The two-dimensional world of Christian imagination. 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 a 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 ours. 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. That's why local residents they live in houses that resemble our dugouts, and the 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 a positive side to this: since it is impossible to tie a knot in the 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 at reverse side- 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 has metallurgy, steam engines, clockwork, musical instruments, rockets, space stations, astronomy, chemistry, cell biology, electricity, books, art and computers. Every scientific field, every mechanism is explained in this way - not by simply copying the things of our world, but by explaining the principles of operation and the inherent limitations. For example, it explains how cells manage to exchange nutrients without 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. That's where he's headed main character. 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 genre science fiction. 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 are about the creation of a virtual world, which, unexpectedly for the creators themselves, exhibits properties that were not intended, beginning to live their own lives.
By the way, much later, when I became interested musical genre Synth Pop, I found albums by 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 was really developing 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 I collected articles on the structure of various things in a 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 most best book of all the science fiction books I've 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.
Let's consider any World. For example, this one. In relation to this World, there is also a Lower World (for example, a two-dimensional
), and a higher World (for example, four-dimensional).
Two-dimensional World.
Two Dimensions...
If three-dimensional reality is considered aggressive, then two-dimensional reality is fear and horror. Those. Consciousnesses residing in the Two-Dimensional World are there in a state of fear and horror. As a result of passing natural development in this dimension, Consciousness gains the experience of Duality and, rising above its own fear, becomes convinced that it is nothing more than suspiciousness, and life is constant work. Also on life experience makes sure that together, at least, it’s more fun. 
Subsequently, from the Consciousnesses that have successfully undergone development, “Grains of Mind” or Souls for the Third World are formed in the Two-Dimensional World Dimensions. IN new form these Consciousnesses are either elements of the “Grain”, or the whole “Grain”, or something else - it depends on Forces of Individual Vibration and other factors.
All this is necessary for Consciousness to be able to withstand the “emotional” scale or scale of sensations, which is so much greater than the similar scale of the previous World that a whole additional dimension had to be created here.
Three Dimensional World.
Dimensions of Three...
We qualitatively Realize the Trinity, with the Eternally indefinite many-sided manifestation of the Third. We reach body mutations up to the fourth level body (minimum). And we move to a World of a higher order.
Four-Dimensional World.
We achieve awareness at the level of a psychological basis that in addition to dad, mom, and child, there are also family units just like yours, and that there are actually an infinite number of them. And every child born into someone’s family is another direct child of yours. Well, the rest will be learned there by those who are destined to live there. And this is neither more nor less - all “Grains”.
And of course, the level of scale of sensations is so high that any microbial brain that gets there prematurely will simply turn into dust. About being there in a state “Black Hole” he need not even dream. So for unprepared types, this is also fraught, not just with loss of health, but with disappearance from the path of development altogether.
Opportunities for interaction.
If we allow the possibility that a Being of a lower World will break through into a higher one, there it can do almost nothing. For example, move the flowerpot, or create an emotional vibration. Creatures of the two-dimensional world in in good condition are able to interact in their own way with three-dimensional Beings. Creatures of the four-dimensional World are also capable of interacting with Creatures of the three-dimensional World, and are for them by the gods, and also have a parental origin relative to Beings of a lower dimension. Even the usual direct appearance of a Being of a four-dimensional World in a three-dimensional one can cause serious damage to the entire World, therefore in such situations you need to act carefully. People today do not see either one or the other, for the simple reason that after brutal rape they have lost almost all their natural abilities.
Many People Those who have even normal sensitivity, not to mention increased sensitivity, somehow hear some voices. This can happen in different ways: as a breakthrough into your thoughts, or an internal dialogue that everyone Human leads constantly, or simply clear and clear voice from outside.
In such situations, one thing is clear for sure, that some Creature is coming into contact with you. It remains to determine who it is and how to behave.
It's simple. The Lower Worlds press on the “bottom”, the upper Worlds appear "above". What do we have “from below”? Of course, emotions! What do we have "above"? Of course, feeling of Love! We conclude: Creatures are more low level behave harshly, incline us to emotional, unfounded, unconsidered actions. They can even give the correct clue several times, and then substitute on a larger scale. In short, you cannot trust them thoughtlessly. To influence, they use a harsh voice, a harsh whisper, a sharp change in actions, and pressure on the “values of the heart.” It’s stupid to be afraid of them: all they can do is try to influence the flow of your thoughts. Those. they are not able to move your arms or legs (THIS IS IMPORTANT), you have complete control over your body, and therefore your thoughts too. Therefore, if you are not comfortable with the course of your thoughts, do not lean in this direction. You are the rightful owner of your body. And it is you who decide where to go, and whether to go there at all. If they start to get impudent, send them to the pujo.
In case of contact with the Beings of the Upper World, a feeling of parenthood arises. Those. From Them comes the Father's or Mother's vibration. These will always help you.
Hells.
Hell is not the Lower World - it is simply a dimension in which those who obeyed their desires during life are “polished.” Errors are “polished” with the help of Karmic transformations.
PS.
And what? Did you think that microbes (in the most common situation) mutate for 5 million years, and one fine day the body suddenly gets tired of it and decides to squander everything accumulated on parties, uncontrollable desires, promiscuous matings, alcohol, cigarettes, drugs, living in the trash?
Cellular memory the body - more powerful than any innate instinct, and this, not even to mention some snotty acquired ones. AND cellular memory exists regardless of whether scientists notice it or are simply not allowed to talk about it. She just is and that's all. And any person understands this, they simply remain silent, or do not have time to say it. Cellular memory- these are not DNA helixes. And when she gets involved, all theories come to their knees.
In short, the whole difficulty is that:
1. U Of people there is no foundation for natural development => cellular memory is not developed.
2. Human Life Form brutally “raped” 75,000 years ago. And this violence, continuing, resulted in destruction Of people, and took the form self-destruction.
3. All psychological corrections that were given in the form of teachings, religions, myths, legends, fairy tales were burned, “hidden” from the mind, “encrypted,” cruelly mutilated, “edited,” deliberately distorted to please the main manipulator.
Based only on this information, it is not difficult to understand what a difficult situation all nations find themselves in, including you and me. Those. each individual has lost almost all levels of his own confidence. Both material and non-material confirmation of one’s rightful Divine origin. Moreover, not just the Divine, but the embodiment of the Supreme God. The symbol of which is the Planet Jupiter. Therefore, for everyone for the future, we form our own own opinion and stick to the chosen direction.
PPP.
Bonus Album. According to the Beacons of Our Life.
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Let's remember what we were taught about measurements and turn to how quantum physics sees it. According to spiritual teachings, there are twenty-one dimensions in the universe.
Let's check how we feel the measurements on different levels consciousness.
1. One dimension has one extension, such as a point and a line.
2. Two dimensions have extensions - this is a plane. It has length and width.
3. Three dimensions have three extensions: length, width and height. Here objects appear in our world, for example, a cube.
4. Four dimensionshave four extensions, here three dimensions are complemented by time. At any moment something is happening around us.
5. Beyond the fourth dimension, in higher dimensions feelings, thoughts, and ideas appear that influence events and actions.
There are many invisible things that affect our lives and the functioning of the worlds. Every action comes from intention! Imagination is already a creation of form, which has all the intentions of movement and germ necessary for fulfillment.
Looking from upper world, the order of measurements changes. The first dimension is intention. The dimensions of imagination, form, time, space, plane and point mean the most extreme dimensions.
Many people have settled on a two-dimensional view of the world. They do not have the courage to think and think about new things that would lead them forward along the path of prosperity. Seems to be the target of someone or some dark forces it was so that a person could not guess what fantastic creature he is. Eventually, man would be able to imagine himself as having creative power. But in what dimension does this creative ability operate?
Let's imagine a two-dimensional world, for example a flat world. Flat people live in this flat world. They have no idea that there are many dimensions, because everything there is two-dimensional. In this flat world, two-dimensional people see only two dimensions.
From the outside, as observers, we see both the two-dimensional and three-dimensional world. We perceive and understand everything that happens there differently. We perceive the same phenomenon as both two-dimensional and three-dimensional.
The case of a three-dimensional rocket speeding through a two-dimensional world:
A three-dimensional rocket flies through a two-dimensional world. What will living two-dimensional creatures see?
A rocket rushing through the world leaves a trail behind it. When touching this world, the tip of the rocket describes a point, then circles, symbols corresponding to the size and, finally, the rocket leaves this two-dimensional world. What will the inhabitants of this two-dimensional world say when watching this? Oh my God! Here, in our world, there were dots, circles and other symbols.
There are, however, people in this world who think differently and have the courage to make themselves heard. A differently thinking two-dimensional creature arriving there will look at the sky, again at the circles and the dot, then dare to look up again, close its eyes and say: there was a three-dimensional rocket here, leaving prints behind it.
Who is right? - we ask.
At their own level of consciousness - everyone. Residents of a one-dimensional world will probably say: a completely crazy creature is talking about something that doesn’t exist. To this, two-dimensional people will say: so abstract, thinks differently, different from us.
If creatures begin to think, they will understand that there are other dimensions beyond the horizon. They will be able to understand that otherwise thinking person, actually right. Socrates was such a dissident person, who on the streets of Athens asked passers-by only questions to think about. The inhabitants began to awaken to consciousness, so the rulers of the city ordered the capture of Socrates and forced him to drink poison. The city fathers were afraid of what would happen if people became self-aware.
A similar thing happened with Jesus, who always makes people think with his spiritual messages. The Romans and elders were horrified by the awakening of people's consciousness, so they killed Jesus. The fact of this terrible crime was distorted by what they began to preach: God sacrificed his son.
Measurements
Our joys and misfortunes experienced in higher dimensions are also visible in lower ones. When bad thoughts, misfortunes or illnesses eat someone up, it can be seen physically. Shadows, projections of higher dimensions are symptoms of the body.
Happiness, spiritual freedom, flight takes the form of a healthy body in visible dimensions.Two-dimensional impressions of bodily symptoms, just like a three-dimensional rocket, are just symbols. The world is more high level, reflected on worlds of a lower level, has the attribute of symbols.
Let someone try to convey, show their feelings, thoughts that form the invisible reality. Everyone knows that it exists, but we carry it invisible within ourselves.
How simple it would be if there were only that which is felt by the five senses. Simple, i.e. “one-dimensional”. A “multilateral” person feels free in higher areas.
Task beyond nine points:
There are nine points in the task. Please connect them with straight lines. This can be done in any order without lifting the pencil, touching each point.
If you can go beyond the nine points in the two-dimensional boundaries, then you not only go from point to point, but you can also go beyond the area limited by the points. The secret of the task is that we do not think within nine points, but are able to go beyond them.
In the process of solving the problem, it seems that we have not yet crossed into another dimension.
In order to look at the solution to our problem from higher dimensions, we must mentally rise above our knowledge and way of seeing. People make any sacrifices to achieve titles and ranks. If only part of these efforts were spent on spiritual and spiritual growth, there would not be so many sick and unhappy people. The representatives and preachers of these noble ideas were the great mystics.
If anyone wants to go beyond a certain way of seeing, supported by two-dimensional and three-dimensional X-rays, ultrasound, CT and MRI images, he must have great courage, have strong faith, fundamental knowledge and will. The idea already in many cases carries the key to the solution - it is the highest dimension of form, which comes from intention.
Have the courage to go beyond the traditions, the familiar, the ingrained? What happens if you connect the dots with four lines? I solved the matrix, since this task already requires free thinking. We not only move into three-dimensional space, but also go beyond it, into the higher realms of thought.
Limited human consciousness acts and thinks on the same plane. Anyone who unexpectedly accomplishes things unimaginable for others deserves to be called a dimensional traveler with his versatility.
Sum of interior angles of a triangle:
(Equator)
The answer to this question modern man with an inferior or even higher education: 180 degrees. This definition is one of the cornerstones of mathematics.
Let's analyze the triangle on the scale of the Earth. It is known that the Earth is not flat; many centuries ago it became known: the Earth is round.
Let's draw two perpendiculars to the Earth's equator. As you can see, 90° + 90°, this is the sum of the angles of a triangle equal to 180°. Now let's follow two perpendiculars that meet at the north pole and another angle is closed there. This latter can be 1°, 30° or even 359°. Let's add the internal angles of the resulting triangle: 90°+90°+30°=210°. This, as you can see, is more than the 180° amount indicated above.
A significant proportion of students today grew up with Euclidean geometry. They think in a plane - that's how they were taught. (Another thing is that the theorems of Euclid and Thales are valid in plane geometry). However, thinking only in the plane will be fatal. If people saw everything and thought only in planes, life would be contained in two dimensions. Of course, those who set out to think in many dimensions sometimes encounter serious problems. Often, even very educated people live with a flat consciousness, i.e. in a limited world.
How will the human psyche react: if one day we go beyond the traditional, specific, flat thinking imposed on us?
When people meet a person who thinks differently, they will immediately condemn him. There is a danger that people will also have to change their views. Some are as attached to ingrained dogma and faith, just as an alcoholic or smoker is to the subject of his passion.
We should think carefully about whether we intend to change our views. Those who embrace the challenge of adventure and travel will become healthier, happier, more hopeful, successful, out-of-the-mundane people.
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 made contact with the two-dimensional world of Arde, a disk-shaped planet inscribed in the outer shell of a vast, shaped hot air balloon space called 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 a fairly complete history of the origin of the flat universe Planiverse. As soon as Martin Gardner's column on Scientific American math games, an article about Planiverse appeared, thousands (not even hundreds) of readers sent 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 an 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's written with a pen researcher, 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 took 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 operating in our three-dimensional universe, the sum of kinetic and potential energy 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 very useful program, and working with her was incredibly interesting. 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.
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 most a big problem for theoretical physicists - how to combine all 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 in a certain 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
Things 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 identify physical meaning, hidden 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. Having studied strange fashions vibrations of strings, reminiscent of carrier particles, they realized that these properties coincide surprisingly exactly 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 amazing Progress achieved during the first revolution of superstring theory, physicists were faced with the fact that the approximate equations used were unable to give the correct answer to a number of important questions, thereby hindering the 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 "drought" was interrupted from time to time important discoveries, but it was clear to everyone that new methods were required that would allow us to 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 searches final theory"became a bestseller and received 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 also see any multidimensional object as two-dimensional, but it will change in amazing ways depending on our relationship with it 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 modern science Are they 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 important role the curvature of the space-time continuum plays into them. 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 options developments of events. 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. Known to a wide audience as the author of popular science books. His Elegant Universe was nominated for a Pulitzer Prize.
In September 2013, Brian Greene came to Moscow at the invitation of the Polytechnic Museum. 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.
