Today I must conclude these lectures on the fourth dimension of space, though I actually would like to present a complicated system in greater detail, which would require demonstrating many more of Hinton's models. All I can do is refer you to his three thorough and insightful books. [Note 47] Of course, no one who is unwilling to use analogies such as those presented in the previous lectures will be able to acquire a mental image of four-dimensional space. A new way of developing thoughts is needed.
Now I would like to develop a real image (parallel projection) of a tessaract. We saw that a square in two-dimensional space has four edges. Its counterpart in three dimensions is the cube, which has six square sides (Figure 42).
Figure 42
The four-dimensional counterpart is the tessaract, which is bounded by eight cubes. Consequently, the projection of a tessaract into three-dimensional space consists of eight interpenetrating cubes. We saw how these eight cubes can coincide in three-dimensional space. I will now construct a different projection of a tessaract. [Note 48]
Imagine holding a cube up to the light so that it casts a shadow on the board. We can then trace the shadow with chalk (Figure 43). As you see, the result is a hexagon. If you imagine the cube as transparent, you can see that in its projection onto a plane, the three front faces coincide with the three rear faces to form the hexagonal figure.
Figure 43
To get a projection that we can apply to a tessaract, please imagine that the cube in front of you is positioned so that the front point \(A\) exactly covers the rear point \(C\). If you then eliminate the third dimension, the result is once again a hexagonal shadow. Let me draw this for you (Figure 44).
Figure 44
When you imagine the cube in this position, you see only its three front faces,—the three other faces are concealed behind them. The faces of the cube appear foreshortened, and its angles no longer look like right angles. Seen from this perspective, the cube looks like a regular hexagon. Thus, we have created an image of a three-dimensional cube in two-dimensional space. Because this projection shortens the edges and alters the angles, we must imagine the six square faces of the cube as rhombuses. [Note 49]
Now let's repeat the operation of projecting a three-dimensional cube onto a plane with a four-dimensional figure that we project into three-dimensional space. We will use parallel projection to depict a tessaract, a figure composed of eight cubes, in the third dimension. Performing this operation on a cube results in three visible and three invisible edges,—in reality, they jut into space and do not lie flat in the plane of projection. Now imagine a cube distorted into a rhombic parallelepiped. [Note 50] If you take eight such figures, you can assemble the structures defining a tessaract so that they interpenetrate and doubly coincide with the rhombic cubes in a rhombic dodecahedron (Figure 45).
Figure 45
This figure has one more axis than a three-dimensional cube. Naturally, a four-dimensional figure has four axes. Even when its components interpenetrate, four axes remain. Thus, this projection contains eight interpenetrating cubes, shown as rhombic cubes. A rhombic dodecahedron is a symmetrical image or shadow of a tessaract projected into three-dimensional space. [Note 51]
Although we have arrived at these relationships by analogy, the analogy is totally valid. Just as a cube can be projected onto a plane,—a tessaract also can be depicted by projecting it into three-dimensional space. The resulting projection is to the tessaract as the cube's shadow is to the cube. I believe this operation is readily understandable.
I would like to link what we have just done to the wonderful image supplied by Plato and Schopenhauer in the metaphor of the cave. [Note 52] Plato asks us to imagine people chained in a cave so that they cannot turn their heads and can see only the rear wall. Behind them, other people carry various objects past the mouth of the cave. These people and objects are three dimensional, but the prisoners see only shadows cast on the wall. Everything in this room, for example, would appear only as two-dimensional shadow images on the opposite wall.
Then Plato tells us that our situation in the world is similar. We are the people chained in the cave. Although we ourselves are four-dimensional, as is everything else, all that we see appears only in the form of images in three-dimensional space. [Note 53] According to Plato, we are dependent on seeing only the three-dimensional shadow images of things instead of their realities. I see my own hand only as a shadow image,—in reality, it is four-dimensional. We see only images of four-dimensional reality, images like that of the tessaract that I showed you.
In ancient Greece, Plato attempted to explain that the bodies we know are actually four-dimensional and that we see only their shadow images in three-dimensional space. This statement is not completely arbitrary, as I will explain shortly. Initially, of course, we can say that it is mere speculation. How can we possibly imagine that there is any reality to these figures that appear on the wall? But now imagine yourselves sitting here in a row, unable to move. Suddenly, the shadows begin moving. You cannot possibly conclude that the images on the wall could move without leaving the second dimension. When an image moves on the wall, something must have caused movement of the actual object, which is not on the wall. Objects in three-dimensional space can move past each other, something their two-dimensional shadow images cannot do if you imagine them as impenetrable—that is, as consisting of substance. If we imagine these images to be substantial, they cannot move past each other without leaving the second dimension.
As long as the images on the wall remain motionless, I have no reason to conclude that anything is happening away from the wall, outside the realm of two-dimensional shadow images. As soon as they begin to move, however, I am forced to investigate the source of the movement and to conclude that the change can originate only in a movement outside the wall, in a third dimension. Thus, the change in the images has informed us that there is a third dimension in addition to the second.
Although a mere image possesses a certain reality and very specific attributes, it is essentially different from the real object. A mirror image, too, is undeniably a mere image. You see yourself in the mirror, but you are also present out here. Without the presence of a third element—that is, a being that moves—you cannot really know which one is you. The mirror image makes the same movements as the original,—it has no ability to move itself but is dependent on the real object, the being. In this way, we can distinguish between an image and a being by saying that only a being can produce change or movement out of itself. I realize that the shadow images on the wall cannot make themselves move,—therefore, they are not beings. I must transcend the images in order to discover the beings.
Now apply this train of thought to the world in general. The world is three-dimensional, but if you consider it by itself, grasping it in thought, you will discover that it is essentially immobile. Even if you imagine it frozen at a certain point in time, however, the world is still three-dimensional. In reality, the world is not the same at any two points in time. It changes. Now imagine the absence of these different moments—what is, remains. If there were no time, the world would never change, but even without time or changes it would still be three-dimensional. Similarly, the images on the wall remain two dimensional, but the fact that they change suggests the existence of a third dimension. That the world is constantly changing but would remain three-dimensional even without change suggests that we need to look for the change in a fourth dimension. The reason for change, the cause of change, the activity of change, must be sought outside the third dimension. At this point you grasp the existence of the fourth dimension and the justification for Plato's metaphor. We can understand the entire three-dimensional world as the shadow projection of a four-dimensional world. The only question is how to grasp the reality of this fourth dimension.
Of course, we must understand that it is impossible for the fourth dimension to enter the third directly. It cannot. The fourth dimension cannot simply fall into the third dimension. Now I would like to show you how to acquire a concept of transcending the third dimension. (In one of my earlier lectures here, I attempted to awaken a similar idea in you.) [Note 54] Imagine that we have a circle. If you picture this circle getting bigger and bigger, so that any specific segment becomes flatter and fatter, the diameter eventually becomes so large that the circle is transformed into a straight line. A line has only one dimension, but a circle has two. How do we get back into two dimensions? By bending the straight line to form a circle again.
When you imagine curving a circular surface, it first becomes a bowl and eventually, if you continue to curve it, a sphere. A curved line acquires a second dimension and a curved plane a third. And if you could still make a cube curve, it would have to curve into the fourth dimension, and the result would be a spherical tessaract. [Note 55] A spherical surface can be considered a curved two-dimensional figure. In nature, the sphere appears in the form of the cell, the smallest living being. The boundaries of a cell are spherical. Here we have the difference between the living and the lifeless. Minerals in their crystalline form are always bounded by planes, by flat surfaces, while life is built up out of cells and bounded by spherical surfaces. Just as crystals are built up out of flattened spheres, or planes, life is built up out of cells, or abutting spheres. The difference between the living and the lifeless lies in the character of their boundaries. An octahedron is bounded by eight triangles. When we imagine its eight sides as spheres, the result is an eight-celled living thing.
When you "curve" a cube, which is a three-dimensional figure, the result is a four-dimensional figure, the spherical tessaract. But if you curve all of space, the resulting figure relates to three-dimensional space as a sphere relates to a plane. [Note 56] As a three-dimensional object, a cube, like any crystal, is bounded by planes. The essence of a crystal is that it is constructed of flat boundary planes. The essence of life is that is constructed of curved surfaces, namely, cells, while a figure on a still higher level of existence would be bounded by four-dimensional structures. A three-dimensional figure is bounded by two-dimensional figures. A four-dimensional being—that is, a living thing—is bounded by three-dimensional beings, namely, spheres and cells. A five-dimensional being is bounded by four-dimensional beings, namely, spherical tessaracts. Thus, we see the need to move from three-dimensional beings to four-dimensional and then five-dimensional beings.
What needs to happen with a four-dimensional being? [Note 57] A change must take place within the third dimension. In other words, when you hang pictures, which are two-dimensional, on the wall, they generally remain immobile. When you see two-dimensional images moving, you must conclude that the cause of the movement can lie only outside the surface of the wall—that is, that the third dimension of space prompts the change. When you find changes taking place within the third dimension, you must conclude that a fourth dimension has an effect on beings who experience changes within their three dimensions of space.
We have not truly recognized a plant when we know it only in its three dimensions. Plants are constantly changing. Change is an essential aspect of plants, a token of a higher form of existence. A cube remains the same,—its form changes only when you break it. A plant changes shape by itself, which means that the change must be caused by some factor that exists outside the third dimension and is expressed in the fourth dimension. What is this factor?
You see, if you draw this cube at different points in time, you will find that it always remains the same. But when you draw a plant and compare the original to your copy three weeks later, the original will have changed. Our analogy, therefore, is fully valid. Every living thing points to a higher element in which its true being dwells, and time is the expression of this higher element. Time is the symptomatic expression or manifestation of life (or the fourth dimension) in the three dimensions of physical space. In other words, all beings for whom time is intrinsically meaningful are images of four-dimensional beings. After three years or six years, this cube will still be the same. A lily seedling changes, however, because time has real meaning for it. What we see in the lily is merely the three-dimensional image of the four-dimensional lily being. Time is an image or projection of the fourth dimension, of organic life, into the three spatial dimensions of the physical world.
To clarify how each successive dimension relates to the preceding one, please follow this line of thought: A cube has three dimensions. To imagine the third, you tell yourself that it is perpendicular to the second and that the second is perpendicular to the first. It is characteristic of the three dimensions that they are perpendicular to each other. We also can conceive of the third dimension as arising out of the next dimension, the fourth. Envision coloring the faces of a cube and manipulating the colors in a specific way, as Hinton did. The changes you induce correspond exactly to the change undergone by a three-dimensional being when it develops over time, thus passing into the fourth dimension. When you cut through a four-dimensional being at any point—that is, when you take away its fourth dimension—you destroy the being. Doing this to a plant is just like taking an impression of the plant and casting it in plaster. You hold it fast by destroying its fourth dimension, the time factor, and the result is a three-dimensional figure. When time, the fourth dimension, is critically important to any three-dimensional being, that being must be alive.
And now we come to the fifth dimension. You might say that this dimension must have another boundary that is perpendicular to the fourth dimension. We saw that the relationship between the fourth dimension and the third is similar to the relationship between the third and second dimensions. It is more difficult to imagine the fifth dimension, but once again we can use an analogy to give us some idea about it. How does any dimension come about? When you draw a line, no further dimensions emerge as long as the line simply continues in the same direction. Another dimension is added only when you imagine two opposing directions or forces that meet and neutralize at a point. The new dimension arises only as an expression of the neutralization of forces. We must be able to see the new dimension as the addition of a line in which two streams of forces are neutralized. We can imagine the dimension as coming either from the right or from the left, as positive in the first instance and negative in the second. Thus I grasp each independent dimension as a polar stream of forces with both a positive and a negative component. The neutralization of the polar component forces is the new dimension.
Taking this as our starting point, let's develop a mental image of the fifth dimension. We must first imagine positive and negative aspects of the fourth dimension, which we know is the expression of time. Let's picture a collision between two beings for whom time is meaningful. The result will have to be similar to the neutralization of opposing forces that we talked about earlier. When two four-dimensional beings connect, the result is their fifth dimension. The fifth dimension is the result or consequence of an exchange or neutralization of polar forces, in that two living things who influence each other produce something that they do not have in common either in the three ordinary dimensions of space or in the fourth dimension, in time. This new element has its boundaries outside these dimensions. It is what we call empathy or sensory activity, the capacity that informs one being about another. It is the recognition of the inner (soul-spiritual) aspect of another being. Without the addition of the higher, fifth dimension—that is, without entering the realm of sensory activity,—no being would ever be able to know about any aspects of another being that lie outside time and space. Of course, in this sense we understand sensory activity simply as the fifth dimension's projection or expression in the physical world.
It would be too difficult to build up the sixth dimension in the same way, so for now I will simply tell you what it is. If we continued along the same line of thinking, we would find that the expression of the sixth dimension in the three-dimensional world is self-awareness. As three-dimensional beings, we humans share our image character with other three-dimensional beings. Plants possess an additional dimension, the fourth. For this reason, you will never discover the ultimate being of the plant in the three dimensions of space. You must ascend to a fourth dimension, to the astral sphere. If you want to understand a being that possesses sensory ability, you must ascend to the fifth dimension, lower devachan or the Rupa sphere, and to understand a being with self-awareness—namely, the human being—you must ascend to the sixth dimension, upper devachan or the Arupa sphere. The human beings we encounter at present are really six-dimensional beings. What we have called sensory ability (or empathy) and self-awareness are projections of the fifth and sixth dimensions, respectively, into ordinary three-dimensional space. Albeit unconsciously for the most part, human beings extend all the way into these spiritual spheres,—only there can their essential nature be recognized. As six-dimensional beings, we understand the higher worlds only when we attempt to relinquish the characteristic attributes of lower dimensions.
I cannot do more than suggest why we believe the world to be merely three-dimensional. Our view is based on seeing the world as a reflection of higher factors. The most you can see in a mirror is a mirror image of yourself. In fact, the three dimensions of our physical space are reflections, material images of three higher, causal, creative dimensions. Thus, our material world has a polar spiritual counterpart in the group of the three next higher dimensions, that is, in the fourth, fifth, and sixth. Similarly, the fourth through sixth dimensions have their polar counterparts in still more distant spiritual worlds, in dimensions that remain a matter of conjecture for us.
Consider water and water that has been allowed to freeze. In both cases, the substance is the same, but water and ice are very different in form. You can imagine a similar process taking place with regard to the three higher human dimensions. When you imagine human beings as purely spiritual beings, you must envision them as possessing only the three higher dimensions of self-awareness, feeling, and time and that these dimensions are reflected in the three ordinary dimensions in the physical world.
When yogis (students of esotericism) want to ascend to knowledge of the higher worlds, they must gradually replace reflections with realities. For example, when they consider a plant, they must learn to replace the lower dimensions with the higher ones. Learning to disregard one of a plant's spatial dimensions and substitute the corresponding higher dimension—namely, time—enables them to understand a two-dimensional being that is moving. What must students of esotericism do to make this being correspond to reality rather than remaining a mere image? If they were simply to disregard the third dimension and add the fourth, the result would be something imaginary. The following thought will help us move toward an answer: By filming a living being, even though we subtract the third dimension from events that were originally three-dimensional, the succession of images adds the dimension of time. When we then add sensory ability to this animated image, we perform an operation similar to the one I described as curving a three-dimensional figure into the fourth dimension. The result of this operation is a four-dimensional figure whose dimensions include two of our spatial dimensions and two higher ones, namely, time and sensory ability. Such beings do indeed exist, and now that I have come to the real conclusion of our study of the dimensions, I would like to name them for you.
Imagine two spatial dimensions—that is, a plane—and suppose that this plane is endowed with movement. Picture it curving to become a sensate being pushing a two-dimensional surface in front of it. Such a being is very dissimilar to and acts very differently from a three-dimensional being in our space. The surface being that we have constructed is completely open in one direction. It looks two-dimensional,—it comes toward you, and you cannot get around it. This being is a radiant being,—it is nothing other than openness in a particular direction. Through such a being, initiates then become familiar with other beings whom they describe as divine messengers approaching them in flames of fire. The description of Moses receiving the Ten Commandments on Mount Sinai shows simply that he was approached by such a being and could perceive its dimensions. [Note 58] This being, which resembled a human being minus the third dimension, was active in sensation and time.
The abstract images in religious documents are more than mere outer symbols. They are mighty realities that we can learn about by taking possession of what we have been attempting to understand through analogies. The more diligently and energetically you ponder such analogies, the more eagerly you submerse yourself in them, the more they work on your spirit to release higher capacities. This applies, for example, to the explanation of the analogy between the relationship of a cube to a hexagon and that of a tessaract to a rhombic dodecahedron. The latter is the projection of a tessaract into the three-dimensional physical world. By visualizing these figures as if they possessed independent life—that is, by allowing a cube to grow out of its projection, the hexagon, and the tessaract to develop out of its projection, the rhombic dodecahedron—your lower mental body learns to grasp the beings I just described. When you not only have followed my suggestions but also have made this operation come alive as esoteric students do, in full waking consciousness, you will notice that four-dimensional figures begin to appear in your dreams. At that point, you no longer have far to go to be able to bring them into your waking consciousness. You then will be able to see the fourth dimension in every four-dimensional being.
The astral sphere is the fourth dimension. Devachan up to Rupa is the fifth dimension. Devachan up to Arupa is the sixth dimension. [Note 59]
These three worlds—physical, astral, and heavenly (devachan)—encompass six dimensions. The still higher worlds are the polar opposites of these dimensions.
PresumabIy, this reference is to Hinton's books Scientific Romances [1886], A New Era of Thought [ 1900], and The Fourth Dimension [ 1904].
Strictly speaking, the depiction of a tessaract in the previous lecture (May 31, 1905) is not a projection but simply an unfolded view. In the present lecture, Steiner proceeds to construct an orthogonal parallel projection of a tessaract in three-dimensional space, taking one of its diagonals as the direction of projection.
Considering the framework formed by the edges of a cube, an oblique parallel projection of the cube onto a plane generally consists of two parallel, non-coinciding cubes and the line segments connecting their corresponding corners (Figure 88: oblique parallel projections of a cube).
Figure 88
If the diagonal AC is selected as the direction of projection, vertices A and C coincide, producing an oblique hexagon and its diagonals. The images of the six individual faces of the cube can be reconstructed from this hexagon by tracing all the possible parallelograms defined by the existing structure of lines. Each of these parallelograms overlaps with two others, and the hexagon's surface is covered twice by the faces of the cube. When the direction of projection is perpendicular to the plane of projection, the resulting image of a cube is a regular hexagon (Figure 89: orthogonal parallel projections of a cube).
Figure 89
Note that the three diagonals of the hexagon also represent the three (zone) axes of the cube. The zone associations belonging to each of these axes—that is, the four faces of the cube that parallel it—appear as four parallelograms or rhombuses with one edge coinciding with the corresponding axis.
’“Earlier in this lecture, Steiner called a distorted or oblique square a "rhombus," which is a parallelogram with four equal sides. The corresponding solid figure, Steiner's "rhombic parallelepiped," is an oblique cube—i.e., a parallelepiped whose edges are all the same length.
If we see the tessaract as the framework formed by its edges, the result of projecting the tessaract into three-dimensional space generally consists of two parallel, displaced oblique cubes and the line segments connecting their corresponding vertices (Figure 90: oblique parallel projections of a tessaract).
Figure 90
When the direction of projection passes through the diagonal A'C, the endpoints A' and C coincide, resulting in a rhombic dodecahedron with four diagonals. In the first figure, it is easy to trace the images of the eight cubes defining the boundaries of a tessaract: they are all the possible parallelepipeds formed by the edges of the existing framework. These parallelepipeds include the original cube, the displaced cube, and the six parallelepipeds that share one face each with the original and displaced cubes. This situation does not change fundamentally when we make the transition to the rhombic dodecahedron, except that in this instance all of the "rhombic cubes" (parallelepipeds) interpenetrate in such a way that they fill the internal space of the rhombic dodecahedron exactly twice, with each parallelepiped including portions of three others.
The four diagonals of the rhombic dodecahedron that appear in the projection of the tessaract are the zone axes of the rhombic dodecahedron's four associations of six faces each. Each such face association consists of all six surfaces that are parallel to a single zone axis. (Note that in a rhombic dodecahedron the axes pass through vertices rather than through the centers of faces, as in a cube.)
These four axes, however, are also the projections of the tessaract's four perpendicular axes in four-dimensional space. A cube's three axes pass through the centers of its square sides. Analogously, a tessaract's axes pass through the middle of the cubes that form its sides. In parallel projection, the middle of a cube is transformed into the middle of the corresponding parallelepiped. As we can ascertain by studying all eight parallelepipeds in a rhombic dodecahedron, the four axes pass exactly through the middles of these parallelepipeds.
A cube's four perpendicular axes are simultaneously the zone axes of its three face associations of four faces each. Similarly, a tessaract's four axes are also the zone axes of four cell associations of six cells each (cell equals the cube forming a side of the tessaract). In the rhombic dodecahedron, the cells belonging to each axis are easy to find: they are the six parallelepipeds with one edge coinciding with that axis.
Plato, The Republic, book 7, 514a-518c. It has not yet been possible to ascertain where Schopenhauer used this metaphor.
Zöllner drew attention to this interpretation of Plato’s cave metaphor in his essay Über Wirkungen in die Ferne [ 1878a], pp. 260ff.
See the lecture of March 24, 1905.
What Steiner seems to mean here by spherical tessaract is not a four-dimensional cube in the narrower sense but rather its topological equivalent, the three-dimensional sphere in four-dimensional space, which is produced by curving and attaching two solid three-dimensional spheres. See Note 11 , Lecture 5.
See Note 9 above and Note 11, Lecture 5.
The remaining text of this lecture incorporates fragments of transcripts quoted in Haase's essay [1916], which helped clarify the meaning.
Exodus 19,—also Exodus 33 and 34.
ln theosophical literature, the three upper regions of the land of spirit were called Arupa regions, in contrast to the four lower, or Rupa, regions. See the editor's note to Rudolf Steiner's Die Grundelemente der Esoterik,'The Basic Elements of Esotericism" (GA 93a), p. 281 ff. On the seven regions of the land of spirit, see Rudolf Steiner, Theosophy (GA 9), 'The Country of Spirit Beings." On the problem of dimensionality in connection with the planes or regions of the spirit world, see also Rudolf Steiner's lecture of May 17, 1905,—his response to questions asked by A. Strakosch on March 11, 1920,—the questions and answers of April 7, 1921 (GA 76), and April 12, 1922 (GA 82); and the lectures of August 19, 29, 22, and 26, 1923 (GA 227).
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