Quantum physics - measurements. The Two-Dimensional World of Christian Imaginations

When I saw this book, Martin Gardner's Mathematical Leisures and Math Puzzles and Fun, which I read when I was still in school, came to mind. I remembered that one of these books described a book about an imaginary two-dimensional country of Flatland. This book was published under the pseudonym A. Square, which can be translated into Russian as "A Certain Square". The main character of the book "Flatland" was a square that lived in this two-dimensional country. I remember exactly that this book was written in the 19th century. But I have never heard of the book "Planiversum". The author's surname reminded me of the surname of the author of a book of puzzles, which was often mentioned in Martin Gardner's books - Dudeney. As I found out later, in the books of Martin Gardner, Henry Ernest Dudeney was mentioned - an Englishman, and the author of this book is Alexander Kivatin Dewdeny - a Canadian. Alexander Kivatin Dewdney is also known as the author of a computer game for programmers - CoreWars, which in Russian is called "Memory Fight".

I didn't expect much from this book. Well, what can you think of a flat world? Since there is one dimension less in this world, there is clearly nowhere to turn around and write something interesting. But I was wrong.

Firstly, the author made a very competent eyeliner to the story. One would have expected the book to start off somewhat ordinary: "Let's imagine a world in which there is no third spatial dimension, what would it be?" Or: “Once upon a time in a flat country there lived flat man". The end of the book is already imagining: “And then I suddenly woke up.” Not interested.

In fact, it all starts with the fact that a teacher at a university gives his students a task - to create a program for modeling a two-dimensional world. It all starts with a model of the planetary system, in which round flat planets revolve around a round flat sun. Further, students began to fill this program with various additional elements- someone modeled the continents and seas, someone modeled the weather, and someone populated this country with two-dimensional living beings. One of the students added a lexical module to this program - it became possible to ask the program to describe the environment.

Further, this program sometimes starts to behave strangely - it writes words that are not in the dictionary, but does not recognize them when these words are used by the operator sitting at the computer. The fact is that the world modeled in the program turns out to be so similar to the real two-dimensional world that it enters into resonance with it, so that through the program it becomes possible to look at the real two-dimensional world. However, the connection with this world is through a local resident named Yndrd, whom the teacher and students call Yendred for convenience.

It was first. And now - secondly. Secondly, the details of the structure of this world are not thoughtlessly copied from our three-dimensional world. The two-dimensional world has its own specifics, and what is familiar to us in the two-dimensional world turns out to be unviable. For example, in this two-dimensional world, the weather is always predictable: the low pressure area forms from the direction of the sun, and the surface wind always blows towards the sun. In the morning the wind blows to the east, where the sun rises, and in the evening it starts to blow to the west, where the sun sets.

It rains in this world, but the rivers do not have a channel: water flows over the surface of the planet, not being able to go around obstacles to the right or left. That is why the inhabitants of the planet do not build houses. If you build a house, then the water flowing from the side of the mountains will reach the house and fill the entire lowland formed by the mountain and the house. Therefore, local residents live in houses that resemble our dugouts, and animals live in burrows. So that the dugout is not flooded, it is clogged as soon as they hear the sound of approaching water.

In this world, the door hinges we are accustomed to cannot exist, and ropes cannot be tied into knots. Door hinges resemble ball joints - the circle is inserted into a semicircular hole, and the door attached to the circle moves up and down. Ropes are usually glued or attached to each other with hooks. However, this also has positive side: since it is impossible to tie a knot on a rope, the ropes never get tangled.

As a boat in this world, you can use a simple stick, the ends of which are bent in one direction. Such a boat cannot turn around - only change direction. A pole is used as a sail, which is installed vertically in the center of the boat. Since the wind always has a predictable direction, in the east every morning you can go by boat to the ocean, and in the evening the wind will blow in the opposite direction - towards the mainland. In the west, the opposite is true - you can go to the ocean in the evening, and return to the mainland in the morning.

Local creatures do not have an internal rigid skeleton, because the skeleton in this case would divide the body into independent parts. All creatures in this world have an external skeleton, like beetles. There is no through digestive tract, because if it were, then the creature would split into two parts. Therefore, through the mouth, both the consumption of food and the removal of digestive waste occur - they are spit out. The circulation, however, still exists. The tissues separate, capture the fluid bubble, and then join. The fluid bubble moves between the tissues in such a way that, in the course of its movement, the tissues are separated, and behind they are connected. It turns out a kind of blood peristalsis.

I will not say anything more about the structure of this world, I will only mention that it has metallurgy, steam engines, clockwork, musical instruments, rockets, space stations, astronomy, chemistry, cell biology, electricity, books, fine arts and computers. Every field of science, every mechanism is explained in this way - not simply by copying the things of our world, but by explaining the principles of operation and the inherent limitations. For example, it explains how cells manage to exchange nutrients without splashing their contents out. Explains how nerve cells transmit signals along intersecting paths without mixing signals. The same problem is explained in relation to the design of computers - how logic gates transmit signals along intersecting paths without mixing signals. Explains how electrical power is supplied to the computer's valves.

From what I said, one might get the impression that the book has no plot and it only says about what and how it works. This is not true.

Spoiler (plot reveal) (click on it to see)

The protagonist Yendred heard about a monk who lives in another country - Vanicle. Vanitsla is located in the east of the mainland, behind the mountains. This is where the main character goes. Before leaving, Yendred went fishing with his father. In the city of Is-Felblt, he visits his uncle, who runs a printing house and prints books. With their uncle's children, they go to the market, where they buy a balloon for travel. Then the younger children go home, and Yendred eldest daughter uncle goes to a music concert. Then Yendred visited the only scientific institute of his country - Punitsla. Along the way, he walks, moves in a balloon, holding it in his hands, makes a flight in a transport balloon and on a rocket. Finally, he reaches a mountain plateau, where he nearly dies in a quarry from a flying kite. Then he finally meets the very monk named Drabk, whom he wanted to meet. The monk then initiates Yendred into secret knowledge, after which Yendred stops communicating, losing interest in the inhabitants of the three-dimensional world.

In some way, this book reminded me of Andrey Rodionov's article "Game is a Serious Matter", which I once read in the science fiction magazine "If". This article began as a regular article describing the classification of computer games. Then the author talks about how he made his computer game. This story flows smoothly into the genre science fiction. Then I still went to school, my skeptical thinking was practically absent and I believed almost everything. It is not surprising that this article made a tremendous impression on me then - I simply did not notice the transition from the journalistic genre to the science fiction genre and took the story about a computer game at face value. Both in this book and in Andrei Rodionov's article, reality smoothly turns into fiction, which gives credibility to the sci-fi component. Both the book and the article deal with the creation virtual world, which, unexpectedly for the creators themselves, shows unforeseen properties, starting to live its own life.

By the way, much later, when I became interested in the musical genre Synth Pop, I found albums by Andrey Rodionov and Boris Tikhomirov. I really like some of the songs from these albums, and at one time I even used the song "Electronic Alarm" as an alarm signal on my phone. I did not immediately connect the musician and the author of that article in my head. And then he did find out that he really developed computer games. For example, one of his games is called Major of Pistols at the Factory. It's funny that the world of this game is also flat. True, in it the main character knows how to mirror himself :)

However, I digress. Let's go back to the Planiversum. The book was not written as a result of individual reflections. At the end of the book, the author explains that for a long time he has been collecting articles about the arrangement of various things in a flat world, which were written for fun by other people. Before writing this art book the author wrote the monograph "Science and Technology in a Two-Dimensional World". Later, an article was written about this monograph ... Martin Gardner. The idea of ​​a rocket plane was thrown to the author by Jeff Raskin, the initiator of the Apple Macintosh project. He also created the lesser-known but very peculiar Canon Cat computer. Before reading this book, I was just thinking about buying Jeff Raskin's book The Interface: New Directions in Computer System Design.

This is perhaps the most best book of all the science fiction books I've read. This book is based on just one fantastic assumption - there is a two-dimensional world inhabited by intelligent living beings, and you can communicate with this world. Here, of course, there is no intensity of emotions, no moral messages, but the book is addictive. I would say that I read it avidly, but in fact, I deliberately diverted from it from time to time, because it takes you to another world that operates according to other laws, but has its own logic. While reading, the thinking is reorganized so much that, being distracted from reading, you feel disorientation - thoughts continue to swarm in your head, which suddenly turn out to be inapplicable to the familiar three-dimensional world. It takes a few seconds to push those thoughts aside and get back to reality.

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Let's remember what we were taught about measurements and turn to how quantum physics sees it. According to spiritual teachings, there are twenty-one dimensions in the universe.

Let's check how we feel the measurements on different levels consciousness.

1. One dimension has one extension, such are the point and the line.

2. Two dimensions have yes extensions - this is a plane. It has length and width.

3. Three dimensions have three extensions: length, width and height. Here objects appear in our world, for example, a cube.

4. Four dimensionshave four extensions, here three dimensions are complemented by time. At any moment, something is happening around us.

5. Outside fourth dimension, feelings, thoughts, ideas appear in higher dimensions, which influence events and actions.

There are many invisible things that affect our lives and the functioning of the worlds. Every action comes from an intention! Imagination is already the creation of form, which has all the intentions of movement and germ needed to carry it out.

Looking from the upper world, the order of measurements changes. The first dimension is intent. The dimensions of imagination, form, time, space, plane and point mean the most extreme dimensions.

Many of the people settled on a two-dimensional view of the world. They lack the courage to think and think about new things that would lead them forward along the path of prosperity. It seems that the purpose of someone or some dark forces was so that a person could not guess what a fantastic creature he is. After all, man could imagine that he had creative power. But in what dimension does this creative ability operate?

Imagine a two-dimensional world, such as a flat world. Flat people live in this flat world. They have no idea that there are many dimensions, because there, they have everything two-dimensional. In this flat world, two-dimensional people see only two dimensions.

From the outside, as observers, we see both a two-dimensional and a three-dimensional world. Everything that happens there, we perceive and realize differently. We perceive the same phenomenon as two-dimensional and three-dimensional.

The case of a 3D rocket flying through a 2D world:

A three-dimensional rocket flies through the two-dimensional world. What will two-dimensional beings see living planes?

A rocket flying through the world leaves a trail behind it. When touching this world, the tip of the rocket describes a point, then circles, symbols corresponding to the size, and finally, the rocket will leave this two-dimensional world. What will the inhabitants of this two-dimensional world say, watching this? Oh my God! Here, in our world, there were dots, circles and other symbols.

There are, however, other people in this world who think differently and have the courage to make themselves heard. Arriving there, otherwise thinking two-dimensional being will look at the sky, again at circles and a point, then again dare to look up, close his eyes and say: there was a three-dimensional rocket, leaving prints behind him.

Who is right? we ask.

At their own level of consciousness - everyone. Residents of a one-dimensional world will surely say: a completely crazy creature speaks of something that does not exist. To this, two-dimensional people will say: so abstract, he thinks differently, different than we are.

If beings begin to think, they will understand that there are other dimensions beyond the horizon. They will be able to understand that he, otherwise thinking person, is actually right. Socrates was such a dissident person, who on the streets of Athens asked passers-by only questions that should be thought about. The inhabitants began to wake up consciousness, so the rulers of the city ordered to seize Socrates and forced him to drink poison. The city fathers were afraid of what would happen if people awakened self-consciousness.

The same thing happened with Jesus, who always makes people think with his spiritual messages. The Romans and the elders were horrified by the awakening of people's consciousness, so Jesus was killed. The fact of this terrible crime was distorted by the fact that they began to preach: God sacrificed his son.

measurements

Our joys, misfortunes, experienced in the higher dimensions, are visible in the lower ones. When bad thoughts, misfortunes or illnesses eat someone up, it can be seen physically. Shadows, projections of higher dimensions are symptoms of the body.

Happiness, spiritual freedom, flight takes the form of a healthy body in visible dimensions.The two-dimensional imprints of bodily symptoms, like the three-dimensional rocket, are just symbols. The world is over high level, reflected on the worlds of a lower level, has the sign of symbols.

Let someone try to convey, show their feelings, thoughts that form an invisible reality. Everyone knows that it exists, but we carry it invisible in ourselves.

How simple it would be if there were only that which is felt by the five kinds of senses. Simple, i.e. "one-dimensional". The "many-sided" person feels free in the higher realms.

Setting beyond nine points:

There are nine points in the task. Please connect them with straight lines. You can do this in any order without lifting your pencil by touching each point.

If you can go beyond the nine points in the two-dimensional boundaries, then you can go not only from point to point, but you can also go beyond the area limited by the points. The secret of the task is that we do not think within the nine points, but are able to go beyond them.

In the process of solving the problem, it seems that we have not yet passed into another dimension.

In order to look at the solution of our problem from higher dimensions, we must mentally rise above our knowledge and way of seeing. People, in order to achieve titles, ranks, make any sacrifices. If only part of these efforts were spent on spiritual and spiritual growth, there would not be so many sick and unhappy people. The representatives and preachers of these noble ideas were the great mystics.

If anyone wants to go beyond a certain way of seeing, supported by 2D and 3D x-ray, ultrasound, ST and MRI images, he must have great courage, strong faith, fundamental knowledge and will. The idea already in many cases carries the key to the solution - this is the highest dimension of form, which comes from intention.

Have the courage to go beyond the traditions, the familiar, the ingrained? What happens if you connect the dots with four lines? I solved the matrix, since this task already involves free thinking. We not only move into three-dimensional space, but also go beyond it, into higher areas of thought.

The limited human consciousness acts and thinks on the same plane. Anyone who unexpectedly accomplishes things that are unimaginable to others deserves to be called a traveler in dimensions with his versatility.

The sum of the interior angles of a triangle:

(Equator)

Answer to this question modern man with lower or even higher education: 180 degrees. This definition is one of the cornerstones of mathematics.

Let's analyze the triangle on the scale of the Earth. It is known that the Earth is not flat, many centuries ago it became known that the Earth is round.

Draw two perpendiculars to the Earth's equator. As you can see 90° + 90°, this is the sum of the angles of the triangle, equal to 180°. Now let's follow the two perpendiculars that will meet at the north pole and one more angle is closed there. This latter may have 1°, 30° or even 359°. Let's add the internal angles of the formed triangle: 90°+90°+30°=210°. This, as can be seen, is greater than the sum of 180° indicated above.

A significant part of students today grew up on Euclidean geometry. They think in a plane - they were taught that way. (Another thing is that the theorems of Euclid and Thales are valid in plane geometry). However, thinking only in the plane will be fatal. If people saw everything, thought only in a plane, life would be enclosed in two dimensions. Of course, those who set out to think in many dimensions sometimes run into serious problems. Often, even very educated people live with a flat consciousness, i.e. in a limited world.

How will the human psyche react: if one day we go beyond the traditional, definite, flat thinking imposed on us?

When people meet a person who thinks differently, they will immediately condemn him. There is a danger that people will also have to change their views. Some are so attached to ingrained dogmas, faith, like an alcoholic or a smoker to the object of his passion.

It is well to consider whether we intend to change our views. Those who take on the challenge of adventure and travel will become a healthier, happier, more hopeful, successful, out-of-the-ordinary person.

You know that what makes the world one-dimensional is that the position in it is determined by one unit of information.

It must also be continuous (or close to continuous from a practical point of view). I have described several examples of dimensions: an income line, infinite, and represented by an infinite straight line; rainbow line, terminal, with bounding walls, represented by a line segment; eolian line of wind directions, finite-periodic, represented by a segment whose left end coincides with the right, or, what is the same, a circle. In passing, I mentioned another example - about the world, infinite in one direction, and finite in the other. In another article, I emphasized that there are many types of dimensions, but the physical dimensions of space have unique and special (and also very obvious) properties that distinguish them from dimensions of another type.

Rice. 1: 2D worlds

What about two-dimensional worlds? It is not surprising that there are many more types of two-dimensional worlds than types of one-dimensional worlds. Several examples of such spaces are shown in Fig. 1. One can imagine a world that is infinite in both directions: a plane (top left). One can imagine a world that is infinite in one direction, and in the other forming either a segment or a circle. Such worlds are naturally called strip and pipe (lower left). One can imagine a finite world in both directions (right side of Fig. 1). And how many possibilities are there! Only in this picture you can see from top to bottom a square, a cylinder (a round part of a can without lids and insides), a disk, a torus (something like a car tire), a sphere (only the surface), a double tire. And these are not all options. If extrapolated into the future, it becomes clear that by the time we get to three dimensions, and go further, we will no longer be able to make such lists.

As with one-dimensional spaces, a position in a two-dimensional space is defined by two pieces of information.

An example of a sphere (to a good approximation) would be the surface of the Earth: any location can be represented by latitude and longitude. An ant walking on a garden hose moves along a two-dimensional pipe, and at any time is located at a certain distance from the faucet and under certain angle to the vertical. A multi-lane highway is essentially a two-dimensional strip with a very long side and a short side: the two pieces of information needed to determine your position are the distance from the beginning of the road and the distance from its right edge.

Consider the income line. "Your income for last year is a specific number in your local currency. It can be positive or negative, large or small; it can be represented as a point on a line, as in Fig. 1, which we will call the "income point". Each dot on the line represents a possible return." If you are married and both you and your spouse have income, two members of your household cash flow can be represented as a two-way plane. The two numbers describing a point on this plane will be your income and your spouse's income.

And here is a tricky example of a torus showing how interesting two-dimensional shapes can be imagined, whose dimensions are not those of physical space. On fig. 3 articles on one-dimensional worlds, we saw that the possible directions of the wind form a one-dimensional world in the form of a circle (or a line that has the same beginning and end). Possible destinations the movements of the sailboat also form a similar circle. But all who have sailed know that it is not necessary to sail in the same direction as the wind blows; if you put the sail at an angle, you can move to the west, even if the wind blows from the north. So if I ask for two pieces of information - which direction the wind is blowing, and which direction my sailboat is heading - they will both be points on a circle. Two units of information, both located on a circle, represent a point on the torus.

Before continuing, I will mention a natural and common confusion. I already hinted at it in the description different worlds, given above. Do not confuse the dimensions of the forms themselves with a certain way of representing these dimensions or forms! The property of a circle is that if you move along it in any direction, you will return to where you started. The circle has nothing inside or outside. Simply representing a circle as a closed curve on a two-dimensional plane looks like it has an inside and an outside. But this is simply a property of the representation of the circle on the plane, and not a property of the circle itself.

Icons came to Russia from Byzantium, and here they truly found a second life. The fact is that by the XI-XII centuries. Byzantine iconography is beginning to rely more and more on the canon, since the Greeks are losing contemplation of the spiritual. In Russia, at that time, people lived who to a high degree preserved the ancient figurative clairvoyance. In pagan times, lower spirits, servants of nature, were revealed in this clairvoyance. When, with the advent of Christianity, Greek icons began to be brought to Russia, what was imprinted in them according to tradition was illuminated for Russians by the true light of the imaginations behind the icon-painting images. It is easy to verify this by comparing at least once the early Russian icons, in which the Byzantine influence strongly affects (12, 13, 16), with the icon, on which only the Byzantine canon is reproduced (160).

There was a deep meaning associated with this circumstance. Glancing over the subsequent development of Byzantium - its decline, decline, after which it fell apart and was absorbed by the Ismailis, you get the impression that to a large extent the meaning of its existence consisted in transmitting to Russia the impulse of Christianity in its eastern form, about which we read above in R. Steiner. In any case, icon painting, being the fruit of the Greek-Latin culture of the era of Christianity, played its special role precisely in the Slavic world, and primarily in Russia.

We have already said that the imaginative supersensible image has the ability to directly influence the astral-etheric shells of a person. Christian imaginings, in the center of which stands Christ Himself, carry a special individualizing power, and it is extremely important that from the very beginning it was that divine "sculptor" who "carved" the Russian individuality from a block of group aura. Eastern Slavs. This is the uniqueness of the development of the Russian world. He appeared to a certain tabula rasa for Christianity.

The Slavs were the last among the European peoples to enter into a cultural-historical process, and from the very beginning it was Christian for them. The Greco-Latin world took the individualizing principle into souls prepared for this by a long period of pagan culture. No wonder Plato and Aristotle are called Christians of pre-Christian times. The medieval world took a lot from the Druidic Mysteries, from Rome, and, moreover, from the very beginning it carried in itself special inclinations for the development of the "I". The Slavic world, the world of the Eastern Slavs, before the adoption of Christianity, was in the cultivation of elementary natural magic. Ancient Slav he lived in close merger with nature, so that the processes that took place in it and in himself were experienced by him as a kind of unity. The spiritual world was open to his vision, but it was the world of elemental spirits of water, forest, animals. For higher contemplations in pagan times, the Mysteries were needed. They are not observed among the Eastern Slavs, with the exception of the northern regions, the Onega region, where some remnants of the Celtic Mysteries of the Trotts could be preserved, which, probably, was the reason for the peculiar development of Novgorod Russia. In the rest of the territory of ancient Russia, clan-family communities lived, carrying a single consciousness, going back to one of the deceased ancestors, to the ancestor - the guardian of the clan.

Coming into this environment, the world of Christian revelations could not but produce in it the action of a very special force. In the atmosphere of worship, in the temple, truly new heavens opened up to the Slav. The walls of the temple were transparent to him, the revelations imprinted on them rose in their true vitality. The form of revelations, conveyed by a line, the color of an icon or a fresco, served as a gateway to the spiritual world, it focused the scattered consciousness, concentrating and individualizing the astral body, turning it into an individual bearer of religious experiences. The form of the imaginative image differs from any form of the physical world in its higher purposefulness and organization, and as such it creates in the world of sensible things in its own image and likeness.

After all, what is an icon in form or color? First of all, its two-dimensionality attracts attention. Its perspective is said to be reversed. Let us explain its essence in the figure:

This perspective has a very special effect on the viewer. As early as the beginning of the 20th century. it was noticed that when the image is given in reverse perspective, the viewer, as it were, takes her space. 76 This is achieved due to the fact that the system of reverse perspective, as convincingly revealed by the artist and art researcher L.F. Zhegin, implies the mobility of the visual gaze, which it sums up in a single visual impression and transfers to the image. Direct perspective leaves the viewer passive, suggests even greater flatness of visual perception than the reverse one, since it does not take into account the binocularity of our vision - it can be viewed with one eye. Reverse perspective reveals to us not only the frontal view of the object, but also its side faces. “In the process of visual merging of individual aspects of an object into one integral image,” writes Zhegin, “the side faces and top of the object unfold, the shape of the object becomes, as it were, dynamic.

Since perspective deviations and multilateral visual coverage are applied not only to moving objects, but also to objects that are obviously immobile - household items, buildings, even mountains, it becomes clear that the artist himself is in motion", 77 as well as the viewer, we add .

Zhegin's book describes many interesting patterns of reverse perspective, but one cannot agree with the author, namely, that reverse perspective is a conscious technique of the ancient painter. No, it arises as a natural result of the transference of supersensible contemplation onto a sensuous flat surface. In imagination, a person merges with a supersensible object, therefore he sees it, or rather, he sees and experiences it from all sides.* And the icon leads the viewer to the same experience. It forces you to enter the imaginative space, in which there is no need to have a third dimension due to the merging of the object and the subject of contemplation together.**

* Children draw in reverse perspective because they are close to imaginative experiences

** It is this phenomenon that threatens to "scald" the soul of the positivist, if he experiences the icon really deeply

It is not easy to understand this phenomenon. Therefore, we turn for help to the developments made by anthroposophists in the field of projective geometry. It is in it that the most interesting results have been achieved, which make it possible to form an idea of ​​the essence of the transition from the three-dimensionality of the sensible world to the two- and even one-dimensionality of the supersensible worlds. Therefore, it is desirable to take its constructions not only mathematically, but also psychologically and meditatively. It is important to go through the process of construction, transition from one form to another, then the corresponding representation will be born.

We will take examples that are elementary for those who have gone through Waldorf pedagogy, but those who have not dealt with such issues may encounter some difficulties even in these examples. However, let's try to use our imagination. Imagine a candle and a triangular sheet of paper (all this can be verified by experience). If the flame of the candle is above the top B of the triangle ABC (Fig. 1a), then the shadow AOC is formed from it. Let's lower the flame of the candle to the same level with peak B (Fig. 1c). Then sides of the shadow triangle (AO and CB) will become parallel, and the vertex O will go to infinity. Let's lower the candle even lower, and we will get a completely incomprehensible picture (Fig. 1c). First of all, it is not clear where the top of the shadow triangle went. In the previous position, we say, she went to infinity, although this can only be thought of mathematically. In a new position, the top of the shadow triangle is found by construction. To do this, continue the diverging sides of the shadow triangle to the right. They intersect at the point O ", which is the vertex we are looking for. It is formed by the rays that left the vertices A and C to the left to infinity and "returned" from the other side. And now we need to imagine that the rays AO, CO and BO ( Fig. 1c) passed through infinity, that is, as if through non-existence, and returned from the other side. Therefore, the shadow on the right is not due to the source of light, and its existence, at the same time, is real, and carries some other features. These features due to its passage through non-existence, if you look at it from the point of view of the sensory world, or through other being, through the spiritual world, if you speak in the language of the science of the spirit.Along with it, the laws that cause its appearance on the left have passed through other being, therefore on the right they already wear a different character - a "white" shadow. All this is by no means idle speculation. Mathematicians smartly operate with the concept of infinity, without delving into its meaning, as, indeed, into the meaning of all others mathematical concepts. This, they say, was done by the ancients, for example, by the Pythagoreans, but now it is the subject of the history of mathematics. However, those who develop mathematical problems from the standpoint of Spiritual Science pay great attention to the interpretation of mathematical concepts. One of the results of their research is the disclosure of the transcendental nature of the concept of infinity. Those wishing to get acquainted with this in more detail can refer to the relevant literature, but we can only touch on this very briefly.

So, we should try to experience these examples from projective geometry, and then we will find that in them, despite their simplicity, we almost "tangibly" come into contact with the supersensible. For a person facing the world of imaginations, the contemplated object comes as if from infinity (like the shadow on the right in our example), having the vanishing point of its perspective in front of itself, because it is due to the subjective perception of the one who contemplates the Perceiving Imagination, if he is placed in the conditions of our example, is located, as it were, to the right of the physical light source. But the imaginative image and the sensory world, as we have already said, are interconnected. The imaginative image is a kind of response to the sensory experience of a person, which has gone through his cognizing spirit (like a shadow) into the supersensible (to infinity) and returned from there, being illuminated by the light of other worlds. At the moment of supersensory perception, a person can (and even should) forget about his sensory experience, but after that it is always possible to make sure that what is contemplated in imaginings is closely connected with the physical world - not as a reflection (this is the area of ​​abstract thinking), but as archetypal images. with images, as the essence with the phenomenon.

In the experience of the supersensible cognizer, self-perception, going from the sensible to the supersensible, summarizes the object into a point of the subjective "I", comprehensively embracing it with inner vision. If this did not happen, then the imagination would not acquire a figurative character, and the contemplator would simply dissolve in it, which is the case with Eastern occultists. European development went a different way. Here the human soul, even before it began to think about the sensible world, had already acquired a tendency towards an individual perception of the supersensible. But for this, the supersensible itself has become a kind of reflection of the sensible, not only in the deep sense when we speak of the laws of nature, but also in its manifestation to the individual human soul. In our example, we can talk about some kind of sensual and intimate phenomenon of the nature of light, when the candle is below the top of the triangle. Then its light, as it were, is divided in two, to the left it acts according to physical laws, and to the right the regularity of its supersensible nature is partially manifested - it does not give a shadow, but "illuminates" it.

But let's continue with our example. Let us now take a straight line and a circle (Fig. 2). If two tangents to the circle are drawn from different points of the line G., and the points of contact are connected by straight lines, then they all intersect at point A. Let us bring the line E closer to the circle. When the line touches it, point A will coincide with the point of contact. With the removal of the straight line b to infinity, the tangents become parallel and the point A coincides with the center of the circle. Tangents in this case, since we are talking about infinity, can, of course, come from any direction.

Let's complicate our example a little, take the plane P and the sphere (Fig. 3). As in the previous example, we will build cones tangent to the sphere from different points of the plane. Obviously, the bases of these cones inside the sphere will have one common point (point A). With the removal of the plane to infinity, the cones will turn into cylinders, and the point A will coincide with the center of the sphere. It is also obvious that a plane at infinity can be thought of in any direction.

Next, take two intersecting planes (Fig. 4). If a straight line is drawn through A and B, the points of their contact with the sphere, then it will be in projective connection with the line of intersection of the CO planes. Similar to what we observed in the two previous examples, with the removal of the line CO to infinity, the line AB will pass through the center of the sphere.

If we expand this example and take several planes that, mutually intersecting, form a pyramid, and inscribe a sphere inside it, then through the points of contact with its faces, we can construct a pyramid inscribed in the sphere (Fig. 5) Each face of the inscribed pyramid is in a projective relationship with the face of the circumscribed pyramid due to the ratio of the straight line ab to CD, da to BC, etc. (according to the law considered in Fig. 3). With the removal of the faces of the circumscribed pyramid to infinity, the inscribed pyramid will shrink to a point in the center of the sphere. A pyramid with faces that have gone to infinity can be thought of as a plane.

Not all figures inside the sphere correspond to those outside it. So, for example, a cube inside a sphere corresponds to an octahedron outside (Fig. 6). However, any figure outside the sphere, receding to infinity, can be taken as a plane by us, since any of its faces can come from any direction.

In these examples of projective geometry, we can feel why three-dimensionality loses its mandatory character with the expansion of physical regularity to infinity. At the same time, physical objects, one might even say, are mysteriously conditioned by what comes to them from "flat" and even "point" infinity. It is easy to feel this if the given examples are not fixed purely rationally, but let them pulsate through geometric construction from visible forms to the correspondences that go to infinity.* Then we can feel why the world of imaginations is two-dimensional. It contains the archetypes that determine the phenomenal world. In our examples, we have shown how they come to the phenomenon.

* After all, the whole Earth is spherical, and the firmament surrounding it, conceived geometrically, is a plane, since it is infinitely distant from it.

The discovery of a direct perspective was undoubtedly a necessary step on the way to the knowledge of sensual reality, where the individual "I" found its support. But in this three-dimensional space there is no place for supersensible objects When, for example, we consider the painting "Annunciation" (10), attributed to the brush of Leonardo ,* then, despite artistic skill, a deep knowledge of perspective, the laws of mechanics (as for the wings of the Angel), which its author possessed, we still look at her with some bewilderment. For it is clear to us that this material angel will not fly anyway, no matter how correctly its wings are drawn from the point of view of mechanics, and if it does fly, then where will it go in this three-dimensionality, where Angels do not live? At the same time, it is not difficult for us to accept the supersensible nature of icon-painting images and, in general, everything written with a deviation from a direct perspective.

* We are inclined to believe that this is the work of his followers or students

Iconography has the goal of depicting, or rather, conveying supersensible reality. Therefore, it is entirely two-dimensional, up to the image of faces with "strange", if you look at it from the point of view of the sensory world, proportions (11). In reality, the faces of the saints, as it were, shine towards us from the spiritual world, extending back to infinity, and forward converging towards the center of our perception. It should be noted that the images of Divine beings are less subject to the laws of reverse perspective. This is probably the effect of their active organizing power, which not only reveals itself, but reduces the consciousness of the one who sees it into the sensible world. These images often have an abstract cosmic character, clothed in a figurative form. Take for example the icon of the XII century. "Our Lady Oranta" (159). If we penetrate into the meaning of its symbolism, where "symbolic signs are associated with what can follow one natural ... one enters here into the other" (see note 72), then we can say that on the icon in the image of the Mother of God all The earth with its spiritual aura. In this aura, the Divine Child opens and is born. It is born as a spiritual Sun in the ethereal world of the Earth, enters into it as a gift, as a sacrifice. And the Earth accepts this sacrifice, which is expressed by the hands of the Mother of God: they form a triangle, their open gesture is directed upwards, in it is the readiness of the Earth to accept this gift, and even a prayer for it, ascending from the Earth to the spirit, which, in turn, is additionally emphasized a triangle formed by the head of the Mother of God and the hands of her hands. The gesture of the baby Jesus speaks of Christ's readiness to embrace the Earth, to unite with it. At the same time, on the icon, he appears as a hypostasis of the Trinity - this is evidenced by the fingers folded in a blessing gesture. What is under the feet of the Mother of God is the firmament of the earthly kingdoms of nature, animated by ethereal forces.

In some contrast - according to the mood - the icon "Annunciation" (12) appears to the icon "Our Lady of Oranta". If in the first icon we experience a kind of gesture of cosmic opening, then in the second icon we experience a cosmic mood of closing. Both moods, taken together, form that primordial phenomenon of polarity, within the boundaries of which the individual principle of a person matures. In the "Annunciation" a cosmic impulse approaches the Earth in the form of a messenger - the Archangel Gabriel. Mother Earth, personified in the image of the Mother of God, in a kind of cosmic introspection experiences the news of the coming appearance of the Son of God and, at the same time, already contains it in herself. The firmament under the feet of the Mother of God expresses the fourfold principle of man: the physical, etheric, astral body and "I".

We see much of the universal-cosmic principle of ancient beauty in the image of St. George on the icon of the XII century. (13). However, his gaze is humanly individual, only turned inward. There inside human spirit opens the Regent of the cosmic intelligentsia Michael, whose earthly reflection is St. George. Therefore, his hair is not just ornamental, but to some extent resembles the surface of the human brain, which is an instrument of thinking consciousness. The sword points to the human "I", but not yet active in the earthly sphere. The realization of its real power will come later, in the XIV-XV centuries, and this will be reflected in the images of the Archangel Michael, where He will take the sword out of its scabbard and show them the way upwards for human consciousness (14). St. George on the icon is so far only a receiver of Michael's impulse on Earth, but he thinks meditatively, which, apart from a glance, is indicated by a spear in his hand - a symbol of the supersensible power of thinking, that is, thinking permeated with will. Thus, St. George appears to us on this icon as a call to prepare our individual consciousness for the acceptance on Earth of all the fullness of thinking both about the Earth and about the spirit, for the acceptance of Sophia, or the cosmic intelligentsia.

A completely different mood is evoked by the icon, which is called the "Archangel with golden hair" (15). This Archangel, like St. George, and cosmic, and individually human, but in a different way. His look is soft, he expresses mercy - that new quality of the human soul, which comes into the world with Christianity. The Archangel represents this property macrocosmically. He is all mercy. There is no in his eyes human limitations, in it is the breadth of the Divine Being, endowed with tremendous good power. Therefore, his mercy is full of strength. This is characteristic of a truly strong being; that's how a person should be. A bandage with a ruby ​​is woven into the hair of the Archangel. This stone is related to the center of intuition in the human brain. Intuition requires a high degree of organization of thinking, the ability for moral fantasy, therefore the Archangel has a different hair ornament, not the same as that of St. George; and therefore the hair is pulled together by a bandage. In the mood that radiates from this Archangel, one can feel something inherent in the Archangel of the Russian people, both "young and ancient" (R. Steiner), whose inspirations are expressed in the best properties of the Russian soul.

Images of Christ Himself on ancient Russian icons are sublime, but also highly individualized. Here is the "Savior Not Made by Hands" of the XII century. (16). The parted hair is markedly differentiated with the help of gilded lines, the so-called assist. This assist is applied only to images of spiritual beings; with its radiant color and fractional structure, it, as it were, reveals the Divine-human nature of the spiritual impulse coming from them. The assist technique came to Russia from Byzantium, and there from Rome. The beard of the icon-painting face is even more individualized than the hair, but it is given without an assist, since the beard is something that comes purely from the human, while the hair of the head is the conductor of the spirit. The gaze of the depicted face is turned not to the viewer, but to the side, to surrounding a person world, where, therefore, we, the people, are also invited to direct our attention. The gaze is focused and strict: "I did not come to bring peace, but a sword" (Matt. 10, 34), that is, the impulse of I-consciousness, which will destroy blood and family ties and call for spiritual brotherhood. Therefore, the face of Christ on the icon appeals to the concentration of the mind, to responsibility for what is happening in the earthly world. No wonder this icon was depicted on the banners of the Russian regiments that came to the Kulikovo field.

But the image of Christ on the icon of the end of the XIII century. (17). Here His gaze expresses a completely different thing, what the New Testament says: "God is love." This view is directed at us, at our inner, where a new, Christian morality is born.

Such images, in addition to revealing the nature of God turned to humanity, are called upon to forge separate, but fundamental qualities in the souls. One need only imagine the world of ancient Russia, where human faces reflected a completely non-individualized consciousness in order to understand what kind of impression these iconic faces evoked. Meeting them was impossible to forget.

Absolutely special mention should be made of the depiction of Christ on the Deesis icon of the 12th century, where He is called Immanuel (18). Christ is depicted surrounded by the Archangels Michael and Gabriel, He has a young face, no beard. From the messages of R. Steiner, we know that Christ was often depicted with a beard because in ancient figurative clairvoyance, through the face of the Son of God, the image of God the Father always seemed to shine through. And a different, higher power of clairvoyance was needed in order for the young face of Christ to be revealed. Looking at this icon, we can tell ourselves that there were high initiates in Russia, although their names remained unknown to us. High is the sphere in which Christ is revealed in the image in which the Archangels see him. A person can rise there only with the help of the most intense spiritual exercises. This is the sphere of intuition, it is indicated by the rubies in the headbands of the Archangels. It was also revealed to the creators of the original image of this icon that they contemplate Christ as He appears in His Second Coming. One day, many will see Him in a similar form. This time has already come with the transition from the era of Gabriel to the era of Michael, that is, starting from the end of the 19th century.

Many icons are dedicated to the events of the earthly life of Christ Jesus. The leading place among them is occupied by "The Nativity of Christ" (19). What is especially important for us to see here is the image of two Jesus boys. One of them, Nathan's Baby, is born in a Bethlehem manger that was set up in a cave. Shepherds come to worship Him, as described in Ev. from Luke. But the icon also depicts a star, which led the Magi, kings from the east, to another Infant; this is mentioned in Ev. from Matthew. This Child, from the royal line of Solomon, is depicted at the bottom right. He is bathed in a golden font by two maids. Another significant icon-painting theme dedicated to the earthly life of Christ Jesus is the "Transfiguration" (20). The icons on this theme reflected the knowledge of the ancient Christian initiates about the dual nature of man - about the so-called upper and lower trinity, which even in pre-Christian times was symbolically depicted as a hexagram, the star of Solomon. Christ on the icon forms the whole theme and embraces it with Himself. At the same time, He Himself, as the Life-Spirit, together with Moses and Elijah, abides in the highest trinity formed by the Spirit-Self, the Life-Spirit, and the Spirit-Man. The apostles Peter, James and John express the lower trinity of the body, soul and spirit (as "I"). * That figure, by which the body is meant, remains in an unconscious state on some icons, on others it closes with all its might from a sudden contemplation. . The one that expresses the spirit tries to look and see what is happening above. The middle figure has an average condition between the extreme ones. In the aura of Christ, in some cases there are three circles, in others - a hexagram and circles.

* But also the threefold soul: sentient, rational, conscious.

Among the various iconographic images of Christ, the "Savior in Power" (21) evokes special amazement with its esoteric depth. The icon originated around the beginning of the 15th century. and is included as a central theme in the Deesis tier. She represents Christ seated on a throne, with an open gospel in her hands. His feet rest on the earth, depicted as a quadrangular stand; this firmament is carried by the Thrones (Thrones), which on the icon look like wheels with wings. Christ is surrounded by an enormous aura in which the entire first Hierarchy is revealed. The aura of Christ himself, in turn, is included in a quadrangular red aura, in the corners of which an angel, an eagle, a calf and a lion are depicted. This is the aura of a person, his supersensible image, and everything depicted on the icon is revealed in it.

On some icons, especially those of the Rublev school (22), another, also red, quadrangular aura is given inside the aura of Christ. This aura expresses the incarnation of Christ, His Resurrection in a human body as a Spirit-man. Then the throne on which He sits and which expresses physical corporeality is given with a white stroke - this is no longer just corporality, but a phantom, or, say, the throne of the Spiritual Man. The firmament under the feet of Christ and on this icon is given as physical. Although it is carried by the Hierarchies, it is not yet enlightened, unlike the body of Jesus. These are earthly kingdoms. The icon, therefore, reflects the inner, mystical experience of the experience of Christ, when He is revealed in the inner man as the Hypostasis of the Divine Trinity, possessing the fullness of the forces of all nine Hierarchies. This is, undoubtedly, the knowledge of Christ in intuition, acquired by the initiates in Russia. It constitutes the center of esoteric Christianity, to which the hopes of men and gods converge. Therefore, this icon is placed in the center of the Deesis tier, and on both sides it is surrounded by other icons, on which, on the left side, the Apostle Peter, the Archangel Michael and the Mother of God bow to Christ in prayer positions, and on the right side, the Apostle Paul, the Archangel Gabriel and John the Baptist . Such an arrangement of figures means that both groups do not form two poles on the sides of Christ, but a circle, inside of which there is a prayer. The figures on the left and right, taken in pairs, represent two sides of any one aspect of the appearance of Christ: the Mother of God and John, Michael and Gabriel, Paul and Peter.

The Deesis rank is truly the pinnacle of Russian Christian esotericism. It is addressed to the soul common man and a student of the Christian Mysteries. The Church, over time, lost its understanding, therefore, already in the 15th century, the Deesis rank was underestimated in its most important part: Christ began to be depicted as a man sitting on a throne, without an aura, as a teacher of the church, and the church fathers were included in the number of forthcoming .

The image of the Mother of God enjoys special reverence in Russia. The attitude of believers to Her is, perhaps, to the closest and really effective spiritual force. She is a representative, a prayer book before God for the human race. She is depicted as such in the Deesis ranks. She is an exponent of the best properties of the human soul. Her love for the Divine Child is the same love that every mother feels for her child, but expanded to love for all humanity. So great is Her Soul. From people, the Mother of God is the greatest in Christian love.

The cult of the Mother of God came to Russia in its cosmic aspect of Sophia, Divine Wisdom. In the first centuries of Christianity, churches of St. Sophia were built throughout Russia. The most significant among them are in Kyiv and Novgorod. This cosmic cult is reflected in such icons as Our Lady of Oranta (159). Its mosaic counterpart is given in the apse Sophia Cathedral in Kyiv. But there are also icons directly depicting St. Sophia, in connection with which we can speak of the continuity of the ancient Mysteries in the era of Christianity (23). Unfortunately, very few of these images have survived. And the cult of St. Sophia itself in later centuries somehow strangely faded and receded into the background. But on the other hand, in Russian religious philosophy in late XIX- early XX centuries. Sophia's theme sounded with renewed vigor. Here is how, for example, Vl. Solovyov wrote about the icon of St. Sophia: “Who does this main, middle and royal face depict, clearly different from Christ, and from the Mother of God, and from angels? The image is called the image of Sophia, the Wisdom of God. But what does this mean? In the 14th century, a Russian boyar asked this question to the Archbishop of Novgorod, but received no answer, he responded with ignorance. Meanwhile, our ancestors worshiped this mysterious face, as the Athenians once did "an unknown god".... The icon of Novgorod Sophia itself is no Greek model does not have - this is a matter of our own religious creativity ...

This Great, regal and feminine Being, who, being neither God, nor the eternal Son of God, nor an angel, nor a holy man, accepts veneration both from the consummator of the Old Testament and from the ancestor of the New - who is it, if not the true, pure and complete humanity, the highest and all-encompassing form and living soul of nature and the universe, eternally united and in a temporal process uniting with the Divine and uniting with Him all that is." 78

The predominant value in Russia received images of the Mother of God with the baby. The first among them is "Vladimir Mother of God" (24). The icon was brought from Byzantium and comes from the 11th century. From the original painting, only faces have been preserved on it, the rest was painted anew in the 15th-16th centuries. She has rich history, and many miracles are associated with it. In terms of significance in the religious life of Russia " Our Lady of Vladimir" can be put on a par with the "Our Lady of Czestochowa" in Poland and the "Ostrobrama Mother of God" in Lithuania. And although the icon of "Our Lady of Vladimir" is now in the museum, for believers it remains a shrine, and not a picture or a museum exhibit.

The Assumption of the Mother of God is included in the Orthodox cult as a great holiday. Among the numerous icons on this subject, we present the most ancient of the now preserved (25). The icon reflects the earthly-cosmic understanding of the Mother of God. The bed on which Her body rests, although remotely, resembles an Egyptian sarcophagus. Around the bed are the Apostles and the Fathers of the Church. They are representatives of the earthly world. Against a golden background, as if from the shining expanses of the macrocosm, the highest aspects of the spirit of the Apostles, the earthly representatives of the circle of the Zodiac, rush to the bed in spiritual auras: through the Apostles, the Earth is united with Heaven, humanity - with the Hierarchies. Within this single earthly-heavenly circle, the resurrected Christ receives the Soul of the Mother of God and passes it on to the Hierarchies. Through them, as if by a kind of spiritual "ladder", She ascends to the Highest Devachan. In other versions of this icon, its meaning is esoterically even more profound. In these versions, the figure of Christ is surrounded by an aura, in which all the Hierarchies are revealed, which indicates the origin of Christ from spheres higher than the spheres of the Hierarchies - from the world of the Divine Trinity. On such icons, the Spirit Self of the Apostles is depicted carried by Angels. This is the Spirit Self that descended upon the Apostles on the day of Pentecost in the form of fiery tongues, and the union with which is predestined for other people as well. And just as the Soul of the Mother of God is perceived by Christ, so, people can hope, the soul of each person will once be perceived when he unties the knots of earthly karma and thereby overcomes earthly gravity. The latter is shown in the form of an Angel driving away with a sword (by the power of the individual spirit of the Mother of God) those claims of earthly forces to influence the posthumous existence of the soul, which they no longer have in relation to the Soul of the Mother of God.

Another great Orthodox holiday dedicated to the Mother of God is "Protection". It is celebrated when the first snow falls and the earth is covered with a white robe, in which one can see the earthly prototype of those most pure robes of the Mother of God, which, like an astral aura, envelop the entire Earth. Here the Heavenly Mother of God connects people and heaven together. Since the icon appeared rather late, the influence of church ideology is felt in it: the church itself as an institution with its hierarchy is depicted on the icon as an intermediary between people and God (26). However, such an idea was not the only one; many Russians saw in the church an image of conciliar humanity.

In addition to spiritual beings and biblical images, we often see on ancient icons the images of Christian ascetics, the fathers of the church, but never secular people. For a trivial understanding of religion, this is an inexplicable fact. In fact, when reading the ancient chronicles, we find in them mainly descriptions of the deeds of princes, and very little is said about spiritual people. At the same time, the chroniclers were monks, and the icon painters were monks. Nevertheless, for the second - the princes allegedly did not exist at all, unless one of them performed a spiritual feat, and a special feat, as was the case, for example, in the case of princes Boris and Gleb. We will understand the reason for this if we remember what has already been said about the origins of the writing of icons. The icon painter, as a rule, was a person immersed inward, into the world of his contemplations. Chroniclers in Russia, on the contrary, often became those people who, earlier than others, began to experience the I-consciousness in themselves. Their mystical experience was weaker than that of the icon painters, but their social experience was all the more significant. Chroniclers are people who, before others, recognized the individual manifestation of the spirit. For icon painters, everything individual was determined by its appearance for the spirit, by what was revealed in contemplation as an individual aura. Naturally, lofty spiritual Essences stood in the foreground here, following them, the inner gaze discerned the individualities of the initiates, thanks to their being overshadowed by the Spirit Self. Common people, as well as princes, who gravitated towards the earthly manifestation of their spirit, that is, as reason, merged into a single mass, overwhelmed by lower desires and passions. They could not appeal to anything high, in the religious sense. The icon, even if not included in its esoteric content, should serve as evidence of the predominance of spirit over matter (11). “The icon,” wrote Prince Yevgeny Trubetskoy, “is not a portrait, but a prototype of the coming temple humanity ... an icon can only serve as a symbolic image of it. What does thinned corporality mean in this image? into the supreme and unconditional commandment." 79 When, from the end of the XVI century. the spiritual ideal of the icon began to be underestimated, then Archpriest Avvakum reproached the icon painters: "... they changed the likeness of them (the saints), they paint the same as they themselves."

So, the saints on the icons are people of a rare fate and the greatest fortitude, which allowed the likeness of God to manifest in them. Therefore, for others, for those who pray, they are role models. Their heads, like those of spiritual beings, are surrounded by a radiant aura (nimbus). The faces, although endowed individual traits, but their flesh is spiritualized, the spirit shines through it, the movement of spiritual currents that occurs in the "lotuses", in the etheric and astral bodies. This was depicted with the help of the so-called "engines", applied to the face with white, and dark "marks".*

* In terms of normal portrait painting it is absolutely impossible to understand these methods.

Nikola Ugodnik enjoys special reverence among the saints in Russia. His image on the icon of the XII century. characterized by a deep combination of individual and spiritual (27). The individual is manifested not only in his eyes, which resemble the expression of the eyes on the icon of the "Savior Not Made by Hands" (16), but the entire study of the face with "engines" suggests that "the saint has an individual aura, he, like an earthly person, is an individuality." Among other saints in Russia, Boris and Gleb are especially famous. About the wonderful icon of the XIV century, which depicts these saints, we will talk in the next essay.

Following Nikola, Boris and Gleb, among the early Christian saints in Russia, Blasius, Flora, Laurus, Paraskeva Friday (28) are honored. Florus and Laurus are dedicated to a special, very remarkable icon called "The Miracle of Florus and Laurus" (29). It depicts the Archangel Michael, giving Flora and Lavra saddled horses: one black, the other white. In the lower part of the icon, three horsemen are driving (grazing?) a herd of horses. On some icons, two of them are enthusiastically talking to each other, and the third follows them. In Orthodoxy, Flor and Laurus are considered the patrons of horse breeding, and it is widely believed that this is what is depicted on the icon. However, it is impossible to agree with this, because then Michael himself would have to be considered the patron of horse breeding - after all, it is He who gives horses to Flor and Laurus. He is the Regent of the cosmic intelligence that descends into human thought. The latter, moving according to the laws of logic, lives by the collision of two principles: positive and negative, thesis and antithesis. 80

* Some guides in museums announce it directly.

The horse is a symbol of human thinking, but the icon painter, of course, depicts not a symbol, but that figurative vision that opens up when contemplating the world of thinking with the help of atavistic clairvoyance. In this regard, we can recall one picture from the circle of mannerist artists, many of whom had glimpses of atavistic clairvoyance. This painting is by Nicolò del Abbate and is called "The Blinding of St. Paul" (30). It shows app. Paul before Damascus when he saw the risen Christ. From the esoteric point of view, the experience of Paul means an unexpected, spontaneous ascent of his ego-consciousness into the supersensible worlds. This is depicted in the form of an ideal horse rearing up. Paul, on the one hand, was ready for such an experience, thanks to his involvement in the Hebrew initiation - therefore the horse is white. On the other hand, he was not prepared for the experience of meeting the Risen One. The experience of the old initiation could not help him here, and therefore he was blinded by the vision, his earthly consciousness was broken, the physical shell fell to the ground, while the spirit soared on high.

As the ego-consciousness develops in Russian souls, Russian saints and clergy begin to appear on icons.

** Boris and Gleb, being princes and not priests, are an exception here.

Icons of saints are usually surrounded by hallmarks depicting scenes from their lives. A saint is a person who draws near to God. Therefore, the history of his life is important. She is a model for others, a life "ladder" leading to God. Everyone must eventually step on it. But if the soul wants to go faster than many others, then its path, full of labor and danger, is expressed by a different icon (31). This "ladder" is a problem for the entire Christian world.

Among the images of saints, a special place is occupied by the icons of St. George. On some, he is depicted to the waist (13), on others - in full growth, often - with the stigma of life, in the fight with a snake. This plot reflects two big problems of the spiritual development of mankind: the struggle with hereditary sin, with the temptation of man by the serpent of paradise, and the struggle with that Ahrimanic dragon, which Michael in the 19th century. overthrown from heaven to Earth and which now nests in human souls. The second of these problems arises on the icons in a kind of insight, foreseeing its further course. St. George is, one might say, the earthly aspect of the Archangel Michael, the image of a Michaelite, a man who completes the cosmic deed of Michael on Earth. Therefore, St. George is the true archetype of the human "I", following the paths of Christian evolution.

Images of St. George we find among many peoples. All of them express different aspects the struggle of man with the Luciferic-Ahrimanic dragon, and often the aspects are one-sided. Such pictures of St. George should be perceived not as a prototype, but as a warning, similar to what is given on the "Heavenly Ladder" icon. Let's look at a few examples. Take Raphael's painting "Saint George" (32). It shows the struggle of the human "I" with the Ahrimanic dragon in the conditions of modern civilization: George is clad in iron knightly armor, and chivalry, as we know, is an expression of material culture (this is its color. - R. Steiner). The spear of thought has broken against the dragon, and its clawed paw is already scratching the horse's belly - thinking of the epoch of the conscious soul. And God knows how it will all end! - The raised sword - the human "I" - must pass big way, before falling on the head of an angry dragon, which meanwhile is not asleep. The human soul - the female image in the picture - flees in horror into the stone, lifeless desert of civilization.

In another painting of the XV century. the human "I" is fighting the Luciferic dragon on the Christian mystical path (33). Out of touch with urban civilization (a city outside the walls without signs of life), thanks to the solitary prayer practice of the soul (female image), the “I” strikes the dragon, but at the same time the dragon’s tail entangles the horse’s leg, that is, it fetters the movement of thinking in the world, although it remains pure and unblemished.

On the Russian icon of the beginning of the XIV century. in general, the whole outcome of the fight with the dragon is decided by the soul (34). Catharsis of the astral body pacifies the dragon and puts a leash on it. The “I” itself does not participate in the struggle, it floats over the struggle of the soul with the dragon, not even suspecting it. * On another icon of the 15th century. the Luciferic nature of the dragon is emphasized: it is depicted in a backward movement (35). White horse gallops forward, and the fight against the dragon is led by one I under the sign of the Sun. - This is Christ fighting the dragon in a man who has not yet become an individual.

< p class="discr">* Let us recall what R. Steiner said about the Russian soul in connection with Faust.

The battle of St. George with a dragon on an icon of the 16th century. (36), where everything is given in the correct ratio for our time. The rider sits on a white horse, calmly and confidently moving forward, that is, the human "I" has reached pure thinking and moves in complete balance, being in its essence nothing more than will. The soul (the female image at the gate) also went through purification and curbed the dragon, which is struck by the pure thought coming from the "I". The balance of the soul is due to its double reliance on the material and the spiritual (hand gesture). The struggle takes place in full view of the city, that is, in the conditions of modern civilization; and although the participation of this civilization is passive, it is also good that it is aware of the dependence of its fate on the outcome of this struggle and is ready to follow the gesture that the soul makes spiritual hero(this gesture is repeated by the king on the city wall).

Design by artist A. Balashova.

When the book The Planiversum first appeared 16 years ago, it took quite a few readers by surprise. The line between the voluntary rejection of distrust and ingenuous acceptance, if it exists, is very thin. Despite the sly, ironic overtones, there were those who wanted to believe that we had come into contact with two-dimensional world Arde, a disk-shaped planet inscribed in the outer shell of a vast, shaped hot air balloon space called the Planiversum.

It is tempting to imagine that both gullible and incredulous readers did so because of the convincing logic and consistency of the cosmology and physics of this infinitely thin universe with its bizarre but strangely efficient organisms inhabiting it. After all, not just an ordinary universe, generated by a game of imagination, opened before them. Planiversum is more than a bizarre, fantastic place, since most of it was "made" by a virtual team of scientists and technologists. Reality - even the pseudo-reality of such a place is much more strange than it seems at first glance.

To begin with, we will try to understand what a flat universe Planiversum is. Understand that two dimensions means two dimensions. If the page of this book is a small piece of the Planiversum, then the curved line drawn on it may turn out to be a segment of a planiversal cord or string, the two free ends of which cannot be connected, because this requires an additional, third dimension, which, so to speak, goes beyond this page. But give us some planiversal glue and we'll stick one end to the other, trapping whatever is inside the loop of lace once the glue dries.

The appendix to the book contains a fairly complete history of the origin of the flat universe Planiversum. As soon as an article about Planiversum appeared in Scientific American in Martin Gardner's column on math games, thousands (not even hundreds) of readers sent letters containing enthusiastic responses and new ideas. Both professional scientists and engineers wrote, and even a few well-informed readers who sent reasonable suggestions.

We wove something homogeneously seamless from these ideas, but we needed a plot - a story, to make it interesting book. A story that would take us on a journey through Arda, a disc-shaped planet floating in the 2D Planiversum universe.

From the preface to the final, the narration is conducted with a serious, even impassive face. It is written with the pen of a scientific worker, whose literary possibilities are constantly under the onslaught of events. The story features a modern deus ex machina - a computer. It was through him that a group of students made their first contact with the 2D universe of Planiversum and its four-armed hero Yendred, whose craving for the "higher" turned into fear when he finally came face to face with it.

The author was surprised and disturbed that so many people took the fiction at face value. The subtext of this fantastic, though very rich in detail, story has gone unnoticed by so many. Neoteny trends rooted in Western culture even before 1984. And of course, the fantastical allegory introduced into the narrative - that is, what makes the book, in the words of the Oxford humanist Graham Stuart, "a Sufi parable" - went completely unnoticed by these readers. The temptation to bring to life the higher (third) dimension as a symbol of the forces lurking on the other side of the apparent reality of our world turned out to be too great to be overcome. The story opens with an old preface waiting for you on the next page.

A. K. Dyudni.

January, 2000

I want to note that I am not so much the author of the book as its compiler, and the main merit in the fact that this book was published belongs to the creature depicted on the first page. His name is Yendred, and he lives in a two-dimensional universe that I have named the Planiversum. The history of the discovery of Planiversum - a world in the reality of which few people could believe, will surely seem interesting to you. I want to tell her.

The first acquaintance with this world took place at our university about a year ago. My students have worked with computer program 2DWORLD, which they themselves wrote over several semesters. Initially, the purpose of the program was to give students the opportunity to practice scientific modeling and programming, but soon 2DWORLD took on a life of its own.

It all started with an attempt to model a two-dimensional model of a physical body. For example, a simple two-dimensional object might be disk-shaped and made up of many two-dimensional atoms.

It has some mass (depending on the type and number of atoms it contains) and can move around in two-dimensional space, such as this page. But, unlike a page, two-dimensional space has no thickness, and the disk cannot go beyond it. Suppose that all objects in this space obey laws similar to those that operate in our three-dimensional world. That is, if we push the disk to the right, it will start moving at a constant speed in the plane that is the continuation of the page. Sooner or later, while continuing to move in this imaginary plane, the object will leave the surface of the Earth, unless, of course, it collides with another similar object.

When such two objects meet, they will experience what physicists call an "elastic collision." In the figure, we see two objects at the moment of greatest deformation, when they collided and are about to roll away from each other. In accordance with the well-known law of physics operating in our three-dimensional universe, the sum of the kinetic and potential energies of two disks before and after the collision remains unchanged. Moving in this way, the discs cannot help but collide. They cannot "dodge" and avoid a collision. In a two-dimensional world, they simply have nowhere to "dodge".

This physical process can be easily displayed on a computer by writing a program that will simulate the behavior of two disks at the moment of collision. Of course, if we take into account that disks are composed of individual atoms, this will complicate the work of the programmer and increase the load on the processor during the execution of the program. But almost any programmer is able to write such a program and display the results on the screen.

Around this time, work began on the 2DWORLD program. In the first semester, students under my guidance not only described in the program a certain set of objects and the law of conservation of energy, but also created a whole system of planets revolving around a star. One of the planets, which they named Astria, won particular popularity among the students. By the end of the first semester, talk began about how to draw a map on this planet and populate it with living beings - the Astrians. I nipped these aspirations in the bud: the semester was coming to an end, and there was nothing left before the exams. And it was unrealistic to implement the idea - my students were not so strong programmers.

In any case, 2DWORLD turned out to be very useful program and it was incredibly interesting to work with her. I especially remember the process of the formation of a galaxy from a chaotic cluster of stars. In short, I came to the conclusion that the project was a success and that I was right when I decided to limit the physical space of the model to two dimensions. Thanks to this, students understood what real modeling is.