The Tessaract: Geometry of Higher Dimensional Reality

GA 324a — 7 June 1905, Berlin

Sixth Lecture

I would like to conclude the lectures on the fourth spatial dimension today if possible, although I would like to demonstrate a complicated system in more detail today. I would have to show you many more models after Hinton; therefore, I can only refer you to the three detailed and spirited books.” Those who do not have the will to form a picture through analogies in the way we have heard it in the past lectures cannot, of course, form a picture of four-dimensional space. It involves a new way of forming thoughts.

I will try to give you a true representation [parallel projection] of the tessaract. You know that in two-dimensional space we had the square, which is bounded by four sides. This is the three-dimensional cube, which is bounded by six squares (Figure 42).

In four-dimensional space, we have the tessaract. A tessaract is bounded by eight cubes. The projection of a tessaract [in three-dimensional space] therefore consists of eight interlocking cubes. We have seen how the [corresponding eight] cubes can be intertwined in three-dimensional space. Today I will show you a [different] way of projecting the tessaract.

You can imagine that the cube, when held up to the light, throws a shadow on the blackboard. We can mark this shadow figure with chalk (Figure 43). You see that a hexagon is obtained. Now imagine this cube transparent, and you will observe that in the hexagonal figure the three front sides of the cube and the three rear sides of the cube fall into the same plane.

In order to get a projection that we can apply to the tessaract, I would ask you to imagine that the cube is standing in front of you in such a way that the front point A covers the rear point C. If you imagine the third dimension, all this would give you a hexagonal shadow again. I will draw the figure for you (Figure 44).

If you imagine the cube like this, you would see the three front surfaces here; the other surfaces would be behind them. The surfaces of the cube appear foreshortened and the angles are no longer right angles. This is how you see the cube depicted so that the surfaces form a regular hexagon. Thus, we have obtained a representation of a three-dimensional cube in two-dimensional space. Since the edges are shortened and the angles are changed by the projection, we must therefore imagine the [projection of the] six boundary squares of the cube as shifted squares, as rhombi.

The same story that I did with a three-dimensional cube that I projected into the plane, we want to do this procedure with a four-dimensional spatial object, which we therefore have to place in three-dimensional space. We must therefore bring the structure composed of eight cubes, the tessaract, into the third dimension [by parallel projection]. With the cube, we obtained three visible and three invisible edges, all of which enter into the space and in reality do not lie within the [projection] surface. Now imagine a cube shifted in such a way that it becomes a rhombicuboctahedron.” Take eight of these figures, and you have the possibility of combining the eight [boundary] cubes of the tessaract in such a way that, when pushed together, they form the eight (doubly covered) rhombicuboctahedra of this spatial figure (Figure 45).

Now you have one more axis here [than in the three-dimensional cube]. Accordingly, a four-dimensional spatial structure naturally has four axes. So if we push it together, four axes still remain. There are eight [pushed together] cubes in this projection, which are represented as rhombicuboctahedra. The rhombicuboctahedron is a [symmetrical] image or silhouette of the tessaract in three-dimensional space.

We arrived at this relationship by means of an analogy, but it is completely correct: just as we obtained a projection of the cube onto a plane, it is also possible to represent the tessaract in three-dimensional space by means of a projection. It behaves in the same way as the silhouette of the cube in relation to the cube itself. I think that is quite easy to understand.

Now I would like to tie in with the greatest image that has ever been given for this, namely Plato and Schopenhauer and the parable of the cave.

Plato says: Imagine people sitting in a cave, and they are all tied up so that they cannot turn their heads and can only look at the opposite wall. Behind them are people carrying various objects past them. These people and these objects are three-dimensional. So all these [bound] people stare at the wall and see only what is cast as a shadow [of the objects] on the wall. So they would see everything in the room only as a shadow on the opposite wall as two-dimensional images.

Plato says that this is how it is in the world in general. In truth, people are sitting in the cave. Now, people themselves and everything else are four-dimensional; but what people see of it are only images in three-dimensional space.

This is how all the things we see present themselves. According to Plato, we are dependent on seeing not the real things, but the three-dimensional silhouettes. I only see my hand as a silhouette; in reality it is four-dimensional, and everything that people see of it is just as much an image of it as what I just showed you as an image of the Tessaract. Thus Plato was already trying to make clear that the objects we know are actually four-dimensional, and that we only see silhouettes of them in three-dimensional space. And that is not entirely arbitrary. I will give you the reasons for this in a moment.

Of course, anyone can say from the outset that this is mere speculation. How can we even imagine that the things that appear on the wall have a reality? Imagine that you are sitting here in a row, and you are sitting very still. Now imagine that the things on the wall suddenly start to move. You will not be able to tell yourself that the images on the wall can move without going out of the second dimension. If something moves there, it indicates that something must have happened outside the wall, on the real object, for it to move at all. That's what you tell yourself. If you imagine that the objects in three-dimensional space can pass each other, this would not be possible with their two-dimensional silhouettes, if you think of them as substantial, that is, impenetrable. If those images, conceived substantially, wanted to move past each other, they would have to go out of the second dimension.

As long as everything on the wall is at rest, I have no reason to conclude that something is happening outside the wall, outside the space of the two-dimensional silhouettes. But the moment history begins to move, I must investigate the source of the motion. And you realize that the change can only come from motion outside the wall, only from motion within a third dimension. The change has thus told us that there is a third dimension in addition to the second.

What is a mere image also has a certain reality, possesses very definite properties, but differs essentially from the real object. You will not be able to deny that the mirror image is also a mere image. You see yourself in the mirror, and you are also there. If there is not a third [that is, an active being] there, then you could not actually know what you are. But the mirror image makes the same movements that the original makes; the image is dependent on the real object, the being; it itself has no ability [to move]. Thus, a distinction can be made between image and being in that only a being can bring about movement and change out of itself. I realize from the shadows on the wall that they cannot move themselves, so they cannot be beings. I have to go out of them if I want to get to the beings.

Now apply this to the world in general. The world is three-dimensional. Take this three-dimensional world for itself, as it is; grasp it completely in your thoughts [for yourself], and you will find that it remains rigid. It remains three-dimensional even if you suddenly think the world frozen at a certain point in time. But there is no one and the same world in two points in time. The world is completely different at successive points in time. Imagine that these points in time cease to exist, so that what is there remains. Without time, no change would occur in the world. The world would remain three-dimensional even if it underwent no change at all. The pictures on the wall also remain two-dimensional. But change suggests a third dimension. The fact that the world is constantly changing, and that it remains three-dimensional even without change, suggests that we have to look for the change in a fourth dimension. We have to look for the reason, the cause of the change, the activity outside the third dimension, and with that you have initially uncovered the fourth of the dimensions. But with that you also have the justification for Plato's image. So we understand the whole three-dimensional world as the shadow projection of a four-dimensional world. The only question is how we have to take this fourth dimension [in reality].

You see, we have the one idea to make it clear to ourselves, of course, that it is impossible for the fourth dimension to fall [directly] into the third. That is not possible. The fourth dimension cannot fall into the third. I would like to show you now how one can, so to speak, get an idea of how to go beyond the third dimension. Imagine we have a circle – I have already tried to evoke a similar idea recently – if you imagine this circle getting bigger and bigger, then a piece of this circle becomes flatter and flatter, and because the diameter of the circle becomes very large at the end, the circle finally turns into a straight line. The line has one dimension, but the circle has two dimensions. How do you get a second dimension from a single dimension? By curving a straight line, you get a circle again.

If you now imagine the surface of the circle curving into space, you first get a shell, and if you continue to do this, you get a sphere. Thus a line acquires a second dimension by curvature and a surface acquires a third dimension by curvature. If you could now curve a cube, it would have to be curved into the fourth dimension, and you would have the [spherical] tessaract.

You can understand the sphere as a curved two-dimensional spatial structure. The sphere that occurs in nature is the cell, the smallest living thing. The cell is limited spherically. That is the difference between the living and the lifeless. The mineral always occurs as a crystal bounded by flat surfaces; life is bounded by spherical surfaces, built up of cells. That means that just as a crystal is built from spheres that have been straightened out, that is, from planes, so life is built from cells, that is, from spheres that have been bent together. The difference between the living and the dead lies in the way they are defined. The octahedron is defined by eight triangles. If we imagine the eight sides as spheres, we would get an eight-limbed living thing.

If you curve the three-dimensional structure, the cube, again, you get a four-dimensional structure, the spherical tessaract. But if you curve the whole space, you get something that relates to three-dimensional space in the same way that a sphere relates to a plane.

Just as the cube, as a three-dimensional structure, is bounded by planes, so every crystal is bounded by planes. The essence of a crystal is the assembly of [flat] boundary planes. The essence of the living is the assembly of curved surfaces, of cells. The assembly of something even higher would be a structure whose individual boundaries would be four-dimensional. A three-dimensional structure is bounded by two-dimensional structures. A four-dimensional being, that is, a living being, is bounded by three-dimensional beings, by spheres and cells. A five-dimensional being is itself bounded by four-dimensional beings, by spherical tessaracts. From this you can see that we have to ascend from three-dimensional to four-dimensional, and then to five-dimensional beings.

We only have to ask ourselves: What must occur in a being that is four-dimensional?* A change must occur within the third dimension. In other words: If you hang pictures on the wall here, they are two-dimensional and generally remain static. But if you have pictures in which the second dimension moves and changes, then you must conclude that the cause of this movement can only lie outside the surface of the wall, that the third dimension of space thus indicates the change. If you find changes within the third spatial dimension itself, then you must conclude that a fourth dimension is involved, and this brings us to the beings that undergo a change within their three spatial dimensions.

It is not true that we have fully recognized a plant if we have only recognized it in its three dimensions. A plant is constantly changing, and this change is an essential, a higher characteristic of it. The cube remains; it only changes its shape when you smash it. A plant changes its shape itself, that is, there is something that is the cause of this change and that lies outside the third dimension and is an expression of the fourth dimension. What is that?

You see, if you have this cube and draw it, you would labor in vain if you wanted to draw it differently at different moments; it will always remain the same. If you draw the plant and compare the picture with your model after three weeks, it will have changed. So this analogy is completely accurate. Everything that lives points to something higher, where it has its true essence, and the expression of this higher is time. Time is the symptomatic expression, the appearance of liveliness [understood as the fourth dimension] in the three dimensions of physical space. In other words, all beings for whom time has an inner meaning are images of four-dimensional beings. This cube is still the same after three or six years. The lily bud changes. Because for it, time has a real meaning. Therefore, what we see in the lily is only the three-dimensional image of the four-dimensional lily being. So time is an image, a projection of the fourth dimension, the organic liveliness, into the three spatial dimensions of the physical world.

To understand how a following dimension relates to the preceding one, please imagine the following: a cube has three dimensions; when you visualize the third, you have to remember that it is perpendicular to the second, and the second is perpendicular to the first. The three dimensions are characterized by the fact that they are perpendicular to one another. But we can also imagine how the third dimension arises from the following [fourth dimension]. Imagine that you would change the cube by coloring the boundary surfaces and then changing these colors [in a certain way, as in Hinton's example]. Such a change can indeed be made, and it corresponds exactly to the change that a three-dimensional being undergoes when it passes into the fourth dimension, when it develops through time. If you cut a four-dimensional being at any point, you take away the fourth dimension, you destroy it. If you do that to a plant, you do exactly the same thing as if you were to make a cast of the plant, a plaster cast. You have captured that by destroying the fourth dimension, time. Then you get a three-dimensional object. If for any three-dimensional being the fourth dimension, time, has an essential significance, then it is a living being.

Now we enter the fifth dimension. You can say to yourself that you must again have a boundary that is perpendicular to the fourth dimension. We have seen that the fourth dimension is related to the third dimension in a similar way to the third dimension being related to the second. It is not immediately possible to visualize the fifth dimension in this way. But you can again create a rough idea by using an analogy. How does a dimension come into being in the first place? If you simply draw a line, you will never create another dimension by simply pushing the line in one direction. Only by imagining that you have two opposing directions of force, which then accumulate at a point, only by expressing the accumulation, do you have a new dimension. We must therefore be able to grasp the new dimension as a new line of accumulation [of two currents of force], and imagine the one dimension coming from the right one time and from the left the next, as positive and negative. So I understand a dimension [as a polar [stream of forces] within itself], so that it has a positive and a negative dimension [component], and the neutralization [of these polar force components] is the new dimension.

From there, we want to create an idea of the fifth dimension. We will have to imagine that the fourth dimension, which we have found expressed as time, behaves in a positive and negative way. Now take two beings for whom time has a meaning, and imagine two such beings colliding with each other. Then something must appear as a result, similar to what we have previously called an accumulation of [opposing] forces; and what arises as a result when two four-dimensional beings come into relation with each other is their fifth dimension. This fifth dimension arises as a result, as a consequence of an exchange [a neutralization of polar force effects], in that two living beings, through their mutual interaction, produce something that they do not have outside [in the three ordinary spatial dimensions together], nor do they have in [the fourth dimension,] time, but have completely outside these [previously discussed dimensions or] boundaries. This is what we call compassion [or feeling], by which one being knows another, thus the realization of the [spiritual and mental] inner being of another being. A being could never know anything about another being outside of time [and space] if you did not add a higher, fifth dimension, [i.e. enter the world of] sensation. Of course, here the sensation is only to be understood as a projection, as an expression [of the fifth dimension] in the physical world.

Developing the sixth dimension in the same way would be too difficult, so I will only indicate it. [If we tried to progress in this way, something could be developed as an expression of the sixth dimension that,] when placed in the three-dimensional physical world, is self-conscious.

Man, as a three-dimensional being, is one who shares his imagery with other three-dimensional beings. The plant, in addition, has the fourth dimension. For this reason, you will never find the ultimate essence of the plant within the three dimensions of space, but you would have to ascend from the plant to a fourth spatial dimension [to the astral sphere]. But if you wanted to grasp a being that has feeling, you would have to ascend to the fifth dimension [to the lower Devachan, to the Rupa sphere]; and if you wanted to grasp a being that has self-awareness, a human being, you would have to ascend to the sixth dimension [to the upper Devachan, to the Arupa sphere]. Thus, the human being as he stands before us in the present is indeed a six-dimensional being. That which is called feeling or compassion, or self-awareness, is a projection of the fifth or sixth dimension into ordinary three-dimensional space. Man extends into these spiritual spheres, albeit unconsciously for the most part; only there can he actually be experienced in the sense indicated last. This six-dimensional being can only come to an idea of even the higher worlds if it tries to get rid of the actual characteristics of the lower dimensions.

I can only hint at the reason why man considers the world to be only three-dimensional, namely because he is conditioned in his perception to see only a reflection of something higher in the world. When you look in a mirror, you also see only a reflection of yourself. Thus, the three dimensions of our physical space are indeed reflections, material copies of three higher, causally creative dimensions. Our material world therefore has its polar [spiritual] counter-image in the group of the three next higher dimensions, that is, in those of the fourth, fifth and sixth dimensions. And in a similar sense, the spiritual worlds that lie beyond this group of dimensions, which can only be sensed, are also polar to those of the fourth to sixth dimensions.

If you have water and you let the water freeze, the same substance is present in both cases; but in form they differ quite substantially. You can imagine a similar process for the three higher dimensions of man. If you think of man as a purely spiritual being, then you have to think of him as having only the three higher dimensions – self-awareness, feeling and time – and these three dimensions are reflected in the physical world in its three ordinary dimensions.

The yogi [secret student], if he wants to advance to a knowledge of the higher worlds, must gradually replace the mirror images with reality. For example, when he looks at a plant, he must get used to gradually substituting the higher dimensions for the lower ones. If he looks at a plant and is able to abstract from one spatial dimension in the case of a plant, to abstract from one spatial dimension and instead to imagine a corresponding one of the higher dimensions, in this case time, then he actually gets an idea of what a two-dimensional, moving being is. To make this being more than just an image, to make it correspond to reality, the yogi must do the following. If he disregards the third dimension and adds the fourth, he would only get something imaginary. However, the following mental image can help: when we make a cinematographic representation of a living being, we remove the third dimension from the original three-dimensional processes, but add the [dimension of] time through the sequence of images. If we then add sensation to this [moving] perception, we perform a procedure similar to what I described earlier as the bending of a three-dimensional structure into the fourth dimension. Through this process you then get a four-dimensional entity, but now one that has two of our spatial dimensions, but also two higher ones, namely time and sensation. Such beings do indeed exist, and these beings - and this brings me to a real conclusion to the whole consideration - I would like to tell you about.

Imagine two spatial dimensions, that is, a surface, and this surface endowed with motion. Now imagine a bent as a sensation, a sentient being that then pushes a two-dimensional surface in front of it. Such a being must act differently and be very different from a three-dimensional being in our space. This flat creature that we have constructed in this way is incomplete in one direction, completely open, and offers you a two-dimensional view; you cannot go around it, it comes towards you. This is a luminous creature, and the luminous creature is nothing other than the incompleteness in one direction.

Through such a being, the initiates then get to know other beings, which they describe as divine messengers approaching them in flames of fire. The description of Mount Sinai, where Moses received the Ten Commandments,® means nothing other than that a being could indeed approach him that, to his perception, had these dimensions. It appeared to him like a human being from whom the third spatial dimension had been removed; it appeared in sensation and in time.

These abstract images in the religious documents are not just external symbols, but powerful realities that man can get to know if he is able to appropriate what we have tried to make clear through analogies. The more you devote yourself diligently and energetically to such considerations of analogies, the more you really work on your mind, and the more these [considerations] work in us and trigger higher abilities. [This is roughly the case when dealing with] the analogy of the relationship of the cube to the hexagon and the tessaract to the rhombic dodecahedron. The latter represents a projection of the tessaract into the three-dimensional physical world. If you visualize these figures as living entities, if you allow the cube to grow out of the projection of the die – the hexagon – and likewise allow the tessaract itself to arise from the projection of the tessaract [the rhombic dodecahedron], then you create the possibility and the ability in your lower mental body to grasp what I have just described to you as a structure. And if, in other words, you have not only followed me but have gone through this procedure vividly, as the yogi does in an awakened state of consciousness, then you will notice that something will occur to you in your dreams that in reality is a four-dimensional entity, and then it is not much further to bring it over into the waking consciousness, and you can then see the fourth dimension in every four-dimensional being.

The astral sphere is the fourth dimension. Devachan to rupa is the fifth dimension. Devachan to arupa is the sixth dimension.

These three worlds, the physical, astral and celestial [devachan], comprise six dimensions. The even higher worlds are completely polar to these.

Mineral Plant Animal Human Arupa Self-consciousness Rupa Sensation Self-consciousness Astral plane Life Sensation Self-consciousness Physical form Life Sensation Self-plan consciousness Form Life Sensation Form Life Form