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The Fourth Dimension
GA 324a

Third Lecture

17 May 1905, Berlin

Today I will continue with the difficult subject we have undertaken to explore. We will need to refer back to the topics I mentioned in the last two lectures. After that, I would like to develop a few basic concepts so that in the two final lectures we will be able to use Mr. Schouten's models to fully grasp both the details of the geometric relationships and theosophy's interesting practical perspectives. [Note 22]

As you know, the reason we tried to envision the possibility of four-dimensional space was to gain at least some idea of the socalled astral realm and still higher forms of existence. I have already pointed out that entering the astral world is initially quite confusing for students of esotericism. Without making a closer study of theosophy and esoteric subjects, at least on a theoretical level, it is extremely difficult to form any idea of the very different nature of the objects and beings that we encounter in the so-called astral world. Let me briefly sketch this difference to show you how great it is.

In the simplest example I mentioned, we have to learn to read all numbers in reverse. Esoteric students who are accustomed to reading numbers only as they are read here in the physical world will not be able to find their way through the labyrinth of the. astral realm. In the astral world, a number such as 467 must be read 764. You must become used to reading each number symmetrically, as its mirror image. This is the basic prerequisite. Applying this rule to spatial figures or numbers is easy enough, but it becomes more difficult when we begin to deal with relationships in time, which also must also be interpreted symmetrically — that is, later events come first and earlier events appear later. Thus, when you observe astral events, you must be able to read them backward, from the end to the beginning. I can only suggest the character of these phenomena, which can seem totally grotesque if you have no idea of what is going on. In the astral realm, the son is there first and then the father, — the egg is there first, and the chicken follows. In the physical world, the sequence is different — birth happens first, and birth means that something new emerges from something old. In the astral world, the reverse takes place. There, the old emerges from the new. In the astral realm, the fatherly or motherly element appears to engulf the son or daughter.

Greek mythologies provide a lovely allegory. The three gods Uranus, Kronos, and Zeus symbolize the three worlds. Uranus represents the heavenly world, or devachan, Kronos the astral world, and Zeus the physical world. It is said of Kronos that he devoured his children. [Note 23]

In the astral realm, offspring are not born but devoured.

The issue becomes even more complex when we consider morality on the astral plane. Morality, too, appears in reverse form, or as its own mirror image. You can imagine how greatly explanations of events there differ from our habitual explanations in the physical world. Imagine, for example, that we see a wild animal approaching us in the astral realm, and it strangles us. That is how it appears to someone who is used to interpreting external events, but we cannot interpret this event as we would in the physical world. In reality, the wild animal is an internal quality, — an aspect of our own astral body is strangling us. The attacking strangler is a quality that is rooted in our own desires. If we have a vengeful thought, for example, the thought may appear in external form, tormenting us as the Angel of Death.

In reality, everything in the astral world radiates from us. We must interpret everything that seems to approach us in the astral world as radiating outward from ourselves (Figure 18). It comes back to us on all sides as if from the periphery, from infinite space. In truth, however, we are confronting only what our own astral body has given off.

raying outward and inward
Figure 18

We interpret the astral world correctly and discover its truth only when we are able to bring the periphery into the center, to construe the periphery as the central element. The astral world appears to bear down on you from all sides, but you must envision it as actually radiating outward from you in all directions. At this point I would like to make you aware of a concept that is very important in esoteric schooling. It appears, ghostlike, in many different works on occult research but seldom is it understood correctly. Once you have achieved a certain level of esoteric development, you must learn what your karma predisposes you to find in the astral world. What joys, sorrows, pain, and so forth can you expect to encounter?

Correct theosophical thinking allows you to realize that in this day and age, your outer life and physical body are nothing more than the result, or intersection, of two streams that converge from opposite directions. Picture one stream coming from the past and one coming from the future. The result is two intermingling streams that join together at all these points (Figure 19). Imagine a red stream flowing from one direction and a blue stream flowing from the other. Now picture four different points where the streams join together. At each of these four points, the red and blue streams interact. This is an image of the interaction of four successive incarnations, — in each incarnation we encounter something coming from one direction and something coming from the other. You might say that one stream always travels toward you and that you bring the other stream along with you. Each human being is the confluence of two such streams.

intermingling streams
Figure 19

To gain an idea of this state of affairs, imagine in this way: As you sit here today, you have a certain sum of experiences. At the same time tomorrow, the sum of these events will be different. Now imagine that the experiences you will possess tomorrow are already there. Becoming aware of them would be like seeing a panorama of events coming toward you in space. Imagine that the stream coming toward you from the future is bringing you the experiences you will have between today and tomorrow. You are supported by the past as the future comes to meet you.

At any point in time, two streams flow together to form your life. One stream flows from the future toward the present and the other from the present toward the future, and an interface occurs wherever they meet. Anything that still remains for us to experience in our life appears in the form of astral phenomena, which make a tremendous impression on us.

Imagine that students of esotericism reach the point in their development when they are meant to see into the astral world. Their senses are opened, and they perceive all their future experiences until the end of this time period as outer phenomena surrounding them in the astral world. This sight makes a great impression on each student. An important level in esoteric schooling is reached when students experience an astral panorama of everything they have yet to encounter up to the middle of the sixth root race, which is how long our incarnations will last. The way is opened to them. Without exception, students of esotericism experience all the remaining outer phenomena they will encounter from the near future to the sixth root race.

When you reach this threshold, a question arises: Do you want to experience all this in the shortest possible time? That is the issue for initiation candidates. As you consider this question, your entire future life appears to you in a single moment in the external panorama characteristic of astral vision. Some people will decide not to set out into the astral realm, while others will feel that they must enter. At this point in esoteric development, which is known as the threshold, or moment of decision, we experience ourselves along with everything we must still live through. This phenomenon, which is known as meeting the "guardian of the threshold," is nothing more than facing our own future life. Our own future lies beyond the threshold.

Another unique quality of the world of astral phenomena is initially quite impenetrable if that world is revealed suddenly, through one of life's unforeseen events. When this happens, there is nothing more confusing than this terrible sight. It is good to know about it in case the astral world suddenly breaks in on you as the result of a pathological event, such as the loosening of the connection between the physical body and ether body or between the ether body and the astral body. Such events can reveal a view of the astral world to people who are quite unprepared for it. These people then report seeing apparitions that they cannot interpret because they do not know that they must read them in reverse. For instance, they do not know that a wild animal attacking them must be interpreted as a reflection of an internal quality. In kamaloka, a person's astral forces and passions appear in a great variety of animal forms.

In kamaloka, recently disembodied individuals who still possess all their passions, drives, wishes, and desires are not a pretty sight. Such people, though they are no longer in possession of physical and etheric bodies, still retain all the astral elements that bind them to the physical world and that can be satisfied only through a physical body. Think of average, modern citizens who never amounted to much in their lives and made no particular effort to achieve religious development. They may not have rejected religion in theory, but in practice — that is, as far as their own feelings were concerned — they threw it out the window. It was not a vital element in their lives. What do such people's astral bodies contain? They contain nothing but urges that can be satisfied only through the physical organism, such as the desire to enjoy tasty food, for example. Satisfying this desire, however, would require taste buds. Or the individual in question may long for other pleasures that can be satisfied only by moving the physical body. Suppose that such urges persist, living on in the astral body after the physical body is gone. We find ourselves in this situation if we die without first undergoing astral cleansing and purification. We still have the urge to enjoy tasty food and so on, but such urges are impossible to satisfy. They cause terrible torment in kamaloka, where those who die without first purifying the astral body must lay their desires aside. The astral body is freed only when it learns to relinquish the desires and wishes that can no longer be satisfied.

In the astral world, urges and passions take on animal shapes. As long as a human being is incarnated in a physical body, the shape of the astral body conforms more or less to the human physical body. When the material body is gone, however, the animal nature of urges, desires, and passions is revealed in the forms they assume. In the astral world, therefore, an individual is a reflection of his or her urges and passions. Because these astral beings can also make use of other bodies, it is dangerous to allow mediums to go into a trance without the presence of a clairvoyant who can ward off evil. In the physical world, the form of a lion expresses certain passions, while a tiger expresses other passions and a cat still others. It is interesting to realize that each animal form is the expression of a specific passion or urge.

In the astral world, in kamaloka, we approximate the nature of animals through our passions. This fact is the source of a common misunderstanding with regard to the doctrine of transmigration of souls taught by Egyptian and Indian priests and teachers of wisdom. This doctrine, which teaches that we should live in ways that do not cause us to incarnate as animals, does not apply to physical life but only to higher life. It is intended only to encourage people to live their earthly lives in ways that will not require them to assume animal forms after death, in kamaloka. For example, someone who cultivates the character of a cat during earthly life appears in the form of a cat in kamaloka. To allow individuals to appear in human form in kamaloka is the goal of the doctrine of transmigration of souls. Scholars who fail to understand the true teachings have only an absurd idea of this doctrine.

Figure 20

We saw that when we enter the astral realm of numbers, time, and morality, we are dealing with a complete mirror image of everything we customarily think and do here on the physical plane. We must acquire the habit of reading in reverse, a skill we will need when we enter the astral realm. Learning to read in reverse is easiest when we take up elementary mathematical ideas such as those suggested in the previous lecture. In the discussions that follow, we will become more and more familiar with these ideas. I would like to begin with a very simple one, namely, the idea of a square. Picture a square as you are accustomed to seeing it (Figure 20). I will draw each of its four sides in a different color. This is what a square looks like in the physical world. Now I will draw a square as it looks in devachan. It is impossible to draw this figure precisely, but I want to give you at least an idea of what a square would look like on the mental plane. The mental equivalent [of a square] is something approximating a cross (Figure 21).

Figure 21

Its main features are two intersecting perpendicular axes — or, if you will, two lines that cross each other. The physical counterpart is constructed by drawing lines perpendicular to each of these axes. The physical counterpart of a mental square can best be imagined as a stoppage in two intersecting streams. Let's imagine these perpendicular axes as streams or forces working outward from their point of intersection, with counter tendencies working in from the opposite direction, from the outside inward (Figure 22). A square arises in the physical world when we imagine that these two types of streams or forces — one coming from within and one coming from outside — meet and hold each other at bay. Boundaries develop where a stoppage occurs in the streams of force.

intersection of cross and square
Figure 22

This image describes how everything on the mental plane relates to everything on the physical plane. You can construct the mental counterpart of any physical object in the same way. This square is only the simplest possible example. If, for any given physical object, you could construct a correlate that relates to that object in the same way that two intersecting perpendicular lines relate to a square, the result would be the image of the physical object in devachan, on the. mental level. With objects other than a square, this process is much more complicated, of course.

Now instead of the square, imagine a cube. A cube is very similar to a square. A cube is a figure bounded by six squares. Mr. Schouten has made an extra model showing the six squares that delineate a cube. Instead of the four boundary lines in a square, imagine six surfaces forming the boundaries. Imagine that the boundary of the stopped forces consists of perpendicular surfaces instead of perpendicular lines and assume that you have three instead of two perpendicular axes. You have just defined a cube. At this point, you probably also can imagine a cube's correlate on the mental level. Again we have two figures that complement each other. A cube has three perpendicular axes and three different directions to its surfaces. We must imagine that stoppage occurs in these three surface directions (Figure 23). The three directions of the axes and the six surfaces, like the square's two axes (directions) and four lines, can be imagined only as opposites of a particular sort.

three-dimensional cross and cube
Figure 23

Anyone who thinks about this subject at all must conclude that in order to imagine these figures, we must first arrive at a concept of opposites that contrast activity and counter-activity, or stoppage. This concept of opposites must enter into our considerations. The examples we used are simple, but by practicing with geometric concepts, we will learn how to construct the mental counterparts of more complicated objects properly, — this activity will show us the way to higher knowledge to a certain extent. You already can imagine the monumental complexity of trying to find the mental counterpart of some other figures. Far greater complications emerge. Just imagine thinking about a human form and its mental counterpart with all its different shapes and activity. You can conceive what a complicated mental structure this would be. My book Theosophy gives approximations of how mental counterparts would look. [Note 24]

In the case of a cube, we have three extensions, or three axes. Two planes, one on each side, are perpendicular to each axis. At this point you need to understand clearly that each surface of a cube, like the human life I described earlier, comes about as the meeting of two streams. You can picture streams moving outward from the midpoint. Imagine one of these axial directions. Space streams outward from the midpoint in one direction and toward the midpoint from the other direction, from infinity. Now envision these streams in two different colors, one red and one blue. At the moment of their meeting, they flow together to create a surface. Thus, we can see the surface of a cube as the meeting of two opposing streams in a surface. This visualization gives us a living idea of the nature of a cube.

A cube is a section of three interacting streams. When you think of the totality of their interaction, you are dealing with six directions rather than three: backward/forward, up/down, and right/left. There are actually six directions. The issue is complicated further by the existence of two types of streams, one moving outward from a point and another moving inward from infinity. This will give you a perspective on the practical applications of higher, theoretical theosophy. Any direction in space must be interpreted as two opposing streams, and any physical shape must be imagined as their result. Let's call these six streams, or directions, \(a\), \(b\), \(c\), \(d\), \(e\), and \(f\). If you could visualize these six directions — and next time we will talk about how to cultivate such mental images — and then eliminate the first and last, \(a\) and \(f\), four would remain. Please note that these remaining four are the ones you can perceive when you see only the astral world.

I have attempted to provide you with some idea of the three ordinary dimensions and of three additional and opposite dimensions. Physical forms arise as a result of the opposing action of these dimensions. If you remove one dimension on the physical level and one on the mental level, however, you are left with the four dimensions that represent the astral world, which exists between the physical and mental worlds.

The theosophical worldview must work with a higher geometry that transcends ordinary geometry. Ordinary geometrists describe a cube as delineated by six squares. We must conceive of a cube as the result of six interpenetrating streams — that is, as the result of a movement and its opposite or as the consequence of interacting opposing forces.

I would still like to give you an example from the natural world of a concept that embodies such a pair of opposites and shows us one of the profound mysteries of the world's evolution. In his The Green Snake and the Beautiful Lily, Goethe speaks of the 'revealed mystery,' one of the truest and wisest phrases ever formulated. [Note 25]

Nature does indeed contain unseen but quite tangible mysteries, including many inversion processes. Let me describe one of them.

Let's compare a human being to a plant. This is not a game, though it looks like one. It points to a profound mystery. Which part of the plant is in the ground? It is the root. Up top, the plant develops stems, leaves, flowers, and fruit. The plant's 'head,' its root, is in the ground, and its organs of reproduction develop above ground, closer to the sun. This can be called the chaste method of reproduction. Picture the whole plant inverted, with its root becoming the human head. There you have the human being, — with the head above and the reproductive organs below, — as the inverse of a plant. The animal occupies the middle and represents an interface. The result of inverting a plant is a human being. Esotericists throughout the ages have used three lines to symbolize this phenomenon (Figure 24).

the inverting of the plant
Figure 24

One line symbolizes the plant, another represents the human being, and a third opposing line corresponds to the animal — three lines that together form a cross. The animal occupies the horizontal position — that is, it crosses what we humans have in common with plants.

As you know, Plato speaks of a universal soul that is crucified on the body of the Earth, bound to the cross of the Earth. [Note 26]

If you envision the world soul as plant, animal, and human being, the result is the cross. Since it lives in these three kingdoms, the world soul is bound to the cross they form. Here you find an extension of the concept of interfaced forces. The plant and the human being represent two complementary, and divergent, but intersecting streams, while the animal, which actually interjects itself into an upward and a downward stream, represents the interface that arises between them. Similarly, kamaloka, or the astral sphere, stands between devachan and the physical world. Between these two worlds, whose relationship is that of mirror images, an interface arises — the world of kamaloka — whose outer expression is the animal kingdom.

Strength is required to perceive this world, but those who already have the appropriate organs of perception will recognize what we see in the interrelationship of these three kingdoms. If you interpret the animal kingdom as emerging from an interface, you will discover the relationship between the plant and animal kingdoms and the animal and human kingdoms. The animal stands perpendicular to the direction of the two other kingdoms, which are complementary, interpenetrating streams. Each lower kingdom serves the next higher one as food. This fact sheds light on the difference between the human-plant relationship and the animal-human relationship. Human beings who eat animals develop a relationship to a condition of interfacing. The real activity consists in the meeting of opposing streams. In making this statement, I am initiating a train of thought that will reappear later in a strange and very different guise.

In summary, we have seen that a square comes about when two axes are cut by lines. A cube comes about when three axes are cut by surfaces. Can you imagine four axes being cut by something? The cube is the boundary of the spatial figure that comes about when four axes are cut.

A square forms the boundary of a three-dimensional cube. Next time we will see what figure results when a cube forms the boundary of a four-dimensional figure.

Questions and Answers

What does it mean to imagine six streams and then eliminate two, and so on?

The six streams must be imagined as two times three: three of them work from the center outward in the directions defined by the three axes, and the other three work in the opposite direction, coming from infinity. Thus, for each axial direction there are two types, — one going outward from the interior and the other moving inward from outside. If we call these two types positive and negative, plus and minus, the result is this:

\(+a\)    \(-a\)
\(+b\)    \(-b\)
\(+c\)    \(-c\)

To enter the astral realm we must eliminate one entire direction of inward and outward streams: \(+a \) and \(-a\), for example.

  1. Mr. Schouten. In all probability, Jan Arnoldus Schouten (1883–1971), a Dutch mathematician from Delft.

    In the archives of the Rudolf Steiner Nachlassverwaltung, there is a letter from Schouten to Steiner. The part that relates to this lecture reads:

    December 1, 1905

    Dear Dr. Steiner,
    Before leaving for home in July of this year, I stopped in to say goodbye to you, but unfortunately you had already left. Consequently, the models you needed for your lecture are still in your possession. Since I intend to give several lectures here on the fourth dimension, could you please send the models to me? These lectures are intended for several lodges, including the one in Delft, which was founded a short time ago.

    Sincerely yours,
    J. A. Schouten
    M. T. S.

    After studying electrical engineering at the technical college in Delft, Schouten practiced his profession for several years in Rotterdam and Berlin. In order to be able to understand the theory of special relativity, Schouten studied mathematics privately and wrote the book Grundlagen der Vektor- und AJfinoranalysis ('The Bases of Vector and Affine Analysis") [1914], which he submitted as his dissertation to the University of Delft. Shortly thereafter, he was named professor at Delft, where he remained until 1943.

    Schouten's book [1914], with a personal dedication by the author, was found in Rudolf Steiner's library. Schouten's mother, H. Schouten (1849–19??) was a member of the Theosophical Society and later of the Anthroposophical Society. To date, only one other indication of a connection between Schouten and Rudolf Steiner has been found, in a letter (also in the Steiner archives) to Rudolf Steiner from Schouten’s mother, dated March 4, 1913. This letter reads in part:

    I was very confident that my son, now that he intends to give up his membership in the Theosophical Society, would become a member of the Anthroposophical Society, but he says that for the moment he cannot do so with a clear conscience because he has not been able to keep up his theosophical studies. He told me that he makes a point of seriously studying everything he undertakes in life, and that because his own academic work is so demanding at the moment that he has almost no time to go out, he is temporarily unable to take up the study of theosophy again. The first draft of his paper has been sent to the Royal Academy. In addition to his private work, he is lecturing weekly on mathematics in Delft and on electricity in Rotterdam. In the week when you will be in the Hague, the Philosophical Society in Amsterdam has asked him to give a lecture on his concepts of non-material mathematics. Praise God, both he and his wife have absorbed the truths of reincarnation and karma. They would like to attend your public lectures, and my son also thought that some of his colleagues might attend if the subject appealed to them. I hope you and my son will find the opportunity to meet.

    Schouten's first paper in the Verslagen en Mededeelingen der Koninglijke Akademie van Wetenschappen appeared in 1917 in volume 26; a paper in the Verhandelingen der Koninglijke Akademie van Wetenschappen te Amersterdam appeared in 1918 in volume 12.

  2. Kronos (not to be confused with Chronos, or Time) is one of the sons of Uranus and Gaia. He married his sister Rhea, who gave birth to three daughters (Hestia, Demeter, and Hera) and two sons (Poseidon and Zeus). Kronos devoured all of them except Zeus, whom Rhea had entrusted to her mother, Gaia. (See Kèrenyi, Die Mythologie der Griechen ["The Mythology of the Greeks"] [1966], volume I, chapter I, sections 1 and 2.)

  3. See Rudolf Steiner's Theosophy.

  4. Johann von Wolfgang Goethe (1749–1832). Meanwhile the golden king said to the man with the lamp, "How many mysteries do you know?" 'Three," answered the old man. "Which is the most important?" 'The revealed one," answered the old man.

  5. Plato (427–347 BC). Timaeus 36b-37a. See also Rudolf Steiner's Christianity as Mystical Fact (GA 8), pp. 65 ff.