The Rudolf Steiner Archive

This lecture is the 4th of 12 lectures from the lecture series entitled, The Human's Connection with the Elemental World, and the 3rd of 4 lectures in the series entitled, Olaf Asteson and the Awakening of the Earth Spirit.

The period from about Christmas to the present date (Jan. 7th) is really an important and significant period of the year, also in an occult connection. It is called “The period of the Thirteen Days.” The remarkable thing is that the importance of these thirteen days is felt by those who through the constitution of their souls have preserved an inkling of the ancient connection of the human soul with the spiritual world, of which we have often spoken. We know that the primitive human being who lives in the country or in a community which is little infected by our town life, preserves more of the connection with the spiritual world which existed in ancient times than one belonging to a town.We find many things in folk-poems regarding experiences of the soul during the period from Christmas Eve to Epiphany, Jan.6th.

This is the time when—after darkness has been greatest over the earth, directly after the winter solstice, when the sun again begins his victorious course,—together with the deepest immersion and subsequent liberation and redemption of nature,—the human soul can also have special experiences if it still has a definite connection with the spiritual world. Those who o longer possess the old clairvoyance, but who in their souls are still connected with the spiritual world, perceive a difference in the abnormal world of dreams at this period of the year. What the soul can then experience is important, because the soul—if it is still susceptible—can then really penetrate best into the spiritual world. To the modern man the course of the year is such that he can no longer distinguish the various seasons of the year; for while the snowstorms rage outside, when the darkness descends about 4 p.m. and it grows light late in the morning, the city man feels the same as in the summer months when the sun develops its greatest power. Man has been torn out of his ancient connection with the Cosmos in which he lived when he was outside in nature. To those however who have kept in touch with nature, what happens at Christmas time is not the same as what takes place at some other time in the year, for example, at midsummer. Whereas at midsummer the soul is most emancipated from what is connected with the spiritual world, at the time when nature has died away the most it is connected with the spiritual world and formerly had special experiences during this time.

Now there is a beautiful folk-poem in the old Norwegian language, a poem which was re-discovered a short time ago and has quickly become popular again owing to the peculiarly sympathetic understanding of the Norwegian people. It treats of a man who was still in connection with the spiritual world,—Olaf Oesteson. What he goes through in the time between Christmas and Epiphany is beautifully described in this poem. At the New Year Festival in Hanover on Jan. 1st, 1912, I tried to put this folk-poem “Olaf Oesteson” into German verse, so that it might come before our souls too. We will begin this evening with the song of Olaf Oesteson, which contains his experiences during the “Thirteen Nights.”

The Dream Song

O listen to my song!
I will sing to thee
Of a certain youth:
This was Olaf Oesteson,
who once slept so long.
Of him I sing to thee.

He went to rest on Christmas Eve.
A deep sleep fell upon him soon,
And he could not awake,
Till on the thirteenth day
The people went to Church.
This was Olaf Oesteson,
Who once slept for so long,
Of him I sing to thee.

He went to rest on Christmas Eve:
Long, long did he sleep!
And he could not awake,
Till the thirteenth day
The birds were spreading their wings.
This was Olaf Oesteson,
Who once slept for so long
Of him will I sing to thee.

Olaf could not awake
Till on the thirteenth day
The sun shone o'er the mountains
Then did he saddle his nimble steed,
And rode to the church in haste.
This was Olaf Oesteson,
Who once slept for so long,
Of him will I sing to thee.

The priest already stood
By the altar saying mass.
When by the church-door Olaf seated himself
To give tidings of many dreams,
Which during this long sleep
Had filled his soul.
This was Olaf Oesteson,
who once slept so long.
Of him I sing to thee.

Young folk and old ones too,
Give heed unto the words
Which Olaf speaks about his dreams.
This was Olaf Oesteson,
who once slept so long.
Of him I sing to thee.

II

I went to rest on Christmas Eve,
A deep sleep soon enveloped me;
And I could not awake,
Till on the thirteenth day
The people went to church.
Brightly shone the moon
And the paths stretched far away.

Up to the clouds was I lifted,
And flung to the depths of the sea.
Mirth cannot fall upon
ONe who follow me.
Brightly shone the moon
And the paths stretched far away.

Up to the clouds was I lifted
And thrust into dismal swamps,
beheld the terrors of hell
And also the light of heaven.
Brightly shone the moon
and the paths stretched far away.

Into earth's depths did I journey,
Where streams of Gods roar frightfully,
To look at them I was not able,
But I could hear their roaring.
Brightly shone the moon
And the paths stretched far away.

My black steed did not neigh,
And my dogs did not bark,
Nor did the bird of morning sing;
There was but one marvel everywhere.
Brightly shone the moon
And the paths stretched far away.

My spirit-land I had to travel o'er
The wide plane of a thorny heath,
And also the nails on my feet.
Brightly shone the moon
And the paths stretched far away.

I came to Gjaller-Bridge
It hangs high up in windy heights;
It is covered with red gold
and has nails with sharp points.
Brightly shone the moon
And the paths stretched far away.

The ghostly snake did strike me,
The ghostly hound did bite me.
The bull, he stood across the way
These are three beasts of the bridge:
Of frightfully evil nature are they.
Brightly shone the moon
And the paths stretched far away.

The dog to bite is ready
The snake desires to sting,
The bull, he threatens violence:
They let none pass across the bridge
Who will not honour truth.
Brightly shone the moon
And the paths stretched far away.

I went across the bridge
Narrow it is, and makes one dismay.
I had to wade through swamps ...
They lie behind me now!
Brightly shone the moon
And the paths stretched far away.

I had to wade in swamps,
Bottomless did they seem.
When I passed o'er the bridge,
I felt earth in my mouth—
Like the dead who lie in the grave.
Brightly shone the moon
And the paths stretched far away.

III

Then I came to waters
In which, like blue flames,
Masses of ice gleaned bright.
And God my purpose guided,
So that I shunned that region.
Brightly shone the moon
And the paths stretched far away.

I turned my steps to the winter-path
On the right could I perceive it.—
I looked as into Paradise,
That threw its light afar.
Brightly shone the moon
And the paths stretched far away.

And the holy Mother of God—
I saw her there in radiance!
She made no travel to Brooksvaline:
Saying that there men's souls are judged!
Brightly shone the moon
And the paths stretched far away.

IV

In other worlds I tarried
Through the length of many nights;
And God alone can know
How much distress of soul I saw
In Brooksvaline, where souls
The Day of Judgment undergo.

A young man there I saw,
Who had recklessly a boy had killed:
Now he must carry him eternally
In his own arms!
He stood deep in mire,
In Brooksvaline, where souls
The Day of Judgment undergo.

Also an old man did I see,
He wore a cloak as though of lead.
Thus was he punished, because he
In avarice had lived on earth.
In Brooksvaline, where souls
The Day of Judgment undergo.

Ad men came into sight
Who carried fiery burdens;
Dishonesty weighs heavily
On their poor souls,
In Brooksvaline, where souls
The Day of Judgment undergo.

Children too could I behold.
With glowing coals beneath their feet.
These worked their parents ill in life,
Which hurt their spirits grievously
In Brooksvaline, where souls
The Day of Judgment undergo.

I was charged to approach that house,
Where witches had to work
In the blood which in life
Had stirred them up to wrath,
In Brooksvaline, where souls
The Day of Judgment undergo.

Down from the North in savage troops,
Come riding wicked spirits,
Led by the Prince of Hell.
In Brooksvaline, where souls
The Day of Judgment undergo.

What came from out of the North
Seemed worst of all;
In front rode he, the Prince of Hell,
On his black charger.
In Brooksvaline, where souls
The Day of Judgment undergo.

But from the South there came
In serene tranquility yet other bands.
Foremost rode St. Michael
At the side of Jesus Christ.
In Brooksvaline, where souls
The Day of Judgment undergo.

The souls laden with sin
Could but in terror tremble!
Their tears ran down in streams—
The result of their bad deeds.
In Brooksvaline, where souls
The Day of Judgment undergo.

In majesty stood Michael there
Weighing the human souls
Upon his scales for sin;
And giving judgment stood
The judge of the world, Jesus Christ.
In Brooksvaline, where souls
The Day of Judgment undergo.

V

Happy is he who in earth-life
Gives shoes to those in need;
He need not walk with naked feet
In the field of thorns.
Thus speaks the tongue of the scales,
And universal truth
Resounds within the spirit-state.

Happy is he who in earth-life
Hath given bread to the poor!
He cannot in that world
Be injured by the dogs.
Thus speaks the tongue of the scales,
And universal truth
Resounds within the spirit-state.

Happy is he, who in earth-life
Hath given corn to the poor!
Threatened he cannot be
By the sharp horns of the bull,
When he must pass the Gjaller-bridge.
Thus speaks the tongue of the scales,
And universal truth
Resounds within the spirit-state.

Happy is he, who in earth-life
Gives clothing to the poor!
The ice-masses in Brooksvaline
Cannot avail to freeze him.
Thus speaks the tongue of the scales,
And universal truth
Resounds within the spirit-state.

VI

Young folk and old ones too,
Give heed unto the words
Which Olaf spake about his dreams.
Long didst thou sleep indeed ...
Wake now, Olaf Oesteson!

The poem itself is old; but as we have already said, it has recently reappeared as if of itself among the Norwegian people and is spreading with great rapidity. The fact of this poem spreading id one among the many things at the present time which shows how people are longing to understand the secrets now being opened up by Theosophy, for the fact that what is here described takes place—or at least could take place a comparatively short time ago—in a soul, is not merely “imagination.” Olaf Oesteson is a type of those people living in the North who, even in the Middle Ages, about the middle of that period, were able to experience literally, one might say, the things mentioned in this poem. When our Norwegian friends gave me this poem on my visit to Christiania the time before last, and wished me to say something about it, it was the fact just mentioned, one of general theosophical interest, which came particularly to notice, but what led up to include this poem in our theosophical understanding we can really penetrate more and more deeply into what comes to light in it. Thus for instance, it was significant to me that Olaf (that is an old Norwegian name) has the surname “Oesteson.” “Oesteson”—the son of what? Of “Oste”; and I tried to find what sort of mother this is the son of. Now of course we might adduce many things—including some that might lead to dispute—about he meaning of the word “Oste” (East): but it would be impossible to-day to explain all that is connected with it. If, however, we take into account all that comes into question, “Olaf Oesteson” means approximately this: One who is still a son of that soul which passes down from generation to generation, and is connected with the blood which is handed on from generation to generation. Thus we have traced this name back to what we have so often spoken of in Theosophy, namely, that in ancient times the old clairvoyance was connected with the relationship of the blood which passes through generations. We might translate “Olaf Oesteson” thus: Olaf, the one born of many generations and who still bears in his soul the characteristics of many generations.

Now when we examine his experiences, it is extremely interesting to notice that what Olaf Oesteson went through while he was asleep for thirteen days, beginning from Christmas Eve, during which time he did not woke was in a sort of psychic state. When we read these verses describing his various experiences with the broad homeliness of the nation, we are reminded of certain descriptions of the first stages of initiation, where we are told that so and so was led to the portal of death. We are shown in many places in the poem that Olaf Oesteson arrives at the portal of death. It is pointed out particularly clearly where he says that he feels like a corpse, even to the earth which feels between his teeth. When we remember that in initiation the etheric body extends beyond the limits of the skin and the neophite becomes larger and larger, so that he lives into the large, into the wide expanse of space, we are told in this poem how Olaf Oesteson descends deeply, feels himself in the depths of the earth and ascends to the clouds. Olaf Oesteson experience what man has to go through after death, for example, in the sphere of the moon. It is poetically described how the moon shines clearly and how the paths stretch far away, then the chasm is described which has to be passed over in the world which lies between the human world and the one leading out into cosmic space. The heavenly bridge connects what is human with what is cosmic. Our attention is then drawn to the beings expressed in the constellations; the bull and the serpent. To one who can look spiritually into the world, the constellations are only the expression of what exists spiritually in space. Then the world of Kamaloca is disclosed in the description of “Brooksvaline.” It describes how there is a sort of recompense, how people have there to experience what they have not acquired here on earth,—but in a compensating way.—We need not, however, go into all the details of the poem. We should not do this at all with poems such as this. We ought to feel they have originated from a frame of mind still closely connected with something which existed in such a people as this, much longer than among nations which lived in the more interior part of the continent or who were connected with the life in cities. In the Norwegian people, which still possesses in its national language many things which border closely upon occult secrets, it is possible to keep souls in touch for a long time with what exists behind outer material phenomena.

Remember who I explained that, parallel with the seasons of the year, there are spiritual facts taking place, how in the spring when the plants spring forth the earth, when everything wakens, as it were, when the days grow longer, we have to recognize what may be called a sort of sleeping of the elementary and higher spirits connected with the earth. In spring, when outwardly the earth awakens, we see that spiritually this is connected with a sort of falling asleep of the earth; and when outer nature dies down again it is connected with an awakening of the spiritual nature of the earth. When about Christmas time outer nature is as though asleep, it is the time when the spiritual part of the earth is most active, and includes elemental, less important beings, as well as great and mighty beings connected with earthly life. It is only when it is observed outwardly that it seems as though we must compare spring with the awakening of the earth and winter to its going to sleep. Seen occultly it is the reverse. The “Spirit of the Earth” which however, consists of many spirits, is awake in winter and asleep in summer. Just as in the human organism the organic and plant activities are most active during sleep, as these forces then work even into the brain, and as the purely organic activity is subdued while the person is awake, so is it also with the earth. When the earth is most active, when everything has sprouted forth, when the sun has reached its zenith about St. John's Day, the Spirit of the earth is asleep. In accord with this occult truth the festival of Christmas, the festival of the awakening of the spirit, was fixed in winter. Things which have been handed down as customs from ancient times often correspond to these occult verities.

Now one who knows how to live with the spirit of the earth celebrates, for example, the festival of St. John in summer, for this festival is a kind of materialistic festival; it celebrates that which is revealed in an outward materialistic form. One who is connected with the Spirit of the Earth, with what lives spiritually in the earth, awakens in his inner being—that is, he sleeps outwardly like Olaf Oesteson—best at Christmas time, during the “Thirteen days.” This is an occult fact, which to occultism signifies exactly the same as, for example, the fact of the outer solstitial point to ordinary materialistic science. Of course materialistic science will consider it to be an obvious thing that in astronomy it should describe the activity of the sun in summer and in winter in a purely external manner, it will consider foolish what to occultists is a fact, namely that the spiritual solstice is at its highest point in winter, that therefore the conditions are then the most favorable for those who wish to come in touch with the Spirit of the Earth and all that is spiritual. Therefore to one who wishes to strengthen his soul's powers it may come about that he can have his best experiences during the thirteen days after Christmas. At that time, without noticing it, experiences come forth from the soul,—although the modern man is emancipated from outer processes, so that occult experiences can come at any time, but in so far as outer conditions can have an influence, the time between Christmas and New Year is most important.

Thus are we reminded by this poem in quite a natural manner, that a great deal of what we are able to relate regarding the period between death and rebirth was known among certain peoples a comparatively short time ago, many knew it from direct experience.

If you are disappointed about what you are about to hear, I would like to say in advance that today I want to discuss very elementary things [about the fourth dimension]. Those who want to delve deeper into this question should be very familiar with the higher concepts of mathematics. I would like to give you some very elementary and general concepts. One must distinguish between the possibility of thinking in a four-dimensional space and reality. Whoever is able to make observations there is dealing with a reality that extends far beyond what we know as the sensual-real. You have to do thought transformations when you go there. You have to let things play into mathematics a little, find your way into the way of thinking of the mathematician.

You have to realize that the mathematician does not take a step without accounting for what arrives at his conclusions. But we must also realize when we deal with mathematics that even the mathematician cannot penetrate a single step [into reality], that he cannot draw any conclusions [that go beyond what is merely possible in thought]. First of all, it is about simple things, but they become more complicated when one wants to arrive at the concept of the fourth dimension. We must be clear about what we mean by dimensions. It is best to examine the various spatial structures in terms of their dimensionality. They lead to considerations that were only tackled in the 19th century by great mathematicians such as Bolyai, Gauss and Riemann.

The simplest spatial size is the point. It has no extension at all; it must be conceived. It is the fixation of an extension in space. It has no dimension. The first dimension is the line. The straight line has one dimension, length. If we move the line, which has no thickness, ourselves, we step out of the one dimension, and the line becomes a surface. This has two dimensions, a length and a width. If we move the surface, we step out of these two dimensions and we get the body. It has three dimensions: height, width, depth (Figure 1).

If you move the body itself, if you move a cube around in space, you will again only get a spatial body. You cannot move space out of itself.

We need to turn to a few other concepts. If you look at a straight line, it has two boundaries, two endpoints A and B (Figure 2).

Let's imagine that we want A and B to touch. But if they are to touch, we have to curve the line. What happens? You cannot possibly remain within the [one-dimensional] line if you want to make A and B coincide. To connect points A and B, we have to step out of the straight line itself, we have to step out of the first dimension and into the second dimension, the plane. In this way, the straight line becomes a closed curve (that is, in the simplest case, a circle) by bringing its endpoints into alignment (Figure 3).

It is therefore necessary to go beyond the first dimension; you cannot remain within it. Only in this way can the circle be created. You can perform the same operation with a surface. However, this only works if you do not remain within the two dimensions. You have to enter the third dimension and then you can turn the surface into a tube, a cylinder. This operation is done in a very similar way to the way we brought two points into coincidence earlier, thereby moving out of the first dimension. Here, in order to bring two boundaries of the surface into coincidence, we have to move into the third dimension (Figure 4).

Is it conceivable that a similar operation could be carried out with a spatial structure that already has three dimensions itself? If you have two congruent cubes, you can slide one into the other. [Now imagine two congruent cubes as the boundaries of a three-dimensional prismatic body.] If you try to make one cube, which is colored red on one side [and blue on the opposite side], fit exactly over the other cube, which is otherwise [geometrically] identical but with the red and blue colors swapped, then you cannot make them fit except by rotating the cube (Figure 5).

Let us consider another spatial structure. If you take the left-hand glove, it is impossible for you to pull the left-hand glove over the right hand. But if you look at the two [mirror-symmetrical] gloves together, like the straight line with the end points A and B, you have something that belongs together. It is then a single entity, with a boundary [that is, with a mirror plane] in the middle. It is very similar with the two symmetrical halves of the human outer skin. 2

How can we now make two [mirror] symmetrical three-dimensional structures coincide? Only if we go beyond the third dimension, as we did with the first and second. We can also put the right or left glove over the left or right hand, respectively, when we walk through four-dimensional space.

[When constructing the third dimension (depth dimension) of the visualization space, we align the image of the right eye with that of the left eye and place it over it.

We now look at an example from Zöllner. We have a circle and a point P outside of it. How can we bring the point P into the circle without crossing the circle? This is not possible if we remain within the plane. Just as one has to go from the second dimension into the third when moving from a square to a cube, we also have to go out of the second dimension here. With a sphere, there is also no possibility of entering [into the interior] without [piercing the surface of the sphere or] going beyond the third dimension.

These are possibilities for thought, but they have a practical significance for the theory of knowledge, [in particular for the problem of the objectivity of the content of perception]. If we realize how we actually perceive, we will come to the following view. Let us first ask ourselves: How do we gain knowledge of bodies through our senses? We see a color. Without eyes, we would not perceive it. The physicist then says: Out there in space is not what we call color, but purely spatial forms of movement; they penetrate through our eye, are captured by the optic nerve, transmitted to the brain, and there, for example, the red arises. One may now ask: Is the red also present when there is no sensation?

Red could not be perceived without the eye. The ringing of a bell could not be perceived without the ear. All our sensations depend on the transformation of forms of motion by our physical and mental apparatus. However, the matter becomes even more complicated when we ask ourselves: Where is the red, this peculiar quality, actually located? Is it in the body? Is it a process of vibration? Outside there is a process of movement, and this continues right into the eye and into the brain itself. There are vibrational [and nervous] processes everywhere, but red is nowhere to be found. Even if you examine the eye, you would not find red anywhere. It is not outside, but it is also not in the brain. We only have red when we, as a subject, confront these processes of movement. So do we have no possibility at all to talk about how the red meets the eye, how a c sharp meets the ear?

The question is, what is this inner [representation], where does it arise? In the philosophical literature of the 19th century, you will find that this question runs through everything. Schopenhauer, in particular, has provided the following definition: The world is our representation. But what then remains for the external body? [Just as a color representation can be “created” by movements, so can] movement can arise in our inner self through something that is basically not moved. Let us consider twelve snapshots of a [moving] horse figure on [the inside of] a [cylinder] surface, [which is provided with twelve fine slits in the spaces between. If we look at the rotating cylinder from the side,] we will have the impression that it is always the same horse and that only its feet are moving. So [the impression of] movement can also arise through our [physical organization] when something is not moving at all [in reality]. This is how we arrive at a complete dissolution of what we call movement.

But what then is matter? If you subtract color, movement [shape, etc., i.e. what is conveyed by sensory perception] from matter, then nothing remains. If we already have the [secondary, i.e. “subjective” sensations [color, sound, warmth, taste, smell] within us, we must also place [the primary sensations, that is, shape and movement,] within us, and with that the external world completely disappears. However, this results in major difficulties [for the theory of knowledge].

Let us assume that everything is outside, how then do the properties of the object outside come into us? Where is the point [where the outside merges into the inside]? If we subtract all [sensory perceptions], there is no outside anymore. In this way, epistemology puts itself in the position of Münchhausen, who wants to pull himself up by his own hair. But only if we assume that there is an outside, only then can we come to [an explanation of] the sensations inside. How can something from the outside enter our inside and appear as our imagination?

We need to pose the question differently. Let us look at some analogies first. You will not be able to find a relationship [between the outside world and the sensation inside] unless you resort to the following. We return to the consideration of the straight line with endpoints A and B. We have to go beyond the first dimension, curve the line, to make the endpoints coincide (Figure 7).

Now imagine the left endpoint A [of this straight line] brought together with the right endpoint B so that they touch at the bottom, so that we are able to return to the starting point [via the coinciding endpoints]. If the line is small, the corresponding circle is also small. If I turn the [initially given] line into a circle and then turn larger and larger lines into circles, the point at which the endpoints meet moves further and further away from the [original] line and goes to infinity.

of the [original] line and goes to infinity. Only at infinity do the [increasingly large] circle lines have their endpoint. The curvature becomes weaker and weaker, and eventually we will not be able to distinguish the circle line from the straight line with the naked eye (Figure 8).

In the same way, when we walk on the Earth, it appears to us as a straight piece, although it is round. If we imagine that the two halves of the straight line extend to infinity, the circle actually coincides with the straight line.

The straight line can be conceived as a circle whose diameter is infinite.

Now, instead of a [geometric] line, imagine something that is real and that connects to a reality. Let us imagine that as the point C [on the circumference of the circle] progresses, cooling occurs, that the point becomes colder and colder the further it moves away [from its starting point] (Figure 9). Let us leave the point within the circle for the time being, and, as it becomes colder and colder, let it reach the lower limit A, B. When it returns on the other side, the temperature increases again. So on the way back, the opposite condition to the one on the way there occurs. The warming increases until the temperature at C is reached again, from which we started. No matter how extended the circle is, it is always the same process: a flow of heat out and a flow of heat in. Let us also imagine this with the [infinitely extended straight] line: as the temperature [on one side increasingly] dissipates, it can rise on the other side. We have here a state that dissipates on one side while it rebuilds on the other.

In this way, we bring life and movement into the world and approach what, in a higher sense, we can call an understanding of the world. We have here two states that are interdependent and interrelated. However, for everything you can observe [sensually], the process that goes, say, to the right has nothing to do with the one that comes back from the left, and yet they are mutually dependent.

We now compare the body of the external world with the state of cooling and, in contrast, our inner sensation with the state of warming. [Although the external world and inner sensation have nothing directly perceptible in common,] they are related to each other, mutually dependent [in an analogous way to the processes described above]. This results in a connection between the external world [and our internal world] that we can support with an image: [through the relationship between] the seal and the sealing wax. The seal leaves behind an exact imprint, an exact reproduction of the seal in the sealing wax, without the seal remaining in the sealing wax [and without any material from the seal being transferred to the sealing wax]. So in the sealing wax there remains a faithful reproduction of the seal. It is quite the same with the connection between the outside world and inner sensations. Only the essential is transferred. One state determines the other, but nothing (material) is transferred.

If we imagine that this is the case with [the connection between the] outside world and our impressions, we come to the following. [Geometric] mirror images in space behave like gloves from the left and right hand. [In order to relate these directly and continuously to each other,] we have to use a new dimension of space to help us. [Now the outside world and the inner impression behave analogously to geometric mirror images and can therefore only be directly related to each other through an additional dimension.] In order to establish a relationship between the outside world and inner impressions, we must therefore go through a fourth dimension and be in a third element. We can only seek the common ground [of the outside world and inner impressions] where we [are one] with them. [One can imagine these mirror images as] floating in a sea, within which we can align the mirror images. And so we come [initially in thought] to something that transcends three-dimensional space and yet has a reality. We must therefore bring our spatial ideas to life.

Oskar Simony has tried to represent these animated spatial structures with models. [As we have seen, one comes] from the consideration of the zero-dimensional [step by step] to the possibility of imagining four-dimensional space. [On the basis of the consideration of mirror-symmetrical bodies, that is, with the help of] symmetries, we can first [most easily] recognize this space. [Another way to study the peculiarities of empirical three-dimensional space in relation to four-dimensional space is to study the knotting of curves and ribbons.] What are symmetry conditions? By intertwining spatial structures, we cause certain complications. [These complications are peculiar to three-dimensional space; they do not occur in four-dimensional spaces.]

Let's do some practical thinking exercises. If we cut a band ring in the middle, we get two such rings. If we now cut a band whose ends have been twisted by 180° and then glued, we get a single twisted ring that does not disintegrate. If we twist the ends of the tape 360° before gluing them together, then when we cut it, we get two intertwined rings. Finally, if we twist the tape ends 720°, the same process results in a knot.

Anyone who reflects on natural processes knows that such convolutions occur in nature; [in reality,] such intertwined spatial structures are endowed with forces. Take, for example, the movement of the Earth around the Sun, and then the movement of the Moon around the Earth. It is said that the Moon describes a circle around the Earth, but [if you look more closely] it is a line that is wrapped around [a circle, the orbit of the Earth], thus a helix around a circular line. And then we have the sun, which rushes through space so fast that the moon makes an additional spiral movement around it. So there are very complicated lines of force extending in space. We have to realize that we are dealing with complicated concepts of space that we can only grasp if we do not let them become rigid, if we have them in a fluid state.

Let us recall what has been said: the zero-dimensional is the point, the one-dimensional is the line, the two-dimensional the surface and the three-dimensional the body. How do these concepts of space relate to each other?

Imagine you are a creature that can only move along a straight line. What would the spatial perceptions of such a being, which itself is only one-dimensional, be like? It would not perceive its own one-dimensionality, but would only imagine points. This is because, if we want to draw something on a straight line, there are only points on the straight line. A two-dimensional being could encounter lines, and thus distinguish one-dimensional beings. A three-dimensional being, such as a cube, would perceive the two-dimensional beings. Man, then, can perceive three dimensions. If we reason correctly, we must say to ourselves: Just as a one-dimensional being can only perceive points, as a two-dimensional being can only perceive one dimension, and a three-dimensional being can only perceive two dimensions, so a being that perceives three dimensions can only be a four-dimensional being. The fact that a human being can define external beings in three dimensions, can [deal with] spaces of three dimensions, means that he must be four-dimensional. And just as a cube can only perceive two dimensions and not its third, it is true that the human being cannot perceive the fourth dimension in which he lives.

Today I want to discuss some elementary aspects of the idea of multidimensional space [among other things, in connection with the] spirited Hinton.

You will recall how we arrived at the concept of multi-dimensional space, having considered the zeroth dimension [last time]. I would like to briefly repeat the ideas of how we can move from two- to three-dimensional space.

What do we mean by a symmetrical behavior? How do I align a red and a blue [flat figure, which are mirror images of each other]?

With two halves of a circle, I can do this relatively easily by sliding the red [half] circle into the blue one (Figure 10).

This is not so easy in the following [mirror]symmetrical figure (Figure 11). I cannot make the red and blue parts coincide [in the plane], no matter how I try to slide the red into the blue.

But there is a way [to achieve this anyway]: if you step out of the board, that is, out of the second dimension [and use the third dimension, in other words, if you] place the blue figure on the red one [by rotating it through the space around the mirror axis].

The same applies to a pair of gloves: I cannot match one with the other without stepping out of [three-dimensional] space. You have to go through the fourth dimension.

Last time I said that in order to develop an understanding of the fourth dimension, you have to make [the relationships in] space fluid, thereby creating conditions similar to those you have when moving from the second to the third dimension. In the last lesson, we created spatial structures out of paper strips that intertwined. Such interweaving causes certain complications. This is not a game, but such inter-weavings occur in nature all the time. Anyone who reflects on natural processes knows that such inter-weavings really do occur in nature. Material bodies move in such intertwined spatial structures. These movements are endowed with forces, so that the forces also intertwine. Take the movement of the earth around the sun and then the movement of the moon around the earth. The moon moves in an orbit that is itself wound around the earth's orbit around the sun. It thus describes a spiral around a circular line. Because of the movement of the sun, the moon describes another spiral around this. The result is very complicated lines of force that extend through the whole space.

The heavenly bodies behave in relation to each other like the intertwined strips of paper [by Simony, which we looked at last time]. We have to keep in mind that we are dealing with complicated spatial concepts that we can only understand if we do not let them become rigid. If we want to grasp space [in its essence], [we must first conceive it as rigid, but then] make it completely fluid again. [You have to go as far as zero]; the [living] point can be found in it.

Let us once again visualize the structure of the dimensions]. The point is zero-dimensional, the line is one-dimensional, the surface is two-dimensional and the body is three-dimensional. The cube has the three dimensions: height, width and depth. How do the spatial structures [of different dimensions] relate to each other? Imagine that you are a straight line, that you have only one dimension, that you can only move along a straight line. If such beings existed, what would their concept of space be like? Such beings would not perceive one-dimensionality in themselves, but would only be able to imagine points wherever they went. Because in a straight line, if we want to draw something in it, there are only points. A two-dimensional being would only encounter lines, so it would only perceive one-dimensional beings. [A three-dimensional being like] the cube would perceive two-dimensional beings, but could not perceive its [own] three dimensions.

Now, humans can perceive their three dimensions. If we reason correctly, we must say to ourselves: Just as a one-dimensional being can only perceive points, a two-dimensional being only straight lines, and a three-dimensional being only surfaces, so a being that perceives three dimensions must itself be a four-dimensional being. The fact that humans can define external beings in terms of three dimensions, can [deal with] spaces of three dimensions, means that they must be four-dimensional. And just as a cube can perceive only two dimensions and not its third, so it is clear that man cannot perceive the fourth dimension in which he lives. Thus we have shown [that man must be a four-dimensional being]. We swim in the sea [of the fourth dimension, like ice in water].

Let us return once more to the consideration of mirror images (Figure 11). This vertical line represents the cross-section of a mirror. The mirror reflects an image [of the figure on the left]. The process of reflection points beyond the two dimensions into the third dimension. [To understand the direct and continuous connection between the mirror image and the original, we have to add a third dimension to the two.

[Now let us consider the relationship between external space and internal representation.] The cube here apart from me [appears as] an idea in me (Figure 12). The idea [of the cube] is related to the cube like a' mirror image to the original. Our sensory apparatus [creates an imagined image of the cube. If you want to align this with the original cube, you have to go through the fourth dimension. Just as the third dimension has to be transitioned to (during the continuous execution of the two-dimensional) mirroring process, our sensory apparatus has to be four-dimensional if it is to be able to establish a [direct] connection [between the imagined image and the external object].

If you only imagined [two-dimensionally], you would [only] have a dream image in front of you, but you would have no idea that there is an object outside. Our imagination is a direct inversion of our ability to imagine [external objects by means of] four-dimensional space.

The human being in the astral state [during earlier stages of human evolution] was only a dreamer, he had only such ascending dream images.” He then passed from the astral realm to physical space. Thus we have mathematically defined the transition from the astral to the [physical-] material being. Before this transition occurred, the astral human being was a three-dimensional being and therefore could not extend his [two-dimensional] ideas to the objective [three-dimensional physical-material] world. But when he [himself] became physical-material, he still acquired the fourth dimension [and could therefore also experience three-dimensionally].

Due to the peculiar design of our sensory apparatus, we are able to align our perceptions with external objects. By relating our perceptions to external things, we pass through four-dimensional space, imposing the perception on the external object.

How would things appear if we could see from the other side, if we could enter into things and see them from there? To do that, we would have to pass through the fourth dimension.

The astral world itself is not a world of four dimensions. But the astral world together with its reflection in the physical world is four-dimensional. Anyone who is able to see the astral world and the physical world at the same time lives in four-dimensional space. The relationship of our physical world to the astral world is a four-dimensional one.

One must learn to understand the difference between a point and a sphere. In reality, this point would not be passive, but a point radiating light in all directions (Figure 13).

What would be the opposite of such a point? Just as there is an opposite to a line that goes from left to right, namely a line that goes from right to left, there is also an opposite to the point. We imagine an enormous sphere, in reality of infinite size, that radiates darkness from all sides, but now inwards (Figure 14). This sphere is the opposite of the point.

These are two real opposites: the point radiating light and infinite space, which is not a neutral dark entity, but one that floods space with darkness from all sides. [As a contrast, this results in] a source of darkness and a source of light. We know that a straight line that extends to infinity returns to the same point from the other side. Likewise, it is with a point that radiates light in all directions. This light comes back [from infinity] as its opposite, as darkness.

Now let us consider the opposite case. Take the point as the source of darkness. The opposite is a space that radiates light from all sides.

As was recently demonstrated [in the previous lecture], the point behaves in this way; it does not disappear [into infinity, it returns from the other side] (Figure 15).

[Similarly, when a point expands or radiates out, it does not lose itself in infinity; it returns from infinity as a sphere.] The sphere, the spherical, is the opposite of the point. Space lives in the point. The point is the opposite of space.

What is the opposite of a cube? Nothing other than the whole of infinite space, except for the piece that is cut out here [by the cube]. So we have to imagine the [total] cube as infinite space plus its opposite. We cannot do without polarities if we want to imagine the world as powerfully dynamic. [Only in this way] do we have things in their life.

If the occultist were to imagine the cube as red, the space around it would be green, because red is the complementary color of green. The occultist not only has simple ideas for himself, he has vivid ideas, not abstract, dead ideas. The occultist must enter into things from within himself. Our ideas are dead, while the things in the world are alive. We do not live with our abstract ideas in the things themselves. So we have to imagine the infinite space in the corresponding complementary color to the radiating star. By doing such exercises, you can train your thinking and gain confidence in how to imagine dimensions.

You know that the square is a two-dimensional spatial quantity. A square composed of four red- and blue-shaded sub-squares is a surface that radiates differently in different directions (Figure 16). The ability to radiate differently in different directions is a three-dimensional ability. So here we have the three dimensions of length, width and radiance.

What we did here with the surface, we also think of as being done for the cube. Just as the square above was made up of four sub-squares, we can imagine the cube as being made up of eight sub-cubes (Figure 17). This initially gives us the three dimensions of height, width and depth. Within each sub-cube, we can then distinguish a specific light-emitting capacity, which results in a further dimension in addition to height, width and depth: the radiation capacity.

You can imagine a square made up of four sub-squares, a cube made up of eight different sub-cubes. And now imagine a body that is not a cube, but has a fourth dimension. We have created the possibility of understanding this through radiative capacity. If each [of the eight partial cubes] has a different radiating power, then if I have only the one cube that radiates only in one direction, if I want to obtain the cube that radiates in all directions, I have to add another one on the left, doubling it with an opposite one, I have to put it together out of 16 cubes.

Next lesson we will have the opportunity to consider how we can think of a multidimensional space.

My dear friends, today I will continue with the difficult chapter we have undertaken to take on. In doing so, it will be necessary to refer to the various things that I have already mentioned in the last two lectures. Then today I would also like to create the basic lines and basic concepts in order to [the more exact geometrical relationships as well as] the interesting practical aspects of theosophy, to make them our own.

You know that we have tried to imagine four-dimensional space in its potentiality for the very reason that we can at least create some kind of concept about the so-called astral realm as well as about the higher realms, about higher existence in general. I have already indicated that entering the astral space, the astral world, is initially something tremendously confusing for the secret disciple. For those who have not studied these things in detail, who have not even studied them theoretically, who have not even studied Theosophy theoretically, it is extremely difficult to even begin to form an idea of the very different nature of the things and entities that confront us in the so-called astral world. Let us once again point out how great this diversity is.

As the simplest thing, I mentioned that we have to learn to read every number symmetrically. The secret student, who is only accustomed to reading numbers as they are read here in the physical world, will not be able to find his way through the labyrinth of the astral. If you have a number in the astral, for example 467, you have to read it as 764. You have to get used to reading everything symmetrically, to seeing everything symmetrically (in a mirror image). That is the basic condition. This is still easy as long as we are dealing with spatial structures or numbers. It becomes more difficult when we come to time relationships. When we come to time relationships, the matter also becomes symmetrical in the astral, in such a way that what comes later appears to us first and what comes earlier appears later. So when you observe astral processes, you also have to be able to read backwards from front to back. These things can only be hinted at, because they sometimes seem quite grotesque to those who have never had an idea of them. In the astral, the son is there first and then the father; in the physical, the egg is there first and then the chicken. In the physical, it is different. In the physical, birth comes first, and then the birth is an emergence of a new thing from an old one. In the astral, it is the other way around. There, the old emerges from the new. In the astral, what is paternal or maternal nature devours what is filial or daughterly nature for the appearance.

In Greek, you have a pretty allegory. The three gods Uranus, Kronos and Zeus symbolize the three worlds. Uranus represents the heavenly world: Devachan; Kronos represents the astral; Zeus the physical. Kronos is said to devour his children.” So in the astral, one does not give birth, but is consumed.

But things get very complicated when we look at the moral aspect of the astral plane. This also appears in a kind of reversal or mirror image. And that is why you can imagine how differently things appear when we interpret things as we are accustomed to interpreting them in the physical. In the astral, for example, we see a wild animal approaching us. This is not to be understood in the same way as in the physical. The wild animal is choking us. This is the appearance that someone who is accustomed to reading things in the same way as external events has. But the wild animal is in truth something that exists within ourselves, that lives in our own astral body and that is choking us. What approaches you as a strangler is rooted in your own desire. So you can experience that when you have a thought of revenge, this thought of revenge appears to you as a strangling angel that approaches you from outside and harasses you.

In truth, everything radiates from us [in the astral realm]. We must regard everything that we see approaching us in the astral as emanating from us (Figure 18). It comes from the sphere, from all sides, as if from infinite space, it penetrates into us. But in reality it is nothing other than what our own astral body sends out.

We only really read the astral [and only then] find the truth when we are able to bring the peripheral into the center, to see and interpret the peripheral as the central. The astral seems to come at you from all sides. Think of it this way: in reality, it is something that radiates out from you in all directions.

I would like to familiarize you with a term that is very important in occult training. It appears in a wide variety of works on occult research, but is rarely understood correctly. Those who have reached a certain level of occult development must learn to see everything that is still karmically predisposed in them – joy, lust, pain, and so on – in the astral outside world. If you think theosophically in the right sense, you will realize that the outer life, our body, in the present age is nothing more than a result, an average of two currents coming from opposite directions and merging into each other .

Imagine a current coming from the past and one coming from the future, and you have two currents that merge and actually intersect at every point (Figure 19). Imagine a red current in one direction and a blue current in the other direction. And now imagine, for example, four different points in this intersection. [Then, at each of these four points, we have] an interaction of these red and blue currents. [This is an image for the interaction of] four successive incarnations, where in each incarnation something comes towards us from one side [and something from the other]. You can always say to yourself, there is a current that comes towards you and a current that you bring with you. Man flows together out of these two currents.

You get an idea of it if you think of it this way. Today you sit here with different experiences, tomorrow at the same hour you will have a different set of events around you. Imagine the events that you will have by tomorrow are already all there. It would then be the same experience as if you were looking at a panorama. It would be as if you were approaching these events, as if these events were coming towards you spatially. So imagine that the stream that is coming towards you from the future brings you these events, then you have the events between today and tomorrow in this stream. You allow the future flowing towards you to be carried by the past.

In every period of time, your life is an intersection of two currents, one from the future to the present and the other from the present to the future. Where the currents meet, a congestion occurs. Everything that a person still has ahead of him must be seen emerging as an astral phenomenon. This is something that speaks an incredibly impressive language.

Imagine that the secret disciple [comes to the point in his development where he] is supposed to look into the astral world, where the senses are opened to him so that he would see emerging around him as outer phenomena in the astral world that which he would still have to experience before the end of the present period. This is a sight that is very powerful for every human being. We must therefore say that it is an important step in the course of occult training for the human being to be confronted with the astral panorama, the astral phenomenon, of what he still has to experience until the middle of the sixth root race, because that is how long our incarnations will last. The path opens up before him. No secret disciple will experience it differently, except that he sees as an external phenomenon what he still has to face in the near future up to the sixth root race.

When the disciple has advanced to the threshold, the question arises: Do you want to live through all this in the shortest conceivable time? Because that is what it is about for the one who wants to receive the initiation. If you think about it, you have your own future life in front of you as an external panorama in a moment. That, in turn, is what characterizes our view of the astral. For one person, it is something that makes them say, “No, I'm not going in there.” For another, on the other hand, it is something that makes them say, “I have to go in there.” This point in the process of development is called the 'threshold', the decision, and the phenomenon that one has there, oneself with everything that one still has to experience and live through, is called the 'guardian of the threshold'. The guardian of the threshold is therefore nothing other than our own future life. It is ourselves. Our own future life lies behind the threshold.

You see in this another peculiarity of the astral world of appearance, namely, that when the astral world is suddenly opened to someone through some event – and such events do occur in life – that person must first face something incomprehensible. It is a terrible sight, which could not be more confusing for those people upon whom, unprepared, the astral world suddenly breaks in through some event. It is therefore eminently good to know what we have now discussed, so that in the event of the astral world breaking in, one knows what to do. It may be a pathological event, a loosening between the physical body and the etheric body or between the etheric body and the astral body. Through such events, a person may be unexpectedly transported into the astral world and gain insights into astral life. If this happens, the person will come and say that he sees this or that apparition. He sees it and does not know how to read it, because he does not know that he has to read symmetrically, that he has to understand every wild animal that approaches him as a reflection of what lies within himself. Indeed, the astral powers and passions of man appear in Kamaloka in the most diverse forms of the animal world.

It is not a particularly beautiful sight to see people in Kamaloka who have just been reaped. At that moment they still have all their passions, urges, desires and cravings. Such a person in Kamaloka no longer has his physical body or etheric body, but in his astral body he still has everything that connected him to the physical world, that can only be satisfied through the physical body. Imagine an average citizen of the present day who has achieved nothing special in his past life and has not made any effort to achieve anything, who has never done much for his religious development, who may not have abandoned religion in theory, but practically, that is, in his feelings and attitudes, has thrown it overboard. In that case it is not a living element in him. What then is in his astral body? There are only things that can be satisfied through the physical organism. For example, he craves palate enjoyment. But the palate would have to be there for that, so that this desire can be satisfied. Or man craves for other pleasures, which can only be satisfied by setting his physical body in motion. Suppose he had such a craving, but the body was gone. Then all this lives in his astral body. This is the situation in which man finds himself when he has died without astral purification and cleansing. He still has the desire for the pleasures of the palate and the other things, but not the possibility of satisfying them. This is what causes the torment and horror of the life in Kamaloka. Therefore, the desire must be laid aside in Kamaloka if man dies without astral purification. Only when this astral body has learned that it can no longer satisfy its desires and wishes, that it must unlearn them, is it freed.

[In the astral world] the instincts and passions take on animal forms. As long as the person is embodied in the physical body, the shape of their astral body is somewhat based on this physical body. But when the outer body is gone, then the instincts, desires and passions, as they are in their animal [nature], come into their own in their own form. So in the astral world, a person is an image of their instincts and passions.

Because these astral beings can make use of other bodies, it is dangerous to let mediums enter into a trance when there is no clairvoyant is present to avert evil.

In the physical world, the lion is a plastic expression of certain passions, the tiger is an expression of other passions, and the cat is an expression of yet other passions. It is interesting to see how each animal is the plastic expression of a passion, of an urge.

In the astral, in Kamaloka, man is therefore approximately similar to [animal nature] through his passions. This is the source of the misunderstanding regarding the doctrine of transmigration of souls that has been attributed to Egyptian and Indian priests and teachers of wisdom. You should live in such a way that you do not incarnate as an animal, says this teaching. But this teaching never speaks of the physical life, but of the higher life, and its only aim was to persuade people on earth to lead such a life that after death in Kamaloka they would not have to develop their animal form. Those who develop the characteristics of a cat will appear in Kamaloka as a cat. The fact that one also appears as a human being in Kamaloka is the meaning of the rules of the doctrine of the transmigration of souls. The true teachings have not been understood by the scholars; they only have an absurd idea of them.

Thus we have to deal with a complete mirror image of what we actually think and do here in the physical world in every area – in the areas of number, time and moral life – when we enter the astral realm. We must get used to reading symmetrically, because we must be able to do so when we enter the astral space.

The easiest way for a person to get used to reading symmetrically is to build on such elementary mathematical ideas as we have hinted at in the previous lecture and as we will get to know more and more in the following discussions. I would like to start with a very simple idea, namely the idea of a square. Imagine a square as you are accustomed to seeing it (Figure 20). I will draw the square so that the four sides are drawn in four different colors.

This is the physical appearance of the square. Now I would like to draw the devachan aspect of the square on the board. It is not possible to do this exactly, but I would like to give you an approximate idea of what a square would look like in the mind. The mental counter-image [of a square] is approximately like a cross (Figure 21).

We are dealing here mainly with two perpendicular intersecting axes. Two lines that pass through each other, and that's it. The physical counter-image is created by drawing perpendicular lines on each of these axes. The physical counter-image of a mental square can best be imagined as a congestion [of two mutually intersecting currents]. Let us imagine these perpendicular axis lines as currents, as forces that act outwards from the point of intersection, and let us imagine countercurrents to these currents, only now in the direction from outside to inside (Figure 22). A square then enters into the physical world by imagining these two types of currents or forces - one from the inside, the other from the outside - as accumulating against each other. The currents of force are thus limited by accumulations.

With this, I have given a picture of how everything mental relates to the physical. Likewise, you can construct the mental counterpart for any physical thing. The square here is only the simplest of examples.

If you could construct a correlative for every physical thing that corresponds to the physical world in the same way that two perpendicular lines correspond to a square, then you would obtain the devachan or mental image for every physical thing. With other things, it is of course much more complicated.

Now imagine a cube instead of a square. The cube is very similar to the square. The cube is a body that is bounded by six squares. Mr. Schouten made these six squares that bound the cube specially. Now, instead of the four bounding lines that are present in the square, imagine six bounding surfaces. Imagine that instead of vertical lines we have vertical surfaces as a kind of congestion, and then assume that you have not two but three axes standing on one another [vertically], and you have the boundary of the cube. Now you can also imagine what the mental correlate of the cube is. You have again two things that challenge each other reciprocally. The cube has three perpendicular axes and three surface directions; we have to think of congestion effects in these three surface directions (Figure 23). We cannot imagine the three axes and the six surfaces, as before the two axes and four lines, in any other relationship than by thinking of a certain contrast.

Anyone who reflects on this will have to admit that we cannot imagine this without forming a certain concept of the opposition, namely the opposition of activity and an obstruction, a counter-activity. You have to introduce the concept of opposition here. The matter is still simple here. By entwining ourselves around geometric concepts, we will be able to construct the mental counter-images of more complicated things in an appropriate way. Then we will find the way and to some extent reach higher knowledge. But you can already imagine the colossal complexity that arises when you think of another body and look for its mental counter-image. Many complicated things come to light. And if you were to imagine another person and their mental counterpart, with all their spatial forms and their activity, you can imagine the complicated mental structure that this produces. In my book 'Theosophy', I was only able to give a rough idea of what mental counter-images look like.

We have three dimensions, three axes in the cube. On each axis we have the corresponding perpendicular planes on both sides. So you must now be clear about the fact that the contrast I have spoken of is to be understood in such a way that you imagine each face of the cube as having come into being in a way similar to the way I described human life earlier, as the intersection of two currents. You can imagine currents emanating from the center point. Imagine space in one axial direction, flowing outwards from the center, and in the other direction, flowing in from infinity, another current. And this [imagine] flowing in two colors, one red, the other blue. At the moment they meet, they will flow into a surface, a surface will arise, so that we can assume the surface of the cube to be the meeting point of two opposing currents in a surface. This gives a vivid idea of what a cube is.

The cube is therefore an intersection of three currents acting on each other. If you think about it, you are not dealing with three, but with six directions: forward-backward, up-down, right-left. So you have six directions. And indeed, that is the case. Then the matter becomes even more complicated by the fact that you have two types of currents: One in the direction of a point, the other coming from infinity. This will give you a perspective on the practical application of the higher, theoretical theosophy. I have conceived every direction in space as two opposing currents. And if you then imagine a physical body, then you have in that physical body the result of these two currents running into each other.

Let us now denote these six currents, these six directions, with six letters a, b, c, d, e, f. If you could visualize these six directions or currents — we will come to being able to do this next time — and you would imagine the first and last, a and f, erased, then you would be left with four. And that is what I now ask you to take into account: these four that remain are the four that you can perceive when you see the astral world alone.

I have tried to give you an idea of the three [ordinary dimensions] and of three [further] dimensions that actually behave in the opposite way. It is through the interaction of these dimensions and their counteraction that physical bodies arise. But if you think a little way away from the physical [dimension] and a little way away from the mental on the other side, you are left with four dimensions. These then represent the astral world existing between the physical and mental worlds.

The theosophist's view of the world is such that it necessarily has to work with a higher sense of geometry that goes beyond ordinary geometry. The ordinary geometer describes the cube as bounded by six squares. We must understand the cube as the result of six currents running into each other, that is, as the result of a movement and its reversal, of the interaction of opposing forces.

I would like to show you another such concept outside in nature, where a real contrast has taken place that contains a deep secret of the world's development before the eyes of man. In the “Fairytale of the Snake and the Lily”, Goethe speaks of the “revealed secret”, and that is one of the truest and wisest words that can be spoken at all. It is true, there are secrets in nature that can be grasped with hands, but are not seen by people. We are dealing with reversal processes in nature in many cases. I would like to show you one such reversal process.

Let us compare humans with plants. When compared to plants, humans behave as follows. What I am about to say is not a game, even if it initially seems like one. It is something that points to a deep mystery. What does a plant have in the ground? Its roots. And upwards it develops stems, leaves, flowers and fruit. The main part of the plant, the root, is in the earth, and the organs of reproduction it develops upwards, towards the sun, which we can call the chaste way of reproducing. Imagine the whole plant turned upside down, with the root becoming the head of a human being. Then you have the opposite of the plant in the human being, who has his head at the top and his reproductive organs at the bottom. And the animal is in the middle of it all, as a stowage. If you turn the plant upside down, you get a human being. That is why the occultists of all times draw this with three lines (Figure 24).

One [line] as the symbol of the plant, one as the [symbol] of the human being, and one in the opposite direction as the [symbol] of the animal – three lines that together form the cross. The animal has the transverse position, it thus crosses what we have in common with the plant.

You know that we speak of an all-soul, of which Plato says that it is crucified to the cosmic body, that it is chained to the cross of the cosmic body.? Imagine the world soul as plant, animal and human being, and you have the cross. By living in these three realms, the world soul is chained to this cross. As a result, you will find the concept of congestion expanded. You will find it expanded by something in nature. Two complementary, diverging, but interlocking currents form plants and humans, with congestion being the animal. Thus, the animal actually places itself between an upward and a downward current. In this way, the Kamaloka [astral sphere] interposes itself between Devachan and the physical world. Thus, something interposes itself between these two symmetrical worlds, between Devachan and the physical world, and acts between them, acting on both sides like a dam. The outer expression of this Kamaloka world is the animal world.

Those who already have organs for this world, which must be grasped with strength, will recognize what we see in the three kingdoms in their mutual relationship to one another. If you understand the animal kingdom as emerging from a congestion, if you understand the three kingdoms as mutual congestion, then you will find the position that the plant kingdom has to the animal kingdom and the animal kingdom to the human kingdom. The animal is perpendicular to the other two directions, and the other two are two complementary currents that merge into each other. The lower realm serves the higher realm as food. This is something that allows a small glimpse of the very different kind of relationship between humans and plants and between animals and humans. Those who feed on animals are therefore related to a congestion.

The real effect consists in the encounter of opposing currents. This is the beginning of a series of thoughts that you may later see in a strange and very different way.

We have seen that the square is created when two axes are intersected by lines. The cube is created by intersecting three axes through surfaces. Can you now imagine four axes intersecting through something? The cube is the boundary of the spatial structure that is created when four axes are intersected.

The square limits the three-dimensional cube. Next time, we will see whose boundary the cube itself is. The cube bounds a four-dimensional structure.

Answering Questions

[What does it mean to] imagine six currents, of which two must be imagined as having been extinguished, and so on?

The six currents must be thought of as two times three currents: three acting from the inside out according to the three axial directions, and the other three as flowing towards these from infinity. For each axial direction, there are thus two types, one going from the inside out, the other coming from the outside in the opposite direction. If we call the two categories positive and negative, plus and minus, we have:

math figur

And of this [in order to get to the astral space, we have to] imagine an entire direction, [for example the] inner and outer flow, erased, so for example +a and -a.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Last time, we tried to get an idea of a four-dimensional space. To visualize it, we reduced it to a three-dimensional one. First, we started by transforming a three-dimensional space into a two-dimensional one. We used colors instead of dimensions. We formed the idea in such a way that a cube appeared in three colors along the three dimensions. Then we laid the boundaries of a cube on the plane, which resulted in six squares in different colors. Through the diversity of colors on the individual sides, we obtained the three different dimensions in two-dimensional space. We had three colors, and with that we had represented the three dimensions.

We then imagined that we were passing a square cube into the third dimension, as if we were passing it through a colored fog and it reappeared on the other side. We imagined that we had pass squares, so that the square cubes move through these squares and are thereby tinged [with the color of the pass square]. This is how we tried to imagine the [three-dimensional] cube [by means of a two-dimensional color representation]. [For the one-dimensional representation of the] surfaces, we thus have two boundary colors and [for the two-dimensional representation of the] cube, three colors. [To represent a four-dimensional spatial structure in three-dimensional space, we must] then add a fourth boundary color.

Now we have to imagine in the same way that a cube, which, analogous to our square, has two different colors as boundary sides, has three different colors in its boundary surfaces. And finally, each cube moves through another cube that has the corresponding fourth color. In doing so, we let it disappear into the fourth color dimension. So, according to Hinton's analogy, we let the respective boundary cubes pass through the new [fourth] color, which then reappears on the other side, emerging in their [original] own color.

Now I will give you another analogy and first reduce the three dimensions back to two, so that we will then be able to reduce four dimensions to three. To do this, we have to imagine the following. The cube can be put together at its boundary surfaces from its six boundary squares; but instead of doing it in succession, as we did recently, it will now be done in a different way. I will also draw this figure (Figure 31). You see, we have now spread out the cube in two systems, each of which lies in the plane and consists of three squares. Now we have to be clear about how these different areas will lie when we actually put the cube together. I ask you to consider the following. If I now want to reassemble the cube from these six squares, I have to place the two sections on top of each other so that square 6 comes to rest on square 5. When square 5 is placed at the bottom, I have to fold up squares 1 and 2, while folding down squares 3 and 4 (Figure 32). In doing so, we get certain corresponding lines that overlap. The lines marked in the figure with the same color [here in the same line quality and in the same number of lines] will coincide. What lies here in the plane, in two-dimensional space, coincides to a certain extent when I move into three-dimensional space.

The square consists of four sides, the cube of six squares, and the four-dimensional area would then have to consist of eight cubes.? We call this four-dimensional area a tessaract [after Hinton]. Now, the point is that these eight cubes cannot simply be reassembled into a cube, but that one of them should always pass through the fourth dimension in the appropriate way.

If I now want to do the same with the tessaract as I just did with the cube, I have to follow the same law. The point is to find analogies of the three-dimensional to the two-dimensional and then of the four-dimensional to the three-dimensional. Just as I obtained two systems of [three squares each] here, the same thing happens with the tessaract with [two systems of four cubes each] when I fold a four-dimensional tessaract into three-dimensional space. The system of eight cubes is very ingeniously devised. This structure will then look like this (Figure 33).

Each time, these four cubes in three-dimensional space are to be taken exactly as these squares in two-dimensional space.

You just have to look carefully at what I have done here. When the cube was folded into two-dimensional space, a system of six squares resulted; when the corresponding procedure is carried out on the tessaract, we obtain a system of eight cubes (Figure 34). We have transferred the observation from three-dimensional space to four-dimensional space. [Folding up and joining the squares in three-dimensional space corresponds to folding up and joining the cubes in four-dimensional space.] In the case of the folded-down cube, [in the two-dimensional plane] different corresponding lines were obtained, which coincided when it was folded up again later. The same occurs with the surfaces of our individual cubes of the tessaract. [When the tessaract is folded down in three-dimensional space, corresponding surfaces appear on the corresponding cubes.] So, for example, in the case of the tessaract, the upper horizontal surface of

cube 1—by observing [mediation] the fourth dimension—with the front face of cube 5.

In the same way, the right face of cube 1 coincides with the front square of cube 4, and likewise the left square of cube 1 with the front square of cube 3 [as well as the lower square of cube 1 with the front square of cube 6]. The same applies to the other cube surfaces. The remaining cube, 7, is enclosed by the other six.

You see that here again we are concerned with finding analogies between the third and fourth dimensions. Just as a fifth square enclosed by four squares remains invisible to the being that can only see in two dimensions, as we saw in the corresponding figure of the previous lecture (Figure 29), so it is the case here with the seventh cube: it remains hidden from the three-dimensional eye. Corresponding to this seventh cube in the tessaract is an eighth cube, which, since we have a four-dimensional body here, lies as a counterpart to the seventh in the fourth dimension.

All analogies lead us to prepare for the fourth dimension. Nothing forces us to add the other dimensions to the usual dimensions [within the mere spatial view]. Following Hinton, we could also think of colors here and think of cubes put together in such a way that the corresponding colors come together. It is hardly possible in any other way [than by such analogies] to give a description of how to think of a four-dimensional entity.

Now I would like to mention another way [of representing four-dimensional bodies in three-dimensional space], which may also give you a better understanding of what we are actually dealing with here. This is an octahedron bounded by eight triangles, with the sides meeting at obtuse angles (Figure 35).

If you visualize this structure here, I ask you to follow the following procedure with me in your mind. You see, here one surface is always intersected by another. Here, for example, in AB, two side surfaces meet, and here in EB, two meet. The entire difference between an octahedron and a cube lies in the angle of intersection of the side surfaces. If surfaces intersect as they do in a cube [at right angles], a cube is formed. But if they intersect as they do here [obtuse], then an octahedron is formed. The point is that we can have surfaces intersect at the most diverse angles, and then we get the most diverse spatial structures."

Now imagine that we could also make the same faces of the octahedron intersect in a different way. Imagine this face here, for example AEB, continued on all sides, and this lower one here, BCF, also (Figure 36). Then likewise the ADF and EDC lying backwards. Then these faces must also intersect, and in fact they intersect here in a doubly symmetrical way. If you extend these surfaces in this way, [four of the original boundary surfaces] are no longer needed: ABF, EBC and, towards the back, EAD and DCF. So of the eight surfaces, four remain. And the four that remain give this tetrahedron, which is also called half of an octahedron. It is therefore half of an octahedron because it intersects half of the faces of the octahedron. It is not the case that you cut the octahedron in half. If you bring the other four faces of the octahedron to the cut, the result is also a tetrahedron, which together with the first tetrahedron has the octahedron as a common intersection. In stereometry [geometric crystallography], it is not the part that is halved that is called the half, but the one that is created by halving the [number of] faces. With the octahedron, this is quite easy to imagine.

If you imagine halving the cube in the same way, that is, if you allow one face to intersect with the corresponding other face, you will always get a cube. Half of a cube is a cube again. I would like to draw an important conclusion from this, but first I would like to use something else to help me.

Here I have a rhombic dodecahedron (Figure 37). You can see that the surfaces adjoin each other at certain angles. At the same time, we can see a system of four wires, which I would like to call axial wires, and which run in opposite directions to each other [i.e. connect certain opposite corners of the rhombic dodecahedron, and are therefore diagonals]. These wires now represent a system of axes in a similar way to the way in which you imagined a system of axes on the cube.

You get the cube when you create sections in a system of three perpendicular axes by introducing blockages in each of these axes.

If the axes are made to intersect at other angles, a different spatial figure is obtained. The rhombic dodecahedron has axes which intersect at angles other than right angles.

The cube reflects itself in half.

But this applies only to the cube. The rhombic dodecahedron, cut in half, also gives a different spatial structure.

Now let us take the relation of the octahedron to the tetrahedron. And I will tell you what is meant by this. This becomes clear when we gradually let the octahedron merge into the tetrahedron. For this purpose, let us take a tetrahedron, which we cut off at one vertex (Figure 38). We continue this process until the cut surfaces meet at the edges of the tetrahedron; then what remains is the indicated octahedron. In this way we obtain an eight-sided figure from a three-dimensional figure bounded by four surfaces, provided we cut off the corners at corresponding angles.

What I have done here with the tetrahedron, you cannot do with the cube.

The cube has very special properties, namely that it is the counterpart of three-dimensional space. Imagine the entire universe structured in such a way that it has three perpendicular axes. If you then imagine surfaces perpendicular to these three axes, you will, under all circumstances, get a cube (Figure 39). That is why, when we speak of the cube, we mean the theoretical cube, which is the counterpart of three-dimensional space. Just as the tetrahedron is the counterpart of the octahedron when I make the sides of the octahedron into certain sections, so the single cube is the counterpart of the whole of space.” If you think of the whole of space as positive, the cube is negative. The cube is the polar opposite of the whole of space. Space has in the physical cube its actually corresponding structure.

Now suppose I would not limit the [three-dimensional] space by two-dimensional planes, but I would limit it in such a way that I would have it limited by six spheres [thus by three-dimensional figures].

I first define two-dimensional space by having four circles that go inside each other [i.e., two-dimensional shapes]. You can now imagine that these four circles are getting bigger and bigger [as the radius gets longer and longer and the center point moves further and further away]; then, over time, they will all merge into a straight line (Figure 40). You then get four intersecting lines, and instead of the four circles, a square.

Now imagine that the circles are spheres, and that there are six of them, forming a kind of mulberry (Figure 41). If you imagine the spheres in the same way as the circles, that they get larger and larger in diameter, then these six spheres will ultimately become the boundary surfaces of a cube, just as the four circles became the boundary lines of a square.

The cube has now been created from the fact that we had six spheres that have become flat. So the cube is nothing more than a special case of six interlocking spheres – just as the square is nothing more than a special case of four interlocking circles.

If you are clear in your mind about how to imagine these six spheres, that they correspond to our earlier squares when brought into the plane, and if you imagine an absolutely round shape passing into a straight one, you will get the simplest spatial form. The cube can be imagined as the flattening of six spheres pushed into each other.

You can say of a point on a circle that it must pass through the second dimension if it is to come to another point on the circle. But if you have made the circle so large that it forms a straight line, then every point on the circle can come to every other point on the circle through the first dimension.

We are considering a square bounded by figures, each of which has two dimensions. As long as each of the four boundary figures is a circle, it is therefore two-dimensional. Each boundary figure, when it has become a straight line, is one-dimensional.

Each boundary surface of a cube is formed from a three-dimensional structure in such a way that each of the six boundary spheres has one dimension removed. Such a boundary surface has therefore been created by the third dimension being reduced to two, so to speak bent back. It has therefore lost a dimension. The second dimension was created by losing the dimension of depth. One could therefore imagine that each spatial dimension was created by losing a corresponding higher dimension.

Just as we obtain a three-dimensional figure with two-dimensional boundaries when we reduce three-dimensional boundary figures to two-dimensional ones, so you must conclude that when we look at three-dimensional space, we have to think of each direction as being flattened out, and indeed flattened out from an infinite circle; so that if you could progress in one direction, you would come back from the other. Thus, each [ordinary] spatial dimension has come about through the loss of the corresponding other [dimension]. In our three-dimensional space, there is a three-axis system. These are three perpendicular axes that have lost the corresponding other dimensions and have thus become flat.

So you get three-dimensional space when you straighten each of the [three] axis directions. If you proceed in reverse, each spatial part could become curved again. Then the following series of thoughts would arise: If you curve the one-dimensional structure, you get a two-dimensional one; by curving the two-dimensional structure, you get a three-dimensional one. If you finally curve a three-dimensional structure, you get a four-dimensional structure, so that the four-dimensional can also be imagined as a three-dimensional structure curved on itself.*

And with that, I come from the dead to the living. Through this bending, you can find the transition from the dead to the living. Four-dimensional space is so specialized [at the transition into three dimensions] that it has become flat. Death is [for human consciousness] nothing more than the bending of the three-dimensional into the four-dimensional. [For the physical body taken by itself, it is the other way around: death is a flattening of the four-dimensional into the three-dimensional.]

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

Our ordinary space has three dimensions: length, width and height. A line extends in one dimension, it has only length. A table is a surface, so it has two dimensions: length and width. A body extends in three dimensions. How does a body of three dimensions come about?

Imagine a shape that has no dimensionality at all: that is the point. It has zero dimensions. When a point moves in one direction, a straight line is created, a one-dimensional shape. If you imagine the line continuing, a surface with length and width is created. Finally, if you imagine the surface moving, it describes a three-dimensional shape. But we cannot use the same method to create a fourth dimension from a three-dimensional object [through movement]. We must try to visualize how we can arrive at the concept of a fourth dimension. [Certain] mathematicians [and natural scientists] have felt compelled to harmonize the spiritual world with our sensual world [by placing the spiritual world in a four-dimensional space], for example, Zöllner.

Imagine a circle. It is closed on all sides in the plane. If someone demands that a coin should come into the circle from outside, we have to cross the circle line (Figure 46). But if you do not want to touch the circle line, you have to lift the coin [into the space] and then put it in. You must necessarily go from the second to the third dimension. If we wanted to conjure a coin into a cube [or into a sphere], we would have to go [out of the third dimension and] through the fourth dimension.'

In this life, the first time I began to grasp what space actually is was when I started to study recent [synthetic projective] geometry. Then I realized what it means to go from a circle to a line (Figure 47). In the most intimate thinking of the soul, the world opens up.

Now let us imagine a circle. If we follow the circle line, we can walk around it and return to the original point. Now let us imagine the circle getting bigger and bigger [holding a tangent line]. In the end, it must merge into a straight line because it flattens out more and more. [When I go through the enlarging circles, I always go down on one side and then come up on the other side and back to the starting point. If I finally move on the straight line, for example to the right into infinity, I have to return from the other side of infinity, since the straight line behaves like a circle in terms of the arrangement of its points. From this we see that space has no end [in the same sense that the straight line has no end, that is, the arrangement of its points is the same as in a closed circle. Accordingly, we must think of infinitely extended space as closed in itself, just as the surface of a sphere is closed in itself]. Thus you have represented infinite space [in the sense of] a circle [or] a sphere. This concept leads us to imagine space in its reality.

If I now imagine that I do not simply disappear [into infinity] and then return [unchanged from the other side], but think to myself that I have a radiating light, this will become weaker and weaker as I move away (seen from a stationary point on the line) and stronger and stronger when I return (with the light from infinity). And if we consider that this light not only has a positive effect, but, as it approaches from the other side here, shines all the more strongly, then you have [here the qualities] positive and negative.

In all natural effects, you will find these two poles, which represent nothing other than the opposite effects of space. From this you get the idea that space is something powerful, and that the forces that work in it are nothing other than the outflow of the power itself. Then we will have no doubt that within our three-dimensional space there could be a force that works from within. You will realize that everything that occurs in space is based on real relationships in space.

If we were to intertwine two dimensions, we would have brought these two into relation. If you want to entwine two [closed] rings, you have to unravel one of them to get the other inside. But now I will demonstrate the inner diversity of space by entwining this structure [a rectangular paper band] twice around itself [that is, holding one end and twisting the other end 360° and then holding the two ends together]. I pin the paper tape together tightly with pins and cut it in half. Now one tape is firmly stuck inside the other. Before that, it was just one tape. So here, by merely intertwining the tapes within the three dimensions, I have created the same thing that I would otherwise have to reach out into the [fourth] dimension to achieve."

This is not a gimmick, but reality. If we have the sun here, and the earth's orbit around the sun here, and the moon's orbit around the earth here (Figure 48), we have to imagine that the earth moves around the sun and therefore the moon's orbit and the earth's orbit are intertwined exactly [like our two paper ribbons]. Now the moon has branched off from the earth [in the course of the earth's development]. This is an internal bifurcation that has occurred in the same way [as the intertwining of our two paper ribbons]. [Through such a way of looking at it] space comes alive in itself.

Now consider a square. Imagine it moving through space in such a way that it forms a cube. Then it must progress within itself.

A cube is composed of six squares, which together form the surface of the cube. To put the cube together [in a clear way], I first place the six squares next to each other [in a plane] (Figure 49). I get the cube again when I put these squares on top of each other. I then have to place the sixth on top by going through the third dimension. Thus I have now laid the cube out in two dimensions. I have transformed a three-dimensional structure by laying it out in two dimensions.

Now imagine that the boundaries of a cube are squares. If I have a three-dimensional cube here, it is bounded by two-dimensional squares. Let's just take a single square. It is two-dimensional and is bounded by four one-dimensional lines. I can expand the four lines into a single dimension (Figure 50). What appears in the one dimension, I will now paint in red [solid line] and the other dimension in blue [dotted line]. Now, instead of saying length and width, I can speak of the red and blue dimensions.

I can reassemble the cube from six squares. So now I go from the number four [the number of side lines of the square] to the number six [the number of the side surfaces of the cube]. If I go one step further, I get from the number six [the number of the side surfaces of the cube] to the number eight [the number of “side cubes” of a four-dimensional structure]. I now arrange the eight cubes in such a way that the corresponding structure is created in three-dimensional space to that which was previously constructed in two-dimensional space (Figure 51) from six squares.

Imagine that I could turn this structure inside out so that I could turn it right way up and put it together in such a way that I could cover the whole structure with the eighth cube. Then I would get a four-dimensional structure in a four-dimensional space from the eight cubes. This figure is called [by Hinton] the tessaract. Its limiting figure is eight cubes, just as the ordinary cube has six squares as its limiting figure. The [four-dimensional] tessaract is therefore bounded by [eight] three-dimensional cubes.

Imagine a creature that can only see in two dimensions, and this creature would now look at the squares laid out separately, it would only see the squares 1, 2, 3, 4 and 6, but never the hatched square 5 in the middle (Figure 52). It is quite the same for you with the four-dimensional structure. [Since you can only see three-dimensional objects, you] cannot see the hidden cube in the middle.

Now imagine the cube drawn on the board like this [so that the outline forms a regular hexagon]. The other side is hidden behind it. This is a kind of silhouette, a projection of the cube into two-dimensional space (Figure 53). This two-dimensional silhouette of a three-dimensional cube consists of rhombi, oblique rectangles [parallelograms]. If you imagine the cube made of wire, you would also be able to see the rhomboid squares at the back. So here you have six interlocking rhomboid squares in the projection. In this way you can project the whole cube into two-dimensional space.

Now imagine our tetrahedron formed in four-dimensional space. If you project this figure into three-dimensional space, you should get four non-intersecting rhombic parallelpipeds. One of these rhombic parallelpipeds should be drawn as follows (Figure 54).

Eight such shifted rhombic cubes would have to be inserted into each other in order to obtain a three-dimensional image of the four-dimensional tessaract in three-dimensional space. Thus, we can represent the three-dimensional shadow image of such a tessaract with the help of eight rhombic cubes that are suitably inserted into each other. The spatial structure that results is a rhombic dodecahedron with four spatial diagonals (Figure 55). Just as in the rhombus representation of the cube, three directly neighboring rhombuses are shifted into each other, so that only three of the six cube surfaces are seen in the projection, only four non-intersecting rhombic cubes appear in the rhombic dodecahedron only four non-intersecting rhombic cubes appear as projections of the eight boundary cubes, since four of the directly neighboring rhombic cubes completely cover the remaining four.'>

We can construct the three-dimensional shadow of a four-dimensional body, but not the tessaract itself. In the same sense, we are the shadows of four-dimensional beings. Thus, as man rises from the physical to the astral, he must develop his powers of visualization. Let us imagine a two-dimensional being who makes an [intense and repeated] effort to vividly imagine such a [three-dimensional] shadow image. When it then surrenders to the dream, then (...).

When you mentally build up the relationship between the third and fourth dimensions, the forces at work within you allow you to see into [real, not mathematical] four-dimensional space.

We will always be powerless in the higher world if we do not acquire the abilities [to see in the higher world] here [in the world of ordinary consciousness]. Just as a person in the womb develops eyes to see in the physical-sensual world, so must a person in the womb of the earth develop [supernatural] organs, then he will be born in the higher world [as a seer]. The development of the eyes in the womb is an [illuminating] example [of this process].

The cube would have to be constructed from the dimensions of length, width and height. The tessaract would have to be constructed from the dimensions of length, width, height and a fourth dimension.

As the plant grows, it breaks through three-dimensional space. Every being that lives in time breaks through the three [ordinary] dimensions. Time is the fourth dimension. It is invisibly contained in the three dimensions of ordinary space. However, you can only perceive it through clairvoyant power.

A moving point creates a line; when a line moves, a surface is created; and when a surface moves, a three-dimensional body is created. If we now let the three-dimensional space move, we have growth [and development]. This gives you four-dimensional space, time [projected into three-dimensional space as movement, growth, development].

[The geometric consideration of the structure of the three ordinary dimensions] can be found in real life. Time is perpendicular to the three dimensions, it is the fourth, and it grows. When you bring time to life within you, sensation arises. If you increase the time within you, move it within yourself, you have the sentient animal being, which in truth has five dimensions. The human being actually has six dimensions.

We have four dimensions in the etheric realm [astral plane], five dimensions in the astral realm [lower devachan] and six dimensions in the [upper] devachan.

Thus the [spiritual] manifoldness swells up to you. The devachan, as a shadow cast into the astral realm, gives us the astral body; the astral realm, as a shadow cast into the etheric realm, gives us the etheric body, and so on.

Time flows in one direction, which is the withering away of nature, and in the other direction it is the revival. The two points where they merge are birth and death.

The future is constantly coming towards us. If life only went in one direction, nothing new would ever come into being. Man also has genius – that is his future, his intuitions, which flow towards him. The processed past is [the stream coming from the other side; it determines] the essence [of how it has become so far].

The subject we are to discuss today will present us with a number of difficulties. Consider the lecture as an episode; it is being held at your request. If you only want to grasp the subject formally in its depth, some mathematical knowledge is necessary. But if you want to grasp it in its reality, you have to penetrate very deeply into occultism. So today we can only talk about it very superficially, only give a suggestion for this or that.

It is very difficult to talk about multidimensionality at all, because if you want to get an idea of what more than three dimensions are, you have to delve into abstract areas, and there the concepts must be very precisely and strictly defined, otherwise you end up in a bottomless pit. And that's where many friends and enemies have ended up.

The concept of multidimensional space is not as foreign to the world of mathematicians as one might think.® In mathematical circles, there is already a way of calculating with a multidimensional type of calculation. Of course, the mathematician can only speak of this space in a very limited sense; he can only discuss the possibility. Whether it really is can only be determined by someone who can see into a multidimensional space. Here we are already dealing with a lot of concepts that, if we grasp them precisely, really provide us with clarity about the concept of space.

What is space? We usually say: there is space around me, I walk around in space — and so on. If you want a clearer idea, you have to go into some abstractions. We call the space in which we move three-dimensional. It has an extension in height and depth, to the right and left, to the front and back, it has length, width and height. When we look at bodies, these bodies are extended for us in this three-dimensional space; they have a certain length, a certain width and height for us.

However, we have to deal with the details of the concept of space if we want to arrive at a more precise concept. Let us look at the simplest body, the cube. It shows us most clearly what length, width and height are. We find a base of the cube that is the same in length and width. If we move the base up, just as far as the base is wide and long, we get the cube, which is therefore a three-dimensional object. The cube is the clearest way for us to learn about the details of a three-dimensional object. We examine the boundaries of the cube. These are formed everywhere by surfaces bounded by sides of equal length. There are six such surfaces.

What is a surface? Those who are not capable of very sharp abstractions will already falter here. For example, you cannot cut the boundaries of a wax cube as a fine layer of wax. You would still get a layer of a certain thickness, so you would get a body. We will never get to the boundary of the cube this way. The real boundary has only length and width, no height. Thickness is eliminated. We thus arrive at the formulaic sentence: The area is the boundary [of a three-dimensional object] in which one dimension is eliminated.

What then is the boundary of a surface, for example of a square? Here we must again take the most extreme abstraction. [The boundary of a surface] is a line that has only one dimension, length. The width is canceled. What is the boundary of a line? It is the point, which has no dimension at all. So you always get the boundary of a thing by leaving out a dimension.

So you could say to yourself, and this is also the line of thought that many mathematicians have followed, especially Riemann,* who has achieved the most solid work here: We take the point, which has none, the line, which has one, the plane, which has two, the solid, which has three dimensions. Now mathematicians asked themselves: Could it not be that formally one could say that one could add a fourth dimension? Then the [three-dimensional] body would have to be the boundary of the four-dimensional object, just as the surface is the boundary of the body, the line is the boundary of the surface, and the point is the boundary of the line. Of course, the mathematician then goes even further to five-, six- and seven-dimensional objects and so on. We have [even arbitrary] “-dimensional objects [where ” is a positive integer].

Now, there is already some ambiguity in the matter when we say: the point has none, the line has one, the plane two, the solid three dimensions. We can now make such a solid, for example a cube, out of wax, silver, gold and so on. They are different in terms of matter. We make them the same size, then they all occupy the same space. If we now eliminate all material, only a certain part of space remains, which is the spatial image of the body. These parts of space are the same [among themselves], regardless of what material the cube was made of. These parts of space also have length, width and height. We can now imagine these cubes as infinitely extended and thus arrive at an infinitely extended three-dimensional space. The (material) body is, after all, only a part of it.

The question now is whether we can simply extend such conceptual considerations, which we make starting from space, to higher realities. In these considerations, the mathematician actually only calculates, and does so with numbers. Now the question is whether one can do that at all. I will show you how much confusion can arise when calculating with spatial quantities. Why? I only need to tell you one thing: Imagine you have a square figure here. I can make this figure, this area, wider and wider on both sides and thus arrive at an area that extends indefinitely between two lines (Figure 56).

This area is infinitely large, so it is >. Now imagine someone who hears that the area between these two lines is infinite. Of course, he thinks of infinity. If you now talk to him about infinity, he may have very wrong ideas about it. Imagine that I now add below [each square one more, so another row of] an infinite number of squares, and I get a [different] infinity that is exactly twice as large as the first (Figure 57). So we have > = 2 + 0,

In the same way I could get: “ = 3 +,

In calculating with numbers, you can just as well use infinity as finiteness. Just as it is true that space was already infinite in the first case, it is just as true that it is 2 + c, 3 - c, and so on. So we are calculating numerically here.

We see that the concept of the infinity of space [which follows from the numerical representation] does not give us any possibility of penetrating deeper [into the higher realities]. Numbers actually have no relation to space at all, they relate to it quite neutrally, like peas or any other objects. You now know that nothing changes in reality as a result of calculation. If someone has three peas, multiplication does not change that, even if the calculation is done correctly. The calculation 3 + 3 = 9 does not give nine peas. A mere consideration does not change anything here, and calculation is a mere consideration. Just as three peas are left behind, [you do not actually create nine peas,] even if you multiply correctly, three-dimensional space must also be left behind if the mathematician also calculates: two-, three-, four-, five-dimensional space. You will feel that there is something very convincing about such a mathematical consideration. But this consideration only proves that the mathematician could indeed calculate with such a multidimensional space; [but whether a multidimensional space actually exists, that is,] he cannot determine anything about the validity of such a concept [for reality]. Let us be clear about that here in all strictness.

Now we want to consider some other considerations that have been made very astutely by mathematicians, one might say. We humans think, hear, feel and so on in three-dimensional space. Let us imagine that there are beings that could only perceive in two-dimensional space, that would be organized so that they always have to remain in the plane, that they could not get out of the second dimension. Such beings are quite conceivable: they can only move [and perceive] to the right and left [and backwards and forwards] and have no idea of what is above and below.

Now it could be the same for man in his three-dimensional space. He could only be organized for the three dimensions, so that he could not perceive the fourth dimension, but for him it arises just as the third arises for the others. Now mathematicians say that it is quite possible to think of man as such a being. But now one could say that this is also only one interpretation. One could certainly say that. But here one must again proceed somewhat more precisely. The matter is not as simple as in the first case [with the numerical determination of the infinity of space]. I am intentionally only giving very simple discussions today.

This conclusion is not the same as the first purely formal [calculative] consideration. Here we come to a point where we can take hold. It is true that there can be a being that can only perceive what moves in the plane, that has no idea that there is anything above or below. Now imagine the following: Imagine that a point becomes visible to the being within the surface, which is of course perceptible because it is located in the surface. If the point only moves within the surface, it remains visible; but if it moves out of the surface, it becomes invisible. It would have disappeared for the surface being. Now let us assume that the point reappears, thus becoming visible again, only to disappear again, and so on. The being cannot follow the point [as it moves out of the surface], but the being can say to itself: the point has now gone somewhere I cannot see. The being with the surface vision could now do one of two things. Let us put ourselves in the place of the soul of this flat creature. It could say: There is a third dimension into which the object has disappeared, and then it has reappeared afterwards. Or it could also say: These are very foolish creatures who speak of a third dimension; the object has always disappeared, perished and been reborn [in every case]. One would have to say: the being sins against reason. If it does not want to assume a continuous disappearance and re-emergence, the being must say to itself: the object has submerged somewhere, disappeared, where I cannot see.

A comet, when it disappears, passes through four-dimensional space.

Everything that has been said so far is as unassailable as it can possibly be. And the confirmation is so simple that it will not even occur to man in his present deluded state to admit it. The answer to the question: Is there something that always disappears and reappears? — is so easy. Just imagine, a feeling of joy arises in you and then it disappears again. It is impossible that anyone who is not clairvoyant will perceive it. Now the same sensation reappears through some event. Now you, just like the surface creature, could behave in different ways. Either you say to yourself that the sensation has disappeared somewhere where I cannot follow it, or you take the view that the sensation passes away and arises again and again.

But it is true: every thought that has vanished into the unconscious is proof that something disappears and then reappears. At most, the following can be objected to: if you endeavor to object to such a thought, which is already plausible to you, with everything that could be objected to from a materialistic point of view, you are quite right. I will make the most subtle objection here, all the others are very easy to refute. For example, one says to oneself: everything is explained in a purely materialistic way. Now I will show you that something can quite well disappear within material processes, only to reappear later. Imagine that some kind of vapor piston is always acting in the same direction. It can be perceived as a progressive piston as long as the force is acting. Now suppose I set a piston that is exactly the same but acting in the opposite direction. Then the movement is canceled out and a state of rest sets in. So here the movement actually disappears.

In the same way, one could say here: For me, the sensation of joy is nothing more than molecules moving in the brain. As long as this movement takes place, I feel this joy. Now, let us assume that something else causes an opposite movement of the molecules in the brain, and the joy disappears. Wouldn't someone who doesn't go very far with their considerations find a very meaningful objection here? But let's take a look at what this objection is actually about. Just as one [piston] movement disappears when the opposite [piston movement] occurs, so the [molecular movement underlying the sensation] is extinguished by the opposite [molecular movement]. What happens when one piston movement extinguishes the other? Then both movements disappear. The second movement also disappears immediately. The second movement cannot extinguish the first without itself being extinguished. [A total standstill results, no movement whatsoever remains.] Yes, but then a [new] sensation can never extinguish the [already existing] sensation [without perishing itself]. So no sensation that is in my consciousness could ever extinguish another [without extinguishing itself in the process]. It is therefore a completely false assumption that one sensation could extinguish another [at all]. [If that were the case, no sensation would remain, and a totally sensationless state would arise.]

Now, at most, it could be said that the first sensation is pushed into the subconscious by the second. But then one admits that something exists that eludes our [immediate] observation.

We have not considered any clairvoyant observations today, but have only spoken of purely mathematical ideas. Now that we have admitted the possibility of such a four-dimensional world, we ask ourselves: Is there a way to observe something [four-dimensional] without being clairvoyant? — Yes, but we have to use a kind of projection to help us. If you have a piece of a surface, you can rotate it so that the shadow becomes a line. Similarly, you can get a point from a line as a shadow. For a [three-dimensional] body, the silhouette is a [two-dimensional] surface. Likewise, one can say: So it is quite natural, if we are aware that there is a fourth dimension, that we say: [Three-dimensional] bodies are silhouettes of four-dimensional entities.

Here we have arrived at the idea of [four-dimensional space] in a purely geometrical way. But [with the help of geometry] this is also possible in another way. Imagine a square, which has two dimensions. If you imagine the four [bounding] lines laid down next to each other [i.e., developed], you have laid out the [boundary figures] of a two-dimensional figure in one dimension (Figure 58). Let's move on. Imagine we have a line. If we proceed in the same way as with the square, we can also decompose it into two points [and thus decompose the boundaries of a one-dimensional structure into

zero dimensions]. You can also decompose a cube into six squares (Figure 59). So there we have the cube in terms of its boundaries decomposed into surfaces, so that we can say: a line is decomposed into two points, a surface into four lines, a cube into six surfaces. We have the numerical sequence two, four, six here.

Now we take eight cubes. Just as [the above developments each consist of] unfolded boundaries, here the eight cubes form the boundary of the four-dimensional body (Figure 60). The [development of these] boundaries forms a double cross, which, we can say, indicates the boundaries of the regular [four-dimensional] body. [This body, a four-dimensional cube, is named the Hinton Tessaract after Hinton.]

We can therefore form an idea of the boundaries of this body, the tessaract. We have here the same idea of the four-dimensional body as a two-dimensional being could have of a cube, for example by unfolding the boundaries.

Dear attendees! The art of eurythmy is still in the early stages of development. One could even call it an experiment in intent. Therefore, it will be permitted to say a few words about the nature of the same before the presentation.

Everything that is being attempted and that will probably be perfected in the future with regard to this eurythmic art is based on Goethe's conception of the world and life. This Goethean view of the world and life is accompanied by a very special artistic attitude and a special concept of art. And it is precisely this that is so special about Goethe: he understood how to bridge the gap between artistic attitude, artistic power and general world view for his own perspective. In this way, it could also be attempted on the basis of Goetheanism, on which we stand with our entire anthroposophically oriented spiritual science; it could be attempted in a very special field – in the field of human movement art – to create something ourselves that will be an expression of Goethe's artistic attitude. Therefore, I ask you not to look at what we can offer today in this direction as if we wanted to compete with any of the arts and art forms that are in some sense related to our eurythmic art. We certainly do not want to do that. We know very well that the art of dance and similar arts, which one might confuse with ours, have now reached such a high level of perfection that we cannot compete at all. But we do not want to compete; rather, our aim is to introduce something fundamentally new into the general artistic development of humanity. And without becoming theoretical, I would like to explain very briefly how our attempt is connected with the greatness of Goethe's world view.

The truly significant, the great and decisive aspect of Goethe's world view has by no means been sufficiently appreciated. Goethe was able to orient his world of ideas, his world of cognitive sensation, in such a way that he could truly make the ascent from the science of the non-living – which basically still includes all of today's science – to a certain knowledge of the living. It only appears to be a theoretical matter when everything points to Goethe's great idea of the metamorphosis of organic beings and a single organic entity. In Goethe's sense, one need only imagine how a single plant comes into being as a living being, how it grows, perfects itself and reaches the summit of its becoming. For Goethe, every single plant leaf – whether a green plant leaf or a colored flower petal – is basically a whole plant, only more simply formed than the whole plant, and again the whole plant is for him only an intricate leaf.

This view, which is tremendously significant, applied to everything that is alive for Goethe. Every living being is formed in such a way that, as a whole, it is the more complicated formation of each of its individual parts; and each individual part, in turn, reveals – in a simpler form – the whole living being. This view can now be applied to the expressions and activities of a living being, and in particular of the highest living being known to man within his world: man himself.

And so, based on Goethe, we can also say: in what human language is, a single element of the entire human nature is also given. In what a person expresses through the larynx and its neighboring organs, speaking from the depths of his soul, something is given that is a single organ expression, a revelation of the human being. For those who are able to see what forces, possibilities for activity and movement are actually present in the human larynx when speaking, especially when speaking artistically, when speaking poetry as well as when singing, for those who can see this and are not limited to looking beyond what the larynx accomplishes in terms of movements, and merely listening to what is accomplished in terms of movements, it is possible for the person to transfer to the whole human being what otherwise only comes to expression in the individual organ - in the larynx and its neighborhood - in speaking.

It is possible to make the whole person a larynx, so that he moves in his limbs as, I would say, the larynx is predisposed to move when a person speaks or sings. One could also say: when one speaks, one is dealing with the wave motion of the air. Sounds are movements of the air. Of course, in everyday life we do not see these movements of the air. Those who look can therefore perceive the possibilities of movement that they can transfer to the whole human being, to his limbs. Then a visible language arises in which the arms and other limbs of the human being move in a lawful way. And through this visible language, the poetic-artistic aspect of language, the song-like aspect of music, is brought to revelation, and a completely new art form arises. This is to be our eurythmy.

What you see here is, in the first instance, nothing other than the human being's laryngeal movement transferred to the whole human being in an artistic way. What is now supposed to be art and must make a corresponding aesthetic impression when it is directly observed, if it is to have an artistic effect when observed directly, has of course arisen from the depths of human nature at its source. Thus one can say: what is simply there in man because he is a human organism should be brought forth from him.

There is nothing artificial in eurythmy. All gestures and pantomime are avoided. Just as in music it is not about expressing something through any old note, but about observing a lawfulness in the succession of notes, so here it is also not about the hand or something similar making any old movement, but about the human limbs making lawful eurythmic movements in succession. Thus everything arbitrary is avoided, and where something still occurs, you can regard it as a sign that something imperfect still exists there. If two people or two groups of people were to represent one and the same thing, they would only differ in the way they presented it, just as two different piano players will play a Beethoven sonata differently.

In eurythmy, everything is modeled on the movements of the larynx and its neighboring organs. But human speech is imbued with warmth of soul, with enthusiasm, with joy, with pain and suffering, with all kinds of inner crises. Everything that resonates through human language as an inner expression of the soul can be expressed by us in the relationships between the various forms, the groups, and through what a person can reveal through movements in space. In the same way, the inner mood of the soul, what penetrates from the depths of the soul to the surface, comes to expression. On the one hand, you will see what visible speech is.

We will let it be accompanied either by music, which is only the other, parallel expression of the same thing, or mainly by recitation, by poetry. In this context, I must note that, while the art of eurythmy is accompanied by poetry, it must be borne in mind that what is today the art of declamation, the art of recitation, is very much in decline. If one wants to accompany the art of eurythmy with poetry, one must go back to the old, good forms of recitation, the art of recitation. It is not a matter of expressing the ordinary narrative, the content of a poem through emphasis, but rather of expressing the actual artistic element through the recitation, apart from the purely narrative, from the content: the rhythm, the rhyme, the artistic vibrancy of a poem, everything that exists outside of the content, in other words - the poetic and musical.

There is little understanding of this today. But one need only remember that Goethe conducted his “Iphigenia” with a baton, and one need only keep in mind that Schiller, before he even brought the prose content of a poem to life in his writing, had a general melody in his soul, that is, he started from the general artistic idea. Today's emphasis on content when reciting is, so to speak, nonsense, it is decadent. It would not be possible to accompany eurythmy with this art of recitation, which only focuses on content. Therefore, we must return to what is little understood by our contemporaries as an art of recitation. But in this way we believe we can emphasize an element in the present that is as artistic as possible through this eurythmic art and thereby bring to life something of Goethe's artistic spirit. Goethe says so beautifully: “When nature begins to reveal her secret to someone, they feel an irresistible longing for her most worthy interpreter: art.” He sees in art a revelation of the secret laws of nature, which would not be revealed without art.

This is particularly evident when we see how man himself, in his movement, becomes the expression of a visible, living language. Goethe says elsewhere: Art consists in a kind of recognition, in that we grasp the essence of things in tangible and visible forms. And the highest of external nature, the human being, is revealed to us when we can visualize what is in his movements and present it to our eyes. Therefore, we feel Goethe's saying so much: “[In that man is placed at the summit of nature, he sees himself again as a whole nature, which in itself has to produce a summit again. To do so, he elevates himself by permeating himself with all perfection and virtue, invoking choice, order, harmony and meaning, and finally rising to the production of the work of art [...].

We believe that through this eurythmic art, which is brought forth from the human being himself, at the same time something is visibly placed before the human eye like an artistic revelation of the mystery of the world, which is expressed in the highest sense in the human being. So far, however, only a beginning of all this exists. We know this very well and we are the harshest critics of the imperfections that still cling to our eurythmic artistic experiment. With this in mind, I ask you to also take in today's presentation. If it finds understanding among our contemporaries, then it will lead to it being further perfected. For however convinced we are that it is still in its infancy today, we are equally convinced that it has such principles within it that it can be brought to such perfection, either by ourselves or by others, that this eurythmic art, among other things, will be able to present itself as fully justified.