. The various transcriptions of the lecture differ substantially at this point, presumably owing to different ways of adapting Steiner's use of color (especially white) on the dark board to white paper.
These models were not found among Steiners belongings after his death. Presumably, they were returned to J. A. Schouten (see the letter to that effect in Note 1 of Lecture 3).
A cube bounded by six surfaces can be created by moving a square with its four edges in three-dimensional space. The six surfaces consist of the initial and final cubes plus the four produced by the movement of the edges. This is immediately apparent in the parallel projection of this movement onto a plane — that is, into two-dimensional space (see Figure 88). Similarly, the movement of a cube with six surfaces in four-dimensional space creates a figure with eight cubes forming its boundaries-the initial and final cubes plus the six created through the movement of the sides-as is easily apparent from a parallel projection of the cube's movement into three-dimensional space (see Figure 90).
Hinton seems to have coined the term tessaract for the four-dimensional figure analogous to the cube. The spelling tessarat also occurs in his works.
Hinton's The Fourth Dimension , chapter XII, contains almost the same reasoning and identical figures.
Goethe, Faust, part I, scene 4, Faust's study, verses 2065ff:
So now we simply spread the cloak
That is to carry both of us through the air.
But do not bring too large a bundle
As you take this daring step.
A little fire-air I shall create
To lift us swiftly from the earth.
Once lightened, we shall quickly rise, —
Congratulations on your new career!
Genesis 1:2. See Rudolf Steiner, Genesis :The Secrets of the Biblical Story of Creation (GA 122), especially the lecture of August 20, 1910.