Quantum physics - measurements. Theory of Worlds

When I saw this book, my memory came back to the books “Mathematical Leisure” and “Math Puzzles and Fun” by Martin Gardner, which I read when I was still in school. I remembered that one of these books described a book about the 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 who lived in this two-dimensional country. I remember exactly that this book was written back in the 19th century. But I’ve never heard of the book “Planiverse”. The author's last name reminded me of the last name of the author of a book of puzzles, which was often mentioned in the books of Martin Gardner - Dudeney. As I found out later, Martin Gardner's books mentioned Henry Ernest Dudeny, an Englishman, and the author of this book is Alexander Keewatin Dewdeny, a Canadian. Alexander Kivatin Dewdney is also known as the author of the computer game for programmers - CoreWars, which in Russian is called “Battle in Memory”.

I didn't expect anything particularly interesting from this book. Well, what can you invent about a flat world? Since there is one less dimension in this world, there is clearly no room to turn around and write something interesting. But I was wrong.

Firstly, the author made a very competent summary of the story. One would expect that the book would begin in some ordinary way: “Let’s imagine a world in which there is no third spatial dimension, what would it be like?” Or: “Once upon a time in a flat country there lived a flat man.” The ending of the book is already imagining: “And then I suddenly woke up.” Not interested.

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

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

That was the first thing. And now - secondly. Secondly, the details of the structure of this world are not mindlessly copied from our three-dimensional world. The two-dimensional world has its own specifics and what is familiar to us turns out to be unviable in the two-dimensional world. For example, in this two-dimensional world, the weather is always predictable: an area of ​​low pressure forms towards 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 begins to blow to the west, where the sun sets.

It rains in this world, but the rivers have no channels: water flows along the surface of the planet, unable to go around obstacles on 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 mountains will reach the house and fill the entire depression formed by the mountain and the house. Therefore, local residents live in houses that resemble our dugouts, and animals live in burrows. To prevent the dugout from flooding, it is sealed immediately as soon as the sound of approaching water is heard.

In this world, the door hinges we are familiar with cannot exist, and ropes cannot be tied into knots. Door hinges resemble ball joints - a circle is inserted into a semicircular hole, and the door attached to the circle moves up and down. Ropes are usually glued together or attached to each other with hooks. However, there is also positive side: Since it is impossible to tie a knot in 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 the direction of movement. 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 organism into independent parts. All creatures in this world have an external skeleton, like beetles. There is no end-to-end digestive tract, because if it had been, the creature would have split into two parts. Therefore, food is consumed and digestive waste is removed through the mouth - they are spat out. Blood circulation, however, still exists. The tissues separate, capture the fluid bubble, and then connect. The fluid bubble moves between the tissues in such a way that as it moves, the tissues are separated, and behind them they are connected. This results in a kind of blood peristalsis.

I won’t say anything more about the structure of this world, I’ll just mention that it contains metallurgy, steam engines, clockwork, musical instruments, rockets, space stations, astronomy, chemistry, cell biology, electricity, books, art and computers. Each scientific field, each mechanism is explained in a similar way - not by simply copying the things of our world, but by explaining the principles of operation and inherent limitations. For example, it explains how cells manage to exchange nutrients without spilling their contents out. Explains how nerve cells transmit signals along paths that intersect each other 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 the signals. Explains how power is supplied to computer valves.

From what I have said, one may get the impression that the book has no plot and all that is written in it is about what and how it works. This is wrong.

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

The main character Yendred heard about a monk who lives in another country - Vanitsla. Vanitsla is located in the east of the mainland, behind the mountains. The main character is heading there. Before setting off, Yendred and his father went fishing. In the city of Is-Felblt, he visits his uncle, who runs a printing house and prints books. They go to the market with their uncle's children, where they buy a hot air balloon for travel. Then the younger children go home, and Yendred and his uncle’s eldest daughter go to musical concert. Then Yendred visited the only scientific institute in his country - Punitsli. Along the way, he walks, moves in a hot air balloon, holding it in his hands, and makes a flight in a hot air balloon and on a rocket. Finally, he reaches a mountain plateau, where he is almost killed in a quarry by a flying kite. Then he finally meets the same monk named Drabk whom he wanted to meet. Next, the monk initiates Yendred into secret knowledge, after which Yendred stops communicating, losing interest in the inhabitants of the three-dimensional world.

In some ways, this book reminded me of Andrei Rodionov’s article “Game is a Serious Business,” which I once read in the science fiction magazine “If.” This article began as an ordinary article describing the classification of computer games. Then the author talks about how he made his computer game. This story smoothly flows into the science fiction genre. Then I was still going to school, I had practically no skeptical thinking and I believed almost everything. It is not surprising that then this article made an amazing impression on me - 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 the article by Andrei Rodionov, reality smoothly turns into fiction, which makes the science fiction component more convincing. Both the book and the article talk about the creation virtual world, which, unexpectedly for the creators themselves, exhibits properties that were not intended, beginning to live its own life.

By the way, much later, when I became interested in the Synth Pop music genre, I found the albums of Andrei 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 Clock” as an alarm on my phone. I didn’t immediately connect the musician and the author of that article in my head. And then I found out that he actually 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 turn himself into a mirror :)

However, I digress. Let's return to the Planiverse. 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 collected articles on the structure of various things in the flat world, which other people wrote for fun. Before writing this fiction book, the author wrote the monograph “Science and Technology in a Two-Dimensional World.” Later, Martin Gardner wrote an article about this monograph. The idea of ​​a rocket plane was given to the author by Jeff Raskin, the initiator of the Apple Macintosh project. He also created the lesser-known, but very unique Canon Cat computer. Before reading this book, I was just thinking about buying Jeff Raskin’s book “Interface: New Directions in Computer Systems Design.”

This is perhaps the best science fiction book I've ever read. This book is based on only 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 periodically deliberately distracted myself from it, because it takes me to another world that operates according to different laws, but has its own logic. While reading, thinking is reorganized so much that when distracted from reading, you feel disorientated - thoughts continue to swarm in your head, which suddenly turn out to be inapplicable to the usual three-dimensional world. It takes a few seconds to put these thoughts aside and return to reality.

This is already the fourth topic. Volunteers are also asked not to forget what topics they expressed a desire to cover, or maybe someone has just now chosen a topic from the list. I am responsible for reposting and promoting on social networks. And now our topic: “string theory”

You've probably heard that the most popular scientific theory of our time, string theory, implies the existence of many more dimensions than common sense tells us.

The biggest problem for theoretical physicists is how to combine all the fundamental interactions (gravitational, electromagnetic, weak and strong) into a single theory. Superstring theory claims to be the Theory of Everything.

But it turned out that the most convenient number of dimensions required for this theory to work is as many as ten (nine of which are spatial, and one is temporal)! If there are more or less dimensions, mathematical equations give irrational results that go to infinity - a singularity.

The next stage in the development of superstring theory - M-theory - has already counted eleven dimensions. And another version of it - F-theory - all twelve. And this is not a complication at all. F-theory describes a 12-dimensional space of more than simple equations than M-theory - 11-dimensional.

Of course, theoretical physics is not called theoretical for nothing. All her achievements exist so far only on paper. So, to explain why we can only move in three-dimensional space, scientists started talking about how the unfortunate remaining dimensions had to shrink into compact spheres at the quantum level. To be precise, not into spheres, but into Calabi-Yau spaces. These are three-dimensional figures, inside of which there is their own world with its own dimension. A two-dimensional projection of such a manifold looks something like this:

More than 470 million such figures are known. Which of them corresponds to our reality is currently being calculated. It is not easy to be a theoretical physicist.

Yes, this seems a little far-fetched. But maybe this is precisely what explains why the quantum world is so different from the one we perceive.

Let's go back a little into history

In 1968, a young theoretical physicist, Gabriele Veneziano, was poring over the many experimentally observed characteristics of the strong nuclear force. Veneziano, who was then working at CERN, the European Accelerator Laboratory in Geneva, Switzerland, worked on this problem for several years until one day he had a brilliant insight. Much to his surprise, he realized that an exotic mathematical formula, invented about two hundred years earlier by the famous Swiss mathematician Leonhard Euler for purely mathematical purposes - the so-called Euler beta function - seemed capable of describing in one fell swoop all the numerous properties of the particles involved in strong nuclear interaction. The property noticed by Veneziano provided a powerful mathematical description of many features of the strong interaction; it sparked a flurry of work in which the beta function and its various generalizations were used to describe the vast amounts of data accumulated from the study of particle collisions around the world. However, in a sense, Veneziano's observation was incomplete. Like a rote formula used by a student who does not understand its meaning or meaning, Euler's beta function worked, but no one understood why. It was a formula that required explanation.

Gabriele Veneziano

This changed in 1970, when Yoichiro Nambu of the University of Chicago, Holger Nielsen of the Niels Bohr Institute, and Leonard Susskind of Stanford University were able to discover the physical meaning behind Euler's formula. These physicists showed that when elementary particles are represented by small, vibrating one-dimensional strings, the strong interaction of these particles is exactly described by the Euler function. If the string segments were small enough, these researchers reasoned, they would still appear like point particles, and therefore would not contradict experimental observations. Although this theory was simple and intuitively attractive, the string description of the strong force was soon shown to be flawed. In the early 1970s. High-energy physicists have been able to peer deeper into the subatomic world and have shown that a number of string-based model predictions are in direct conflict with observational results. At the same time, there was a parallel development of quantum field theory—quantum chromodynamics—which used a point model of particles. The success of this theory in describing the strong interaction led to the abandonment of string theory.
Most particle physicists believed that string theory had been consigned to the trash bin forever, but a number of researchers remained faithful to it. Schwartz, for example, felt that “the mathematical structure of string theory is so beautiful and has so many amazing properties that it must surely point to something deeper” 2 ). One of the problems physicists had with string theory was that it seemed to provide too much choice, which was confusing. Some configurations of vibrating strings in this theory had properties that resembled the properties of gluons, which gave reason to truly consider it a theory of the strong interaction. However, in addition to this, it contained additional interaction carrier particles that had nothing to do with the experimental manifestations of the strong interaction. In 1974, Schwartz and Joel Scherk of France's École Technique Supérieure made a bold proposal that turned this apparent disadvantage into an advantage. After studying the strange vibration modes of the strings, reminiscent of carrier particles, they realized that these properties coincide surprisingly closely with the supposed properties of the hypothetical particle carrier of gravitational interaction - the graviton. Although these "minuscule particles" of gravitational interaction have yet to be detected, theorists can confidently predict some of the fundamental properties that these particles should have. Sherk and Schwartz found that these characteristics are exactly realized for some vibration modes. Based on this, they suggested that the first advent of string theory failed because physicists overly narrowed its scope. Sherk and Schwartz announced that string theory is not just a theory of the strong force, it is a quantum theory, which, among other things, includes gravity).

The physics community reacted to this suggestion with great reserve. In fact, according to Schwartz's memoirs, “our work was ignored by everyone” 4). The paths of progress were already thoroughly cluttered with numerous failed attempts to combine gravity and quantum mechanics. String theory had failed in its initial attempt to describe the strong force, and it seemed pointless to many to try to use it to achieve even greater goals. Subsequent, more detailed studies in the late 1970s and early 1980s. showed that string theory and quantum mechanics have their own, albeit smaller, contradictions. It seemed that the gravitational force was again able to resist the attempt to integrate it into a description of the universe at the microscopic level.
That was until 1984. In a landmark paper that summarized more than a decade of intensive research that had been largely ignored or rejected by most physicists, Green and Schwartz established that the minor inconsistency with quantum theory that plagued string theory could be allowed. Moreover, they showed that the resulting theory was broad enough to cover all four types of forces and all types of matter. Word of this result spread throughout the physics community, with hundreds of particle physicists stopping work on their projects to take part in an assault that seemed to be the final theoretical battle in a centuries-long assault on the deepest foundations of the universe.
Word of Green and Schwartz's success eventually reached even the first-year graduate students, and the previous gloom was replaced by an exciting sense of participation in a turning point in the history of physics. Many of us stayed up late into the night, poring over the hefty tomes of theoretical physics and abstract mathematics that are essential to understanding string theory.

If you believe scientists, then we ourselves and everything around us consists of an infinite number of such mysterious folded micro-objects.
Period from 1984 to 1986 now known as "the first revolution in superstring theory". During this period, more than a thousand papers on string theory were written by physicists around the world. These works conclusively demonstrated that the many properties of the standard model, discovered through decades of painstaking research, flow naturally from the magnificent system of string theory. As Michael Green noted, “The moment you are introduced to string theory and realize that almost all the major advances in physics of the last century have flowed—and flowed with such elegance—from such a simple starting point, clearly demonstrates the incredible power of this theory.”5 Moreover, for many of these properties, as we will see below, string theory provides a much more complete and satisfactory description than the standard model. These achievements convinced many physicists that string theory could deliver on its promises and become the ultimate unifying theory.

Two-dimensional projection of a three-dimensional Calabi-Yau manifold. This projection gives an idea of ​​how complex the extra dimensions are.

However, along this path, physicists working on string theory again and again ran into serious obstacles. In theoretical physics, we often have to deal with equations that are either too complex to understand or difficult to solve. Usually in such a situation, physicists do not give up and try to obtain an approximate solution to these equations. The situation in string theory is much more complicated. Even the derivation of the equations itself turned out to be so complex that so far only an approximate form of them has been obtained. Thus, physicists working in string theory find themselves in a situation where they have to look for approximate solutions to approximate equations. After several years of amazing progress made during the first superstring revolution, physicists were faced with the fact that the approximate equations used were unable to correctly answer a number of important questions, thereby hindering further development of research. Without concrete ideas for moving beyond these approximate methods, many physicists working in the field of string theory experienced a growing sense of frustration and returned to their previous research. For those who remained, the late 1980s and early 1990s. were a testing period.

The beauty and potential power of string theory beckoned to researchers like a golden treasure locked securely in a safe, visible only through a tiny peephole, but no one had the key that would unleash these dormant forces. The long period of “dryness” was interrupted from time to time by important discoveries, but it was clear to everyone that new methods were required that would go beyond the already known approximate solutions.

The stalemate ended with a breathtaking talk given by Edward Witten in 1995 at a string theory conference at the University of Southern California—a talk that stunned a room filled to capacity with the world's leading physicists. In it, he unveiled a plan for the next stage of research, thereby ushering in the “second revolution in superstring theory.” String theorists are now working energetically on new methods that promise to overcome the obstacles they encounter.

For the widespread popularization of TS, humanity should erect a monument to Columbia University professor Brian Greene. His 1999 book “The Elegant Universe. Superstrings, Hidden Dimensions, and the Quest for the Ultimate Theory” became a bestseller and won a Pulitzer Prize. The scientist’s work formed the basis of a popular science mini-series with the author himself as the host - a fragment of it can be seen at the end of the material (photo Amy Sussman/Columbia University).

clickable 1700 px

Now let's try to understand the essence of this theory at least a little.

Start over. The zero dimension is a point. She has no size. There is nowhere to move, no coordinates are needed to indicate the location in such a dimension.

Let's place a second one next to the first point and draw a line through them. Here's the first dimension. A one-dimensional object has a size - length, but no width or depth. Movement within one-dimensional space is very limited, because an obstacle that arises on the way cannot be avoided. To determine the location on this segment, you only need one coordinate.

Let's put a dot next to the segment. To fit both of these objects, we will need a two-dimensional space with length and width, that is, area, but without depth, that is, volume. The location of any point on this field is determined by two coordinates.

The third dimension arises when we add a third coordinate axis to this system. It is very easy for us, residents of the three-dimensional universe, to imagine this.

Let's try to imagine how the inhabitants of two-dimensional space see the world. For example, these two men:

Each of them will see their comrade like this:

And in this situation:

Our heroes will see each other like this:

It is the change of point of view that allows our heroes to judge each other as two-dimensional objects, and not one-dimensional segments.

Now let’s imagine that a certain volumetric object moves in the third dimension, which intersects this two-dimensional world. For an outside observer, this movement will be expressed in a change in two-dimensional projections of the object on the plane, like broccoli in an MRI machine:

But for an inhabitant of our Flatland such a picture is incomprehensible! He can't even imagine her. For him, each of the two-dimensional projections will be seen as a one-dimensional segment with a mysteriously variable length, appearing in an unpredictable place and also disappearing unpredictably. Attempts to calculate the length and place of origin of such objects using the laws of physics of two-dimensional space are doomed to failure.

We, inhabitants of the three-dimensional world, see everything as two-dimensional. Only moving an object in space allows us to feel its volume. We will see any multidimensional object as two-dimensional, but it will be amazingly change depending on our relationship with him or time.

From this point of view it is interesting to think, for example, about gravity. Everyone has probably seen pictures like this:

They usually depict how gravity bends space-time. It bends... where? Exactly not in any of the dimensions familiar to us. And what about quantum tunneling, that is, the ability of a particle to disappear in one place and appear in a completely different one, and behind an obstacle through which in our realities it could not penetrate without making a hole in it? What about black holes? What if all these and other mysteries of modern science are explained by the fact that the geometry of space is not at all the same as we are used to perceiving it?

The clock is ticking

Time adds another coordinate to our Universe. In order for a party to take place, you need to know not only in which bar it will take place, but also the exact time of this event.

Based on our perception, time is not so much a straight line as a ray. That is, it has a starting point, and movement is carried out only in one direction - from the past to the future. Moreover, only the present is real. Neither the past nor the future exists, just as breakfasts and dinners do not exist from the point of view of an office clerk during his lunch break.

But the theory of relativity does not agree with this. From her point of view, time is a full-fledged dimension. All events that have existed, exist and will exist are equally real, just like the sea beach is real, regardless of where exactly the dreams of the sound of the surf took us by surprise. Our perception is just something like a spotlight that illuminates a certain segment on a straight line of time. Humanity in its fourth dimension looks something like this:

But we see only a projection, a slice of this dimension at each individual moment in time. Yes, yes, like broccoli in an MRI machine.

Until now, all theories worked with a large number of spatial dimensions, and the temporal one was always the only one. But why does space allow multiple dimensions for space, but only one time? Until scientists can answer this question, the hypothesis of two or more time spaces will seem very attractive to all philosophers and science fiction writers. And physicists, too, so what? For example, American astrophysicist Itzhak Bars sees the root of all troubles with the Theory of Everything as the overlooked second time dimension. As a mental exercise, let's try to imagine a world with two times.

Each dimension exists separately. This is expressed in the fact that if we change the coordinates of an object in one dimension, the coordinates in others may remain unchanged. So, if you move along one time axis that intersects another at a right angle, then at the intersection point the time around will stop. In practice it will look something like this:

All Neo had to do was place his one-dimensional time axis perpendicular to the bullets' time axis. A mere trifle, you will agree. In reality, everything is much more complicated.

Exact time in a universe with two time dimensions will be determined by two values. Is it difficult to imagine a two-dimensional event? That is, one that is extended simultaneously along two time axes? It is likely that such a world would require specialists in mapping time, just as cartographers map the two-dimensional surface of the globe.

What else distinguishes two-dimensional space from one-dimensional space? The ability to bypass an obstacle, for example. This is completely beyond the boundaries of our minds. A resident of a one-dimensional world cannot imagine what it is like to turn a corner. And what is this - an angle in time? In addition, in two-dimensional space you can travel forward, backward, or even diagonally. I have no idea what it's like to pass through time diagonally. Not to mention the fact that time underlies many physical laws, and it is impossible to imagine how the physics of the Universe will change with the advent of another time dimension. But it’s so exciting to think about it!

Very large encyclopedia

Other dimensions have not yet been discovered and exist only in mathematical models. But you can try to imagine them like this.

As we found out earlier, we see a three-dimensional projection of the fourth (time) dimension of the Universe. In other words, every moment of the existence of our world is a point (similar to the zero dimension) in the period of time from the Big Bang to the End of the World.

Those of you who have read about time travel know what an important role the curvature of the space-time continuum plays in it. This is the fifth dimension - it is in it that four-dimensional space-time “bends” in order to bring two points on this line closer together. Without this, travel between these points would be too long, or even impossible. Roughly speaking, the fifth dimension is similar to the second - it moves the “one-dimensional” line of space-time into a “two-dimensional” plane with all that it implies in the form of the ability to turn a corner.

A little earlier, our particularly philosophically minded readers probably thought about the possibility of free will in conditions where the future already exists, but is not yet known. Science answers this question this way: probabilities. The future is not a stick, but a whole broom of possible scenarios. We will find out which one will come true when we get there.

Each of the probabilities exists in the form of a “one-dimensional” segment on the “plane” of the fifth dimension. What is the fastest way to jump from one segment to another? That's right - bend this plane like a sheet of paper. Where should I bend it? And again correctly - in the sixth dimension, which gives this entire complex structure “volume”. And, thus, makes it, like three-dimensional space, “finished”, a new point.

The seventh dimension is a new straight line, which consists of six-dimensional “points”. What is any other point on this line? The whole infinite set of options for the development of events in another universe, formed not as a result of the Big Bang, but under other conditions, and operating according to other laws. That is, the seventh dimension is beads from parallel worlds. The eighth dimension collects these “straight lines” into one “plane”. And the ninth can be compared to a book that contains all the “sheets” of the eighth dimension. This is the totality of all the histories of all universes with all the laws of physics and all the initial conditions. Period again.

Here we hit the limit. To imagine the tenth dimension, we need a straight line. And what other point could there be on this line if the ninth dimension already covers everything that can be imagined, and even that which is impossible to imagine? It turns out that the ninth dimension is not just another starting point, but the final one - for our imagination, at least.

String theory states that it is in the tenth dimension that strings vibrate—the basic particles that make up everything. If the tenth dimension contains all universes and all possibilities, then strings exist everywhere and all the time. I mean, every string exists both in our universe and in any other. At any time. Straightaway. Cool, huh?

Physicist, string theory specialist. Known for his work on mirror symmetry, related to the topology of the corresponding Calabi-Yau manifolds. Wide audience known as the author of popular science books. His Elegant Universe was nominated for a Pulitzer Prize.

In September 2013, to Moscow by invitation Polytechnic Museum Brian Greene arrived. A famous physicist, string theorist, and professor at Columbia University, he is known to the general public primarily as a popularizer of science and the author of the book “The Elegant Universe.” Lenta.ru spoke with Brian Greene about string theory and the recent difficulties that the theory has faced, as well as quantum gravity, the amplituhedron and social control.

Literature in Russian: Kaku M., Thompson J.T. “Beyond Einstein: Superstrings and the quest for the final theory” and what it was The original article is on the website InfoGlaz.rf Link to the article from which this copy was made -

Design by artist A. Balashova.

When the book Planiverse first appeared 16 years ago, it took quite a few readers by surprise. The line between a voluntary suspension of disbelief and simple-minded 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 disc-shaped planet embedded in the outer shell of a vast, balloon-shaped space called the Planiverse.

It is tempting to imagine that both gullible and distrustful readers did so because of the convincing logic and consistency of the cosmology and physics of this infinitely subtle universe with its bizarre, yet strangely efficient organisms. After all, what opened before them was not just an ordinary universe generated by a play of imagination. The planiverse is more than whimsical, fantastic place, since most of it was “made” by a virtual team of scientists and technologists. The reality - even the pseudo-reality of such a place is much stranger than it seems at first glance.

First, let's try to understand what the flat universe Planiverse is. Understand that two dimensions mean two dimensions. If the page of this book represents a small piece of the Planiverse, then the curved line drawn on it may turn out to be a piece of a planiverse 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 of this page. But give us some planiversal glue and we'll glue one tip to the other, trapping whatever ends up inside the lace loop once the glue dries.

The appendix to the book contains quite full story origin of the flat universe Planiverse. Since Martin Gardner's Math Games column in Scientific American published an article about the Planiverse, thousands (not even hundreds) of readers have sent in letters containing enthusiastic responses and new ideas. Professional scientists and engineers wrote, and even a few well-informed readers sent in reasonable suggestions.

We wove these ideas into something homogeneous and seamless, but we needed a plot - a story - to make it work interesting book. A story that would take us on a journey through Arda, a disc-shaped planet floating in the two-dimensional universe of the Planiverse.

From the preface to the end, the story is told with a serious, even impassive face. It was written by the pen of a scientist whose literary possibilities are constantly under the pressure of events. The story features a modern deus ex machina - a computer. It was with his help that a group of students made first contact with the two-dimensional universe of the Planiverse 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 accepted the fiction at face value. The subtext of this fantastic, albeit very rich in details, story went unnoticed by many. Neoteny trends have taken root in Western culture even before 1984. And of course, the fantastic allegory introduced into the narrative - that is, what makes the book, in the words of the Oxford humanist Graham Stewart, a “Sufi parable”, went completely unnoticed by these readers. The temptation to bring to life a higher (third) dimension as a symbol of the forces lurking beyond the obvious reality of our world turned out to be too great to overcome. The story opens with an old preface waiting for you on the next page.

A. K. Dewdney.

January 2000

I would like to note that I am not so much the author of the book as its compiler, and the main credit for the fact that this book saw the light belongs to the creature depicted on the first page. His name is Yendred, and he lives in a two-dimensional universe that I call the Planiverse. The story of the discovery of the Planiverse, a world whose reality few could believe, will probably seem interesting to you. That's what I want to tell you.

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

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

It has some mass (depending on the type and number of atoms it contains) and can move in a two-dimensional space such as this page. But, unlike a page, two-dimensional space has no thickness, and the disk cannot extend beyond its boundaries. Let's assume 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 begin to move at a constant speed in a plane that is an extension of the page. Sooner or later, 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 that operates in our three-dimensional universe, the sum of the kinetic and potential energies of the two disks before and after the collision remains unchanged. Moving in this way, the disks 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 represented 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 consist of individual atoms, this will complicate the programmer's work and increase the load on the processor during program execution. But almost any programmer can write such a program and display the results on the screen.

This is roughly where work on the 2DWORLD program began. In the first semester, students under my guidance not only described a certain set of objects and the law of conservation of energy in the program, but also created an entire system of planets revolving around a star. One of the planets, which they named Astria, gained particular popularity among students. Towards the end of the first semester, conversations began about drawing a map on this planet and populating it with living creatures - 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 impossible to implement the idea - my students were not that strong programmers.

In any case, 2DWORLD turned out to be a very useful program, and it was incredibly interesting to work with it. I especially remember the process of 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.

What is the Laboratory of Nanoptics and Plasmonics known for? If we try to describe its activities in one sentence, then behind nanooptics and plasmonics there are biosensors, nanolasers, single-photon sources, metasurfaces and even two-dimensional materials. The laboratory cooperates with universities and research centers many countries and continents. Among Russian partners we can highlight groups from Moscow State University, Skoltech and ITMO University. The laboratory plans not only Scientific research and development, but also their commercialization, as well as the organization of the first large-scale conference in Russia on two-dimensional materials.

The head of the laboratory is Valentin Volkov, visiting professor from the University of Southern Denmark in Aalborg. The laboratory was organized in 2008 on the initiative of Anatoly Gladun and Vladimir Leiman, professors of the MIPT Department of General Physics, and Phystech graduates Sergei Bozhevolny and Alexander Tishchenko had a great influence on its formation. She is now part of the Center for Photonics and Two-Dimensional Materials at the Physics and Technology School of Fundamental and Applied Physics.

« We take approaches that have worked well in practice in some areas of research and transfer them to new areas of research. For example, we took copper, which has proven itself well in electronics, combined it with two-dimensional materials and dielectrics, and it turned out that with its help in nanooptics you can do everything that was done before, but much better and cheaper", - argues Valentin Volkov.

Head of the laboratory Valentin Volkov

The laboratory deals with both theory and experiment. It has the most modern equipment for near-field research - aperture and non-aperture near-field optical microscopes. They make it possible to study the distribution of electromagnetic fields along the surfaces of micro- and nano-sized samples at distances much smaller than the wavelength of light, with a spatial resolution of up to 10 nm. A range of instruments ranging from spectral ellipsometry to Raman spectroscopy is used to analyze materials and samples. Experimental studies are accompanied by theoretical studies and numerical simulations. Objects for research are also manufactured directly in the laboratory and the Collective Use Center of MIPT.

Much attention in the laboratory is paid to the use of nanomaterials in optics. It all started with graphene and carbon nanotubes (together with colleagues from Japan and the USA), and now they are working with transition metal dichalcogenides, tellurene and germanium-based compounds. Just this year, scientists launched a facility for CVD synthesis of two-dimensional materials. The laboratory categorically disagrees with the common statement in Russia that two-dimensional materials are just a fashion, and considers them as a key building material for nanophotonics, and also agrees with the words of Andrei Geim that the next 50 years will not be enough to study them. According to Fabio Pulizzi, editor-in-chief of Nature Nanotechnology, who recently visited the laboratory, 30% of the publications in his journal are work related to one degree or another with two-dimensional materials. The competition here is very high, but this is what is needed at Phystech.

Biosensors and graphene

One of important areas laboratories - highly sensitive biosensors for pharmacology and medical diagnostics. It is directly related to plasmonics - we're talking about about plasmonic biosensors - but this is where biology comes into play. This type of work requires other qualifications.

« My colleagues specifically studied biology and chemistry in order to begin this difficult task with a new background. Biology and chemistry integrate well with our interest in the practical use of two-dimensional materials"- says Valentin Volkov.

A recent achievement of the laboratory is the creation of graphene biosensor chips for commercial biosensors based on surface plasmon resonance. The developed chips demonstrate significantly higher sensitivity compared to sensor chips currently on the market. Increased sensitivity is achieved by replacing standard connecting layers with graphene (or graphene oxide), characterized by a record surface area. An additional advantage of the development is the use of copper as a plasmonic metal instead of gold, which is standard for such chips, which has significantly reduced their cost, primarily due to the compatibility of copper with standard technological processes.

Single-photon sources and nanolasers

The laboratory is also conducting research on the creation of truly electrically pumped single-photon light sources - devices that emit single photons when transmitted electric current. The transition to such single-photon technologies will not only make it possible to increase the energy efficiency of existing information processing and transmission devices by more than a thousand times, but will also open the way to the creation of various quantum devices. Another related task in this area is the creation of coherent sources of optical radiation operating at room temperature from miniature power sources, the dimensions of which are only hundreds of nanometers. Such compact devices are in demand in optogenetics, medicine and electronics.

Conference in Sochi, robots in Denmark

This year, Valentin Volkov will organize a session on two-dimensional materials at the Third International Conference “Metamaterials and Nanophotonics” (METANANO-2018). The conference will be attended by scientists who are leaders in their fields, and will be opened by FOPF graduate (1982) and Nobel laureate Andrei Geim. The laboratory staff also has a more ambitious goal - holding an annual large-scale conference on two-dimensional materials in Russia.

This summer, laboratory students will go on an internship at the Danish company Newtec, with which the laboratory has been collaborating for several years. The company is not directly related to science - it develops and produces high-tech robotic complexes for sorting vegetables and fruits - however, it has a very powerful research department, including a complex of laboratories for the study of two-dimensional materials. This company uses graphene to create hyperspectral cameras for high-speed diagnostics of sorted vegetables and fruits. Joint research with the Danes not only helps the laboratory master new technologies and approaches to working with two-dimensional materials, but also allows us to look at the world of research and development from a completely different angle. This cannot be learned at university.

Icons came to Rus' from Byzantium, and here they truly found a second life. The fact is that by the XI-XII centuries. Byzantine icon painting begins to rely more and more on the canon, as the Greeks lose their contemplation of the spiritual. In Rus' at that time there lived people who highly preserved 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 Rus', what was imprinted in them according to tradition was illuminated for the Russians by the true light of the imagination behind the iconographic images. It is not difficult to verify this by comparing at least once early Russian icons, in which Byzantine influence is strongly felt (12, 13, 16), with an icon on which only the Byzantine canon is reproduced (160).

There was a deep meaning associated with this circumstance. Looking at the subsequent development of Byzantium - its decline, decline, after which it fell into pieces and was absorbed by the Ismailis, you get the impression that to a large extent the meaning of its existence was to transfer to Rus' the impulse of Christianity in its eastern form, which we read about above in R. Steiner. In any case, icon painting, being the fruit of the Greco-Latin culture of the Christian era, 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 imaginations, in the center of which stands Christ Himself, carry within themselves a special individualizing force, and it is extremely important that from the very beginning it was the divine “sculptor” who “carved” Russian individuality out of the block of the group aura of the Eastern Slavs. This is the uniqueness of the development of the Russian world. He appeared as a kind of tabula rasa for Christianity.

The Slavs were the last among European peoples to enter into the cultural-historical process, and from the very beginning it was Christian for them. The Greco-Latin world received the individualizing principle into souls prepared for this by a long period of pagan culture. It is not for nothing that Plato and Aristotle are called Christians of pre-Christian times. The medieval world absorbed a lot from the Druid Mysteries, from Rome, and, in addition, initially carried within 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. The ancient Slav lived in close fusion with nature, so that the processes taking 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 the elemental spirits of water, forest, and 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 Trots could have been preserved, which probably determined the peculiar development of Novgorod Rus'. In the rest of the territory ancient Rus' clan-family communities lived, carrying within themselves 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 help but produce in it a completely special force. In the atmosphere of worship, in the temple, truly new heavens opened up for the Slav. The walls of the temple were transparent to him, the revelations imprinted on them stood out in their true vitality. The form of revelations, conveyed by the line, color of an icon or fresco, served as a gateway to the spiritual world; it focused 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 purposiveness and organization, and as such it creates in the world of sensory things in its own image and likeness.

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

This perspective has a very special effect on the viewer. Back at the beginning of the 20th century. it was noticed that when the image is given in reverse perspective, the viewer seems to accept its 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 summarizes in a single visual impression and transfers to the image. Direct perspective leaves the viewer passive and suggests even greater flatness visual perception, than the opposite, 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 an object, but also its side faces. “In the process of visual merging of individual aspects of an object into one holistic image,” writes Zhegin, “the side edges 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 obviously motionless objects - 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 thing we cannot agree with the author on is that reverse perspective is a conscious technique of the ancient painter. No, it arises as a natural result of the transfer of supersensible contemplation to the sensory flat surface. In imagination, a person merges with a supersensible object, therefore he sees it, or rather, sees it and experiences it from all sides.* And the icon leads the viewer to the same experience. It forces one to enter an 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 into one.**

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

** This phenomenon threatens to “scald” the soul of a positivist if he experiences the icon really deeply

Understanding this phenomenon is not entirely easy. Therefore, we turn for help to the developments that have been made by anthroposophists in the field of projective geometry. It was in it that the most interesting results were achieved, allowing one to form an idea of ​​the essence of the transition from the three-dimensionality of the sensory world to the two- and even one-dimensionality of the supersensible worlds. Therefore, it is advisable to take its construction not only mathematically, but also psychologically and meditatively. It is important to experience the process of construction, transition from one form to another, then the corresponding idea will be born.

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

So, we should try to experience these examples from projective geometry, and then we will discover that in them, despite their simplicity, we almost “tangibly” come into contact with the supersensible. For a person turned to the world of imagination, 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, for it is determined by the subjective perception of the one who contemplates the Perceiver of imagination, if he is placed in the conditions of our example, is located 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 a person’s sensory experience, which went through his knowing spirit (like a shadow) into the supersensible (into infinity) and returned from there, being illuminated by the light of other worlds. At the moment of supersensible perception, a person can (and even should) forget about his sensory experience, but after that there is always the opportunity to be convinced that what is contemplated in imagination is closely connected with the physical world - not as a reflection (this is the area of ​​abstract thinking), but as prototypes with images, as an essence with a phenomenon.

In the experience of a supersensory cognizer, self-perception, coming from the sensory to the supersensible, summarizes the object at the point of the subjective “I”, comprehensively covering 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 what happens among the Eastern occultists. European development took a different path. Here the human soul, even before it began to reflect on the sensory world, had already acquired a tendency to individual perception supersensible. But for this, the supersensible itself became a kind of reflection of the sensory, not only in that deep sense when we talk about the laws of nature, but also in its manifestation to the individual human soul. In our example, we can talk about a certain sensual-secret phenomenon of the nature of light, when the candle stands below the top of the triangle. Then its light is, as it were, 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. Now take a straight line and a circle (Fig. 2). If we draw two tangents to the circle from different points of the straight line G, and connect the tangent points by straight lines, then they will all intersect at point A. Let us begin to bring the straight line E closer to the circle. When the straight line touches it, point A will coincide with the point of contact. As line b moves to infinity, the tangents will become parallel and point A will coincide with the center of the circle. The tangents in this case, since we are talking about infinity, can naturally 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 construct 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 (i.e. A). As the plane moves to infinity, the cones will turn into cylinders, and point A will coincide with the center of the sphere. Moreover, it is also obvious that a plane located 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 tangency with the sphere, then it will be in a projective connection with the line of intersection of the planes CO. Similar to what we observed in the two previous examples, with the removal of straight line CO to infinity, straight 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 its contact with the 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 edge of the described pyramid due to the relationship of straight line ab with CD, da with BC, etc. (according to the law discussed in Fig. 3). With the removal of the edges of the described pyramid to infinity, the inscribed pyramid will shrink to a point in the center of the sphere. A pyramid with edges extending to infinity can be thought of as a plane.

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

In these examples of projective geometry we can feel why three-dimensionality loses its obligatory character with the expansion of physical law to infinity. At the same time, physical objects, one might say, are even mysteriously determined by what comes to them from “flat” and even “point” infinity. This is easy to feel if the given examples are not fixed purely rationally, but let them pulsate through geometric construction from visible forms to the correspondences that determine them, stretching into infinity.* Then we will be able to feel why the world of imagination is two-dimensional. It contains hidden prototypes that determine the phenomenal world. Using our examples, we showed how they come to the phenomenon.

* In the end, the whole Earth has the shape of a sphere, and the vault of heaven surrounding it, geometrically conceived, is a plane, since it is infinitely distant from it.

The discovery of direct perspective was undoubtedly a necessary step on the path to understanding sensory reality, where the individual “I” found its support. But in this three-dimensional space there is no place for supersensible objects. When we, for example, consider the painting “The Annunciation” (10), attributed to the brush of Leonardo ,* then, despite artistic skill, deep knowledge of perspective, the laws of mechanics (regarding the Angel’s wings) that its author possessed, we still look at it with some bewilderment. For it is clear to us that this material angel will still not fly, no matter how correctly from the point of view of mechanics his wings are drawn, and if he does fly, then where will he go in this three-dimensionality, where Angels do not live? At the same time, it is not difficult for us to accept the supersensual nature of iconographic images and, in general, everything written with a deviation from 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, right down to the depiction of faces with “strange” proportions, if you look at it from the point of view of the sensory world (11). In fact, the faces of the saints seem to shine towards us from the spiritual world, spreading backward into infinity, and moving forward towards the center of our perception. It should be noted that images of Divine beings are less subject to the laws of reverse perspective. This probably reflects their active organizing power, which not only reveals itself, but brings the consciousness of the person seeing it into the sensory world. These images are often of an abstract cosmic nature, clothed in figurative form. Let's take for example icon XII V. "Our Lady of Oranta" (159). If we penetrate into the meaning of its symbolism, where “symbolic signs are associated with what can follow one natural... one is included here in the other” (see note 72), then we can say that in the icon in the image of the Mother of God the whole The earth with its spiritual aura. In this aura the Divine Child is revealed and born. It is born as the spiritual Sun in the etheric world of the Earth, enters 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 upward, 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 further emphasized a triangle formed by the head of the Mother of God and her hands. The gesture of the baby Jesus speaks of Christ’s readiness to embrace the Earth and unite with it. At the same time, on the icon he appears as a hypostasis of the Trinity - this is indicated 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 etheric forces.

In a certain contrast - in mood - to the icon of "Our Lady of Oranta" stands the icon of the "Annunciation" (12). 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 closure. Both moods, when taken together, form the primordial phenomenon of polarity, within the boundaries of which the individual beginning of a person matures. In the Annunciation, the cosmic impulse approaches the Earth in the form of a messenger - Archangel Gabriel. Mother Earth, personified in the image of the Mother of God, in some kind of cosmic introspection experiences the news of the coming appearance of the Son of God and, at the same time, already contains it within 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 an icon of the 12th century. (13). However, his gaze is humanly individual, only turned inward. There, inside the human spirit, the Regent of the cosmic intelligentsia, Michael, is revealed, whose earthly reflection is St. Georgy. 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 indicates the human self, but not yet active in the earthly sphere. The realization of its real power will come later, in the 14th-15th centuries, and this will be reflected in the images of the Archangel Michael, where He will draw the sword from its scabbard and show them the path upward to human consciousness (14). St. George in the icon is still only a recipient of Michael’s impulse on Earth, but he thinks meditatively, which, in addition to his gaze, is indicated by the 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 to accept on Earth the fullness of thinking about both the Earth and the spirit, to accept Sophia, or the cosmic intelligentsia.

A completely different mood is evoked by the icon, which is called “Archangel of Golden Hair” (15). This Archangel, like St. George, both cosmic and individually human, but in a different way. His gaze is soft, he expresses mercy - that new quality of the human soul that comes into the world with Christianity. The Archangel represents this property macrocosmically. He is all mercy. In his view there is no human limitation, in it is the breadth of the Divine Being, endowed with enormous good power. Therefore, his mercy is full of power. This is characteristic of a truly powerful being; This is how a person should be. A headband with a ruby ​​is woven into the Archangel's hair. This stone is related to the center of intuition in the human brain. For intuition you need high degree organization of thinking, the ability for moral imagination, which is why the Archangel has a different hair pattern, not the same as St. George; and therefore the hair is pulled together by the bandage. In the mood that radiates from this Archangel, one can feel something inherent in the Archangel of the Russian people, at the same time “young and ancient” (R. Steiner), whose inspirations are expressed in the best properties of the Russian soul.

The images of Christ Himself on ancient Russian icons are sublime, but also highly individualized. Here is the "Savior Not Made by Hands" of the 12th century. (16). The parted hair is distinctly differentiated with the help of gilded lines, the so-called assist. This assist is applied only to images of spiritual beings; with its shining color and fractional structure, it seems to reveal the Divine-human nature of the spiritual impulse coming from them. The assisting technique came to Russia from Byzantium, and there from Rome. The beard of the iconographic 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 a conductor of the spirit. The gaze of the depicted face is directed not at the viewer, but to the side, at surrounding a person a world where we, humans, are thus invited to direct our attention. The gaze is focused and stern: “I did not come to bring peace, but a sword” (Matthew 10:34), that is, an impulse of self-consciousness that will destroy blood ties and call for spiritual brotherhood. Therefore, the face of Christ on the icon calls for concentration of mind, for responsibility for what is happening in the earthly world. It is not for nothing that 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 late 13th century. (17). Here His gaze expresses something completely different, what is said in the New Testament: “God is love.” This look is directed at us, at our inner being, where a new, Christian morality is born.

Such images, in addition to the fact that they reveal the nature of God addressed to humanity, are called upon to forge separate but fundamental qualities in souls. One only needs to imagine the world of ancient Rus', where human faces reflected a completely non-individualized consciousness, in order to understand what kind of impression these iconographic faces evoked. It was impossible to forget meeting them.

Something special should be said about 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 youthful face and no beard. From the messages of R. Steiner we know that Christ was often depicted with a beard because in ancient figurative clairvoyance the image of God the Father always seemed to shine through the face of the Son of God. And another, more high strength clairvoyance so that the young face of Christ is revealed. Looking at this icon, we can tell ourselves that there were high initiates in Rus', 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, and is indicated by the rubies in the headbands of the Archangels. It was also revealed to the creators of the pristine image of this icon that they were contemplating Christ as He appears in His Second Coming. One day many will see Him in a similar form. This time has already arrived 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 the two Jesus boys. One of them, Nathan's Baby, is born in a Bethlehem manger, which was built in a cave. Shepherds come to worship Him, as described in Ev. from Luke. But the icon also depicts a star that led the Magi, the kings from the east, to another Baby; this is spoken of in Ev. from Matthew. This Child, from Solomon's royal line, is pictured below on the right. He is washed in a golden font by two maids. Another significant iconographic theme dedicated to the earthly life of Christ Jesus is the “Transfiguration” (20). The icons on this topic reflected the knowledge of 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 in the form of a hexagram, the star of Solomon. Christ in the icon forms the entire theme and embraces it with Himself. At the same time, He Himself, as the Life-Spirit, together with Moses and Elijah, dwells 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 body, soul and spirit (as “I”).* The figure, by which the body is meant, on some icons remains in an unconscious state, on others it struggles to hide from the sudden outbreak of contemplation . The one who expresses the spirit tries to look and see what is happening above. The figure is average and the condition is average between the extremes. In the aura of Christ, in some cases three circles are given, in others - a hexagram and circles.

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

Among the various iconographic images of Christ, “Savior in Powers” ​​(21) evokes particular amazement for its esoteric depth. The icon appeared around the beginning of the 15th century. and is included as a central theme in the Deesis rite. She represents Christ seated on the throne, with the open Gospel in his hands. His feet rest on the earth's firmament, depicted in the form of a quadrangular stand; this firmament is carried by Thrones (Thrones), which on the icon look like wheels with wings. Christ is surrounded by a huge aura in which the entire first Hierarchy is revealed. The aura of Christ himself, in turn, is included in a quadrangular aura of red color, at the corners of which an Angel, an eagle, a calf and a lion are depicted. This is a person’s aura, his supersensible image, and it is in it that everything depicted in the icon is revealed.

On some icons, especially from the Rublev school (22), inside the aura of Christ there is another, also red, quadrangular aura. This aura expresses the incarnation of Christ, His Resurrection in human body as a Spiritual Man. Then the throne on which He sits and which expresses physical corporeality is given a white stroke - this is no longer just corporeality, but a phantom, or, say, the throne of the Spiritual Man. The firmament under the feet of Christ and in 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, thus, reflects the inner, mystical experience of Christ, when He is revealed in the inner man as the Hypostasis of the Divine Trinity, possessing the fullness of the powers of all nine Hierarchies. This is undoubtedly the knowledge of Christ in intuition, acquired by initiates in Rus'. It constitutes the focus of esoteric Christianity; the hopes of people and Gods converge towards it. Therefore, this icon is placed in the center of the Deesis rank, and on both sides it is surrounded by other icons, on which - on the left side, Apostle Peter, Archangel Michael and the Mother of God bow to Christ in prayer positions, and on the right - Apostle Paul, Archangel Gabriel and John the Baptist . This arrangement of figures means that both groups form not two poles on the sides of Christ, but a circle within which the person praying is located. The figures on the left and right, taken in pairs, represent two sides of any one aspect of the appearance of Christ: Our Lady and John, Michael and Gabriel, Paul and Peter.

The Deesis rite is truly the pinnacle of Russian Christian esotericism. It is addressed to the soul as common man, and a student of the Christian Mysteries. The Church, over time, lost its understanding of it, therefore, already in the 15th century, the Deesis order was downgraded 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 those present. .

The image of the Mother of God is especially revered in Rus'. The attitude of believers towards Her is as, perhaps, the closest and most truly effective spiritual force. She is an intercessor, 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 qualities 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 the people, Our Lady is the greatest in Christian love.

The cult of the Mother of God came to Rus' in its cosmic aspect of Sophia, Divine Wisdom. In the first centuries of Christianity, churches of St. Sophia were built throughout Rus'. The most significant among them are in Kyiv and Novgorod. Reflections of this cosmic cult are such icons as “Our Lady of Oranta” (159). Its mosaic analogue is given in the apse of St. Sophia Cathedral in Kyiv. But there are also icons depicting St. Sophia itself, in connection with which we can talk about the continuity of the ancient Mysteries in the era of Christianity (23). Unfortunately, very few such images have survived. And the cult of St. Sophia itself in later centuries somehow strangely faded and faded into the background. But in Russian religious philosophy in late XIX- early 20th century Sofia's theme began to sound with renewed vigor. This 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 the angels? The image is called the image of Sophia, the Wisdom of God. But what does this mean? Back in XIV century one Russian boyar asked this question to the Novgorod archbishop, but did not receive an answer - he responded with ignorance. Meanwhile, our ancestors worshiped this mysterious person, as the Athenians once did to the “unknown god"... The icon of Novgorod Sophia itself is of no Greek model does not have - this is a matter of our own religious creativity...

This Great, royal and feminine Being, who, being neither God, nor the eternal Son of God, nor an angel, nor a holy man, receives veneration from the consummator Old Testament and from the ancestor of the New - who is it, if not the true, pure and complete humanity itself, the highest and all-embracing form and living soul of nature and the Universe, eternally united and in a temporary process uniting with the Divine and uniting with Him everything that is.” 78

Images of the Mother of God and Child received predominant importance in Rus'. The first among them is “Our Lady of Vladimir” (24). The icon was brought from Byzantium and dates back to the 11th century. From the original painting, only the faces have been preserved; the rest was painted anew in the 15th-16th centuries. It has a rich history and many miracles are associated with it. By value in religious life In Russia, the “Our Lady of Vladimir” can be placed on a par with the “Our Lady of Czestochowa” in Poland and the “Our Lady of Ostrobram” in Lithuania. And although the icon of “Our Lady of Vladimir” now stands in the museum, for believers it remains a shrine, and not a painting or a museum exhibit.

The Assumption of the Mother of God is a major holiday in the Orthodox cult. Among the numerous icons on this topic, we present the most ancient of those currently preserved (25). The icon reflects the earthly and cosmic understanding of the Mother of God. The bed on which Her body rests, although distantly, resembles an Egyptian sarcophagus. The Apostles and Church Fathers stand around the bed. They are representatives earthly world. Against a golden background, as if from the shining distances of the macrocosm, the highest aspects of the spirit of the Apostles, so to speak, 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 risen Christ receives the Soul of the Mother of God and transfers it to the Hierarchies. Through them, as if along a kind of spiritual “ladder,” She ascends to the Higher Devachan. In other versions of this icon, its meaning is esoterically even more profound. In these variants, the figure of Christ is surrounded by an aura in which all 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 who descended on the Apostles on the day of Pentecost in the form of fiery tongues and union with which is predetermined for other people. And just as the Soul of the Mother of God is received by Christ, so - people can hope - the soul of every person will one day be received 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 big Orthodox holiday dedicated to the Mother of God is the Intercession. 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 Our Lady of Heaven unites people and heaven together. Since the icon appeared quite 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 a mediator between people and God (26). However, such a view was not the only one; many Russians saw in the church an image of conciliar humanity.

Besides spiritual beings and biblical images We often see on ancient icons the images of Christian ascetics, church fathers, but never secular people. For a trivial understanding of religion, this is an inexplicable fact. In fact, 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, chroniclers were monks, and icon painters were monks. However, for the latter, it was as if princes did not exist at all, unless one of them committed 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, in the world of his contemplation. On the contrary, chroniclers in Rus' often became those people who, earlier than others, began to experience self-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 to the spirit, by what was revealed in contemplation as an individual aura. Naturally, high spiritual Beings stood in the foreground here; after them, the inner gaze distinguished the individualities of the initiates, thanks to their overshadowing by the Spirit Self. The common people, as well as the 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 a religious sense. The icon, even if not included in its esoteric content, should serve as evidence of the predominance of spirit over matter (11). “An icon,” wrote Prince Evgeny Trubetskoy, “is not a portrait, but a prototype of the future temple humanity... an icon can only serve as a symbolic image of it. What does thinned physicality mean in this image? This is a sharply expressed denial of the very biologism that erects the saturation of the flesh into the highest and unconditional commandment." 79 When will late XVI V. The spiritual ideal of the icon began to be lowered, then Archpriest Avvakum reproached the icon painters: “... they have changed the likeness of them (the saints), they are painting the same as themselves.”

So, the saints on the icons are people of rare destiny and greatest power spirit, which allowed God-likeness to manifest itself in them. Therefore, for others, for those who pray, they are role models. Their heads, like those of spiritual beings, are surrounded by a shining aura (halo). Although the faces are endowed individual traits, but their flesh is spiritualized, the spirit shines through it, the movement of spiritual currents occurring in the “lotuses”, in the etheric and astral bodies. This was depicted with the help of so-called “engines” applied to the face with whitewash, and dark “marks”.*

* From the point of view of ordinary portraiture, it is completely impossible to understand these techniques.

Nikola Ugodnik enjoys special veneration among the saints in Rus'. His image on an icon of the 12th century. characterized by a deep combination of the individual and the spiritual (27). Individuality 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 also the entire elaboration of the face with “engines” suggests that “the saint has an individual aura, he and how earthly man there is individuality." Among other saints in Rus', Boris and Gleb are especially famous. We will talk about the wonderful 14th century icon on which these saints are depicted in the next essay.

Following Nikola, Boris and Gleb, among the early Christian saints in Rus', Blasius, Florus, Laurus, and Paraskeva Pyatnitsa are honored (28). A special, very remarkable icon is dedicated to Flora and Laurus, which is called “The Miracle of Flora and Laurus” (29). It depicts Archangel Michael giving Flora and Laurus saddled horses: one black, the other white. In the lower part of the icon, three horsemen are chasing (grazing?) a herd of horses. On some icons, two of them are enthusiastically talking to each other, and the third follows them. In Orthodoxy, Florus 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 Flora and Laurus.* But from the messages of Spiritual Science, we know which “horse breeding” is the patron saint of Michael. He is the Regent of the cosmic intelligence that descends into human thinking. The latter, moving according to the laws of logic, lives in the collision of two principles: positive and negative, thesis and antithesis. 80

* Some museum guides directly announce this.

The horse is a symbol of human thinking, but the icon painter, of course, depicts not a symbol, but the figurative vision that opens when contemplating the world of thinking with the help of atavistic clairvoyance. In this regard, we can recall one painting from the circle of Mannerist artists, many of whom had glimpses of atavistic clairvoyance. This painting is by Nicolo del Abbate and is called “The Blinding of St. Paul” (30). It depicts the ap. Paul before Damascus when he saw the risen Christ. From an esoteric point of view, Paul's experience signifies the 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 - that is why the horse is white. On the other hand, he was not ready 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 broke, his physical shell fell to the ground, while his spirit soared into the heights.

As self-consciousness develops in Russian souls, Russian saints and clergy begin to appear on icons.** What distinguishes them from the saints of the Byzantine period of Christianity is a greater inward focus, mystical depth, the predominance of the spiritual over the spiritual (161).

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

Icons of saints are usually surrounded by marks depicting scenes from their lives. A saint is a person who has approached God. That's why his life story is important. She is a model for others, a life “ladder” leading to God. Everyone must step on it eventually. But if the soul wants to go faster than many others, then its path, full of labor and dangers, is expressed by another 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 waist-deep (13), on others - at full height, often with the marks of life, in the fight with the serpent. This plot reflects two big problems of the spiritual formation of humanity: the fight against hereditary sin, the temptation of man by the paradise serpent, and the fight against that Ahrimanic dragon, which Michael in the 19th century. cast down from heaven to Earth and which now nests in human souls. The second of these problems appears on the icons in a kind of insight, a prediction of 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 work of Michael on Earth. Therefore St. George is the true prototype of the human “I”, following the paths of Christian evolution.

Images of St. We find George among many nations. They all express different aspects of man’s struggle with the Luciferic-Ahrimanic dragon, and often one-sided aspects. Such images of St. George should be perceived not as a prototype, but as a warning, similar to what is given on the “Heavenly Staircase” icon. Let's look at a few examples. Let's take Raphael's painting "Saint George" (32). It shows the struggle of the human “I” with the Ahrimanic dragon in the conditions modern civilization: George is clad in iron knightly armor, and knighthood, as we know, is an expression material culture(this is her color. - R. Steiner). The spear of thought broke on the dragon, and its clawed paw is already scratching the horse’s belly - the thinking of the era of the conscious soul. And God knows how it will all end! - The raised sword - the human “I” - must travel a long way before it falls on the head of the enraged dragon, who meanwhile does not sleep. Human soul- the female image in the picture - runs in horror into the stone, lifeless desert of civilization.

In another painting from the 15th century. the human self struggles with 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 immaculate.

On a Russian icon from the early 14th century. In general, the entire outcome of the fight against the dragon is decided by the soul (34). The 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 above the struggle of the soul with the dragon, without even suspecting it.* On another 15th century icon. The luciferic nature of the dragon is emphasized: he is depicted in a backward movement (35). White horse gallops forward, and the fight against the dragon is waged by one Self under the sign of the Sun. - It is Christ who fights the dragon in a person who has not yet become an individual.

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

The battle of St. was given perfect. George with a dragon on an icon of the 16th century. (36), where everything is brought into the correct ratio for our time. The rider sits on a white horse, calmly and confidently moving forward, that is, the human “I” has achieved pure thinking and moves in complete balance, being in its essence nothing more than will. The soul (the female image at the gate) also underwent 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 support on the material and spiritual (hand gesture). The struggle takes place in full view of the city, that is, in the conditions of modern civilization; and even though the participation of this civilization is passive, it is also good that it realizes the dependence of its fate on the outcome of this struggle and is ready to follow the gesture that the soul of the spiritual hero makes (this gesture is repeated by the king on the city wall).