Euclidean space

Euclidean space(also Euclidean space) - in the original sense, a space whose properties are described by the axioms of Euclidean geometry. In this case, it is assumed that the space has dimension 3.

In the modern sense, in a more general sense, it can denote one of the similar and closely related objects defined below. Usually the -dimensional Euclidean space is denoted, although not quite an acceptable notation is often used.

,

in the simplest case ( Euclidean norm):

where (in Euclidean space, you can always choose a basis in which this very simplest version is correct).

2. Metric space corresponding to the space described above. That is, with a metric entered by the formula:

,

Related definitions

• Under Euclidean metric the metric described above, as well as the corresponding Riemannian metric, can be understood.

• By local Euclidean we usually mean that each tangent space of a Riemannian manifold is a Euclidean space with all the ensuing properties, for example, the possibility (in terms of the smoothness of the metric) to introduce coordinates in a small neighborhood of a point in which the distance is expressed (up to some order of magnitude ) as described above.

• A metric space is also called locally Euclidean if it is possible to introduce coordinates on it in which the metric is Euclidean (in the sense of the second definition) everywhere (or at least on a finite domain) - which, for example, is a Riemannian manifold of zero curvature.

Examples of

Spaces can serve as illustrative examples of Euclidean spaces:

A more abstract example:

Variations and generalizations

see also

Links

Wikimedia Foundation. 2010.

• Linear space
• Convex functional

See what "Euclidean space" is in other dictionaries:

Euclidean space- a finite-dimensional vector space with a positive definite scalar product. It is direct. generalization of the usual three-dimensional space. In E. p. There are Cartesian coordinates, in which the scalar product (xy) of vectors x ... Physical encyclopedia

Euclidean space- space, the properties of which are studied in Euclidean geometry. In a broader sense, Euclidean space is an n-dimensional vector space in which the scalar product is defined ... Big Encyclopedic Dictionary

Euclidean space- space, the properties of which are described by the axioms of Euclidean geometry. Simplistically, you can define Euclidean space as a space on a plane or in a three-dimensional volume, in which rectangular (Cartesian) coordinates are given, and ... ... The beginnings of modern natural science

Euclidean space- see Multidimensional (n-dimensional) vector space, Vector (linear) space ... Economics and Mathematics Dictionary

euclidean space- - [L.G. Sumenko. The English Russian Dictionary of Information Technology. M .: GP TsNIIS, 2003.] Subjects information Technology in general EN Cartesian space ... Technical translator's guide

euclidean space- space, the properties of which are studied in Euclidean geometry. In a broader sense, Euclidean space is an n-dimensional vector space in which the scalar product is defined. * * * Euclidean space Euclidean ... ... encyclopedic Dictionary

Euclidean space- space, the properties of which are studied in Euclidean geometry. In a broader sense, E. n. Is called. n dimensional vector space, in which the scalar product is defined ... Natural science. encyclopedic Dictionary

Euclidean space- space, the properties of which are described by the axioms of Euclidean geometry. In a more general sense, an E. p. Is a finite-dimensional real vector space Rn with scalar product (x, y), x, in a swarm in appropriately chosen coordinates ... ... Encyclopedia of Mathematics

Euclidean space- (in mathematics) space, the properties of which are described by the axioms of Euclidean geometry (See Euclidean geometry). In a more general sense, an E. n. Is called an n-dimensional Vector space, in which it is possible to introduce some special ... ... Great Soviet Encyclopedia

Euclidean space- [by the name of other Greek. mathematician Euclid (Eukleides; 3rd century BC)] space, including multidimensional, in which it is possible to enter coordinates x1, ..., xn so that the distance p (M, M) between points M (x1 ..., xn) and M (x1, .... xn) maybe ... ... Big Encyclopedic Polytechnic Dictionary

Corresponding to such a vector space. In this article, the first definition will be taken as a starting point.

N (\ displaystyle n)-dimensional Euclidean space is denoted E n, (\ displaystyle \ mathbb (E) ^ (n),) the notation is also often used (if it is clear from the context that the space has a Euclidean structure).

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✪ 04 - Linear Algebra. Euclidean space

✪ Non-Euclidean geometry. Part one.

✪ Non-Euclidean geometry. Part two

✪ 01 - Linear algebra. Linear (vector) space

✪ 8. Euclidean spaces

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Formal definition

To define a Euclidean space, it is easiest to take the concept of the scalar product as the main one. Euclidean vector space is defined as a finite-dimensional vector space over the field of real numbers, on the vectors of which a real-valued function is given (⋅, ⋅), (\ displaystyle (\ cdot, \ cdot),) having the following three properties:

An example of Euclidean space - coordinate space R n, (\ displaystyle \ mathbb (R) ^ (n),) consisting of all possible tuples real numbers (x 1, x 2,…, x n), (\ displaystyle (x_ (1), x_ (2), \ ldots, x_ (n)),) dot product in which is defined by the formula (x, y) = ∑ i = 1 n x i y i = x 1 y 1 + x 2 y 2 + ⋯ + x n y n. (\ displaystyle (x, y) = \ sum _ (i = 1) ^ (n) x_ (i) y_ (i) = x_ (1) y_ (1) + x_ (2) y_ (2) + \ cdots + x_ (n) y_ (n).)

Lengths and angles

The scalar product given on the Euclidean space is sufficient to introduce the geometric concepts of length and angle. Vector length u (\ displaystyle u) defined as (u, u) (\ displaystyle (\ sqrt ((u, u)))) and denoted | u | ... (\ displaystyle | u |.) The positive definiteness of the dot product guarantees that the length of a nonzero vector is nonzero, and bilinearity implies that | a u | = | a | | u | , (\ displaystyle | au | = | a || u |,) that is, the lengths of proportional vectors are proportional.

Angle between vectors u (\ displaystyle u) and v (\ displaystyle v) is determined by the formula φ = arccos ⁡ ((x, y) | x | | y |). (\ displaystyle \ varphi = \ arccos \ left ((\ frac ((x, y)) (| x || y |)) \ right).) It follows from the cosine theorem that for a two-dimensional Euclidean space ( euclidean plane) this definition angle coincides with the usual one. Orthogonal vectors, as in three-dimensional space, can be defined as vectors, the angle between which is equal to π 2. (\ displaystyle (\ frac (\ pi) (2)).)

The Cauchy - Bunyakovsky - Schwarz inequality and the triangle inequality

In the definition of the angle given above, there is one space left: in order to arccos ⁡ ((x, y) | x | | y |) (\ displaystyle \ arccos \ left ((\ frac ((x, y)) (| x || y |)) \ right)) was determined, it is necessary that the inequality | (x, y) | x | | y | | ⩽ 1. (\ displaystyle \ left | (\ frac ((x, y)) (| x || y |)) \ right | \ leqslant 1.) This inequality really holds in an arbitrary Euclidean space, it is called the Cauchy - Bunyakovsky - Schwartz inequality. This inequality, in turn, implies the triangle inequality: | u + v | ⩽ | u | + | v | ... (\ displaystyle | u + v | \ leqslant | u | + | v |.) The triangle inequality, together with the length properties listed above, means that the length of the vector is the norm on the Euclidean vector space, and the function d (x, y) = | x - y | (\ displaystyle d (x, y) = | x-y |) defines the structure of a metric space on the Euclidean space (this function is called the Euclidean metric). In particular, the distance between elements (points) x (\ displaystyle x) and y (\ displaystyle y) coordinate space R n (\ displaystyle \ mathbb (R) ^ (n)) is given by the formula d (x, y) = ‖ x - y ‖ = ∑ i = 1 n (x i - y i) 2. (\ displaystyle d (\ mathbf (x), \ mathbf (y)) = \ | \ mathbf (x) - \ mathbf (y) \ | = (\ sqrt (\ sum _ (i = 1) ^ (n) (x_ (i) -y_ (i)) ^ (2))).)

Algebraic properties

Orthonormal bases

Conjugate spaces and operators

Any vector x (\ displaystyle x) Euclidean space defines a linear functional x ∗ (\ displaystyle x ^ (*)) on this space, defined as x ∗ (y) = (x, y). (\ displaystyle x ^ (*) (y) = (x, y).) This mapping is an isomorphism between Euclidean space and

Chapter nineteen. ROME

The Russian, moreover, is much more irritable, because he goes into discord between the living and living other peoples.

A. Ivanov. Letters. End of 1846

Already in his name alone, centuries and peoples were intertwined. Who, without worry, will enter this city. During its existence, how many cultures and capitals collapsed, how many peoples were conceived, developed and died out within its walls, or again scattered around their kurens and camps, remembering, as in a dream, the Eternal City.

Even in the impregnable fortresses of Roman construction, a new toothy culture began to gnaw through its strongholds with a mouse; from the holes of the catacombs, from secret undergrounds, with the talismans of fish, cross and lamb, she crawled out, unkempt and unwashed, and wedged into the power of Rome with its legions, cohorts and law. Still timid churches dedicated to the new are glued to the ruins, it is impossible to understand which one, whether a person, whether God and his propagandists. And for the first time, perhaps, in history, the foundations of tribes, races, families and peoples wavered from the slogan "love your neighbor as yourself."

In the catacombs, there is one horror from affection and cursing, from believers and unbelievers: the first warriors, martyrs for the crucified lamb were inspired here, tempered at secret gatherings. Here the state power caught them and dragged them for fun for a bored street in the Colosseum, where the worshipers of the crucified had to show their ability to die in order to nourish the paths of a new culture and a tricky kingdom, not of this world with blood.

Hence the rebels, ragamuffins carried hatred and destruction to the clear, felt beauty of Greco-Roman. They smeared the walls of the quarries with illiterate daubs, distorted, miserably imitated the greatest in figurative completeness models: disfiguring Zeus, Hera and Apollo, they denoted their delirious babble about the crucified carpenter and his mother. They attached Egyptian wings and regal halos of Indian images to the gods that had been established for centuries. They stole the Greek pentagram in order to give it meaning to neither one nor the other: either a Man with a capital letter, reigning supreme over the world, or he is on the leash of heavenly tyranny.

This duality of differently tuned pilgrims buzzes with catacombs, muddies, unnerves and confuses observation.

From here, across Rome and its environs, the duumvirs of a new world outlook - Paul and Peter - will warm up and merge in a common victory.

The Basilica of Paul will jump out of the walls of Rome, Peter will reign in the center of the city, and Michelangelo and Bramante will cover him with the tiara of the dome as the completion of the wisdom of human behavior.

When leaving Rome on the Appius road, not without a second thought, the meeting of Christ with Peter by the monument-basilica of Quo-Vadis is documented as a justifying reference for the future in general about human weakness.

On the double sound of Greco-Rome and Christianity, all works of art in Rome will play out and then spread throughout Europe in all sorts of variations.

A new consciousness is growing from the catacombs to the Vatican. The human forces liberated by him during this time create Gothic, jump out of Roman programs and assignments. The East, through Constantinople and the Gallic West, attacks from two sides the building of Rome, which is striving to embrace itself, but Rome withstands the onslaught of Gothic, agitated by its own ideas. In order to give vent to passions, in order to unload the rebellious forces of the sharp-eyed savages that have poured through predetermined programs, Rome throws a spark into young, incandescent organisms: oh crusades, about the enemies of the Holy Sepulcher, about the Saracens, and pushes the East and West with their foreheads. Led by adventurers, hordes of enthusiastic ragamuffins are cleverly heading towards the more politically more than the Saracens, a dangerous enemy of Rome - Constantinople. The army, stunned by the treasures of Constantine, smashes brocade and gold, Greek idols and their Christian monuments to ashes and turns the capital of the East into heaps of ruins.

The barbarians of the East will again build up the Bosphorus beauty, and the barbarians of the West again, ricocheting from Palestine, will bring destruction into it, until the Ottomans take the Bosphorus into their own hands and until Hellenism from his homeland spreads throughout Europe and calms the excitement of the Gothic ...

Since that time, the Eternal City has been the only center of the transmission of Hellenic and its own, Roman Catholic, culture. The fires of the Inquisition will heat up the too fervent fantasies of the last of the Gothic excitement.

Then the singers of a balanced, tranquil world-consciousness will appear. Gourmets, popes, princes and merchants will celebrate the artisans, and Giotto, Masaccio, Donatello, Leonardo, Buonarroti and Raphael will unfold their spells over the pacified West, and will answer them with an echo from our Belozerie, from Scandinavia and from the New World.

Leonardo da Vinci, once noticed by The Last Supper, will everywhere be influencing his Italian comrades: he will soften the insolence of the chisel and Buonarroti's painting, alarm Raphael Sanzio; later he will leave the riddles to resolve the smiles of his characters and the canons of his mathematical compositions that are disastrous for imitators.

In the Vatican, one head cannot digest all the visual material accumulated over the centuries. These separate completeness of different cycles in their mass create disputes, noises and a figurative cacophony: sometimes sensual, then contemplative, then naturalistically pathological marbles wore me out to the tendons, and they deprived me of sleep and a healthy appetite. You come to Raphael the Vatican after them as if to rest, this gentle clarity, childish genius playfulness with color and shape, now carelessly cheerful, now thoughtful and sad, like a child who smashed a toy and wrinkled his forehead capriciously, which makes him even sweeter - it disarms you, loosens tense muscles. As perfect in his own strength, Raphael is not afraid of other people's thoughts or compositional canons. It would not be scary and it would be easy to live in Raphael's picturesque space: not a single character would offend you and would accept you into his environment, and you would not have the thought of disturbing their thoughts, but neither Moses, nor David, nor Pieta would fit in there. Michelangelo, and if by some mistake they got there, they would darken and disperse both Parnassus and the School of Athens, and the Fire in Borgo would really flare up, and the frightened Donna would scramble in sobs. Velata.

The very color of Raphael, without problems, without calculation, was drawn up and put in place of the picture in such a way and where it ought to be, and you cannot imagine it differently, and even if it was I, the beholder, invented it.

Not old Michelangelo in the Sixtus Chapel.

He is seething and angry. The plafond, by the time of its work, is separated by tens of years from the "Last Judgment", but the young and old master remains relentless. The Medici tomb in Florence echoes the painting of the Plafon, but in the same museum, there is an unfinished Madonna with a book, a round marble high relief, where, above all, the master's technique is visible, nervously aggravating the work. This marble is consulted with the "Pieta" in Santa Croce and with the "Last Judgment".

The pictorial innovation that Michelangelo introduces is his reverse perspective of the painting ...

Here, on the wall of the chapel, the entire Renaissance overflowed. The story and its meanings are crumpled into the master's fist, crumpled up and thrown on the wall in writhing balls of a man wriggling in chaos.

It is he who, on the ceiling of the creation of man, favorably, dismissively accepts the fact of his creation. There is neither Christian nor pagan in this picture, nor race, nor nation - this is a diagram of a person in the Michelangelo worldview, to whom the master does not give a single indulgence, not a single sentiment ... And this is Buonarroti's terrible Last Judgment. There is nothing and nowhere to warm up in the picture: it will throw you over the abyss, over Apollo - Hercules, administering judgment (this is the construction of his reverse perspective), and fly into the horror of space or fall when you cannot fly! ..

Leonardo defended himself by analyzing the outside of man natural phenomena, - Michelangelo put all the kinetics of nature in a person and opposed himself to it.

Hence - the tragedy of loneliness, which crushes, strangles the viewer from Michelangelo's "Last Judgment".

For the first time, perhaps, with such force in world history, the problem of personality arises, which will fill the entire further course of social relations in the West with content.

The Italian Renaissance was hard on the suppression of the Middle Ages and the victory over the Gothic!

I was returning in an omnibus from the vicinity of Rome. I amused myself by observing my compatriots, a newlywed couple. The pair were separated by seats. His wife, a blonde with a Russian pretty face, understandable, perhaps, only to ourselves: so everything on these faces can be knocked down somehow - in a hurry, a nose stuck in a hurry, eyebrows raised out of place by mistake. The eyes began to be painted blue, but there was not enough paint, and they put either ocher in there, or they simply removed the blue paint with an unclean blot of spapyr, and a gray-blue-brown spot with green was left - and their color cannot be described. Lips - from two different faces: the upper one is very cheerful, with enthusiasm, and the lower one is crying, capricious, in a word, no typical certainty, but there is more than enough touchingness, familiar from birth!

The newlywed woman was sitting opposite me, and her husband - across one passenger, on my bench. She made some complex projections with her eyes and head: having established the direction of her husband's line of sight, the woman turned her gaze to the object that her husband's eyes were fixing. To do this, she unceremoniously bent her head at the neighbors ... Obviously, the man's line of sight rested on the brunette at the window opposite ...

Russians, thanks to the Chineseness of their language, which is impenetrable for Europe, abroad usually abuse it.

Kolya, take your eyes off this girl! - Loudly, for the whole carriage, said his wife.

My dear, I'm really looking out the window at the ruins of Caracalla ... - said the husband reasonably.

What is her name - does not concern me, but please look out the window to the right!

Kolya obediently turned his head to the right. He was a well-meaning, serious man, judging by his appearance, and I did not see in him an intention to examine female faces, but he was unlucky: to the right, he buried himself directly in the red-haired beauty with cornflower-blue eyes against the background of a window with ruins. The wife gasped.

Turn away from there and look directly at me! she commanded. The husband made a half turn with his head ...

My dear, you are sitting right on the wall, and I do not see any historical places ... I am wasting time and money, - he said calmly, but already with slight chagrin, without changing the frontal setting of his head.

Nikolay, you are cunning, you even now glanced at my neighbor! Let's go to! - She took her husband's hand and led him to the platform to get off at the first stop.

I, who did not yet know family ties, laughed at this scene, somehow vividly and naively emerged against the majestically gloomy background of my observations.

On the same evening I met this couple in the cafe del Greca.Here she was sitting in the corner, and Kolya turned to face her and said to her, unexpectedly for his balanced appearance, with pathos and with a tremor in his voice:

There, 'he said, gesturing with his hand behind him, but without turning his head,' sat Gogol, Nikolai Vasilyevich, surrounded by Russian youth. On that chair, to the right of my left ear from you, my dear, sat Ivanov, Alexander Andreevich ... This is an artist who has been painting the same picture for twenty years ... Bryullov, Bruni, Jordan - they all sat, surrounding the great creator " Dead souls". And he said ...

Again you have a tie on your side, Kolya ... Well, well! The husband tugged at the tie to Adam's apple.

He spoke, my dear, about the art that only here, in Italy, gives ...

Kolya, do you love me?

Yes, - said the husband.

No, really, Kolya, tell me: do you love?

I assure you, my dear.

I seriously ask you, forget everything that is in your head, and really answer me: do you love me or not?

The husband, widening his eyes, said thoughtfully and reasonably:

Well, listen, dear, why would I marry you if I didn't love you ?! - and threw up his hands. Everyone would agree to his arguments, but I saw from behind the "Corriere della Sera", which I covered myself with, how the woman's lower lip became agitated and tears appeared in her eyes.

Kolya wiggled his mustache in bewilderment.

Well, what can I say that she looked like our spring landscape with a freshly blossoming birch tree, that, looking at her anti-classical face, I recalled everything young and simple in my homeland ... All this is cloying, and therefore I will not continue further comparisons, even if they are only implied, but Kolya, at least, I admit, did not suspect about these similarities of his wife, but, nevertheless, it was impossible to answer her standard and classically posed questions in such a protocol manner; Well, tell me, he, at least one word: "dear!" - with exclamation mark, or even simpler, one interspace: "Ah ..." - and an ellipsis, yes, finally, without saying anything, I would kiss, unfortunate, her hand on a count of questions ... Moreover, her hand was beautiful and not spoiled by manicures ...

Beautiful is not the right word, because it will lead us back to the Vatican Museum - this is our hand from the Russian plain, neither the Greeks nor the Romans took part in its making, but, calloused or sleek, this hand has its own special expressiveness.

I've looked into my hands a lot over my life, sometimes neither a face nor a figure will tell as much about its owner as this instrument. I delimited my hands according to peoples, according to professions and according to the characters of individuals.

The type of hand, I think, develops according to two of its working gestures: grasping with the palm in an embrace and with fingers, as if auxiliary, then the main action is in the pads of the palm and in the last phalanx thumb The second gesture, when only fingers are dealing with the object, leaving an auxiliary palm and wrist. These are the basic, primitive acts of the hand, on which it develops to the most complex gestures. Taken together, these gestures cause the middle finger to protrude above the index, ring and small and the rebound of the thumb, forming an angle with the index, and its length, which varies on the big phalanx of the same index finger. Complex gestures process each pad with touch and tenacity and complete the back ends of the fingers with nails of the most bizarre shapes. Nails form their own shape, obeying protective and familiarization functions when they meet with objects, point-blank, without bending the fingers. The scratching gesture, still observed in children, I think, does not play a special role in the formation of the nail.

With an increase in the culture of the hand, its action is concentrated in it itself, the wrist and forearm play only the role of support.

As for the hands in works of art, I compared them by epoch and with my observations of the living.

The Greek reversal of the figure subjugates the hand and gives it a decorative, decorative character. The formula for the axial relationship of the head, neck, torso and limbs, ingeniously found by the Greeks, after the profile movement of Egypt and cave sketches, gives new, happy opportunities for the horizontal rotation of the figure around itself, and on this movement a composition is built, although very conditional, but perfect in clarity tasks: if the Egyptian, profile, longitudinal movement pulls the viewer and does not indicate the end of the point where this pull will stop, then the Greek task is to circle the viewer around the statue and let him fully perceive the gesture given by the master. Hence the difference between the hands: the Egyptian cutting and flashing gesture in one plane of the fingers extended with the palm and the gesture of the Greek hand indicating the circular completion ...

But I jumped too far out of the Café del Greco and from my compatriots with my hand.

Well, of course, you worry about all kinds of Raphael as your family, but you answer me like a sentry on a clock ... '' She wiped her eyes with a lace handkerchief and rapped out:

Look back or to the sides - now it doesn't concern me at all!

From the spasms of Kolya's neck, I guessed that it was now that eight Italian carabinieri would not have turned his head one iota away from his wife's face.

I must confess that this, perhaps, the first squabble in the young nest was already beginning to sadden me, but the rationality of my husband became clear with a glimpse of at least a tiny insanity:

Sweetheart, angel ... - and ellipsis. And he could not continue further. The wife had already smiled for the good word, but the husband continued to build on his thought: - Today is such a day, and you grieve yourself on the eve of our New Year ...

It probably happened at the same time that my dear caught herself, and I cursed at the news, and cursed, probably loudly, because the young woman looked at me carefully and asked if I was really Russian, and if so, we would celebrate the New Year together.

In half an hour we were friends so much that I already knew the main secret of the nest, which explained much of the whims of the "darling", but in no way diminished her simplicity and spontaneity.

They were unhappy with the hotel - I suggested they move to Porta Rossa, where I lived. The next day we already had dinner at the same table of the table d'hote Jacomino, the cheerful, resourceful owner of the hotel.

I brought up this episodic memory for many reasons. One thing that I became more comfortable during these two weeks, while our acquaintance lasted. This nest simplified my impressions outwardly. Several excursions, done together, made a healthy mess and confusion, which the "darling" was able to produce, it would seem, with things that were inaccessible in importance.

First of all, my still life was a failure. If the Florentine Benedetta was still showing me the way to Fornarina, this one finally declared that painting is not worth doing for such boring things as my work.

In museums, where I imagined myself to be useful to her by my explanations, hoping to introduce a wild plant into the gardens of tropical cultures, here everything changed at once: she took the initiative and guidance into her own hands. Kolya and I were in her reins. She, if you please see, had to exhaust "the whole essence and ins and outs," as she put it, of the pictures, and therefore all the characters, their relatives, and scandalous, and everyday, and romantic relationships had to be revealed by her. In my entire life, not so many anecdotes, stories and gossip have passed through my head as they have passed in a few days. Kolya was an excellent assistant in the search for explanatory material, and with two hands we tried to fully satisfy the desire of our leader. We got to the bottom, for example, to Fornarina's relatives, opened their addresses. We got to the bottom of the culprit of the fire in Borgo. Under what documentary circumstances was Raphael expelled from the dungeon Peter, was there a real angel, or was it all set up by Christian conspirators. As for the portraits of the beauties, it was extremely difficult to work with them to get to the bottom of who they lived with and why they cheated on such and such ... ...

For the Colosseum, we had to find a synodikon of almost all martyrs torn to pieces by beasts. To depict their deaths in their faces, together with Kolya, to re-enact the scenes at exact historical places, if they were not completely assimilated by our leader.

In a word, a week later we men were completely insane: we got so into the giblets of someone else's life that we spoke and thought in their environment and habits. We caught each other in gestures and expressions. The cheat offended us completely, we, in addition to whether our tormentor needed this or that information, were looking for it for ourselves; we have already entered the search rage ...

And she, as if nothing had happened, counted the days of growing motherhood in her and once declared that she was terribly tired of all these ruins and Rafaeli and that she herself had become terribly smart and disgusting ... And that Rome is a city of the dead, to whom do not allow to die in peace ... And gave up excursions. In her room, she began to sew and cut caps and undershirts ...

In order to somehow ventilate ourselves from the overload of information and get rid of it, we began to read Eugene Onegin to her in turn, since this volume of Pushkin happened to me.

Once climbing into the Capitol, at the cage of a she-wolf, I met a man with a beard and almost shouted with pleasure, seeing him alive: it was a hunter, dead removed from a steamer in Brindisi. We met as old acquaintances.

From this meeting, I enter the world of modern artistic life, into impressionism and neo-impressionism of Italian youth, a little, as it seemed to me then, talented, but stormy. Fragmentation of paint into points and lines and, on the other hand, a still life of altered objects, shifted, repeated in separate parts, surround me.

Segantini is their leader of classic pointillism. From France, through Delacroix, Gustave Moreau, Renoir, Cézanne, Van Gogh and Gauguin, Italians took their innovations and inspirations to shake off the hypnosis of their own old people. They did not have their own contemporary tradition: academicism in Italy was in its complete degeneration. The branches of the French Academy and academies of other countries brought here youth specially trained for state art, but among them Italian official school was the weakest.

The only thing that Protestants-modernists could do was, through the heads of their own and visiting academism, to link with the Impressionists and with the "wild" of France. France was now a hotbed and refuge for new quests in art. But what happened there painlessly, systematically, through Delacroix and later, then in Italy accumulated in a fundamental question connecting art with everyday life, politics and Catholicism. It was much more difficult for them to get out of the reins of the Renaissance. In addition, the French were helped by the Gothic, on the models of which many leading painters passed their schools, and the Italian youth did not have this either.

Partial tests of the forties in Italy to rush to the Ravenna samples ended in unsuccessful stylistics.

These years also coincide with the political upheaval in Italy, and, which is very significant for me, with a turning point in the work of A. Ivanov. Ivanov experienced a reassessment of his Renaissance gods. The need for new expressiveness turned his attention to the East. For a short time and purely by chance, thanks to an order for Moscow, he stayed in Byzantium and threw himself over to Egypt. The quarrel with Gogol, the essence of which remained unclear, and the death of Gogol - all this for Ivanov is escalated to one point: refusal to continue his large canvas and by the beginning of a new period, which left us with his last watercolor compositions of a new artistic rise and strength.

The relationship between Ivanov and Gogol and the intertwining of their fates even in the same Cafe del Greco worried me about their obscurity. Whether Gogol dreamed of seeing Ivanov unfold in a modern plot of the genre of genre that was childishly hated by Ivanov, which captured the Parisians and reached our Fedotov, or with his greedy, insatiable eye that turned nature for him, through the enormous Ivanov technique, Gogol wanted to carry out a still life, about which he sharply hinted he in his literary works?

All this is unclear, not investigated and annoyingly overlooked, but one thing is clear that Ivanov stubbornly resisted something in Gogol, but he also grew stronger and crystallized out of this resistance.

So, if the Italians did not get out of the tenacious paws of Leonardo and Michelangelo with a breakthrough with Byzantine, then Ivanov, on the contrary, became even stronger in the new systems of compositions.

The French modernists, oddly enough, all the time, in the most seemingly risky innovations, were equal to the Italian Renaissance - take, for example, Cezanne, who until his last days copied the Italian paintings of the Louvre to clarify the composition of chiaroscuro and the design of the Renaissance painting ...

The battle in Italy rose along the entire artistic line, but for me then it seemed more of a verbal and theoretical battle. Marinetti, then still a young man, and himself, it seemed to me, did not clearly imagine the idea of ​​the eternal fluidity of creativity. At least I was confused by their arguments at the time:

Old men to the place of the goddess! - okay, I agreed.

To merge with modernity, to reveal the poetry of a machine, Wright's experiments with flights, and so on: I still believed, according to Tolstoy's What Is Art, that in general, poetry is a feeling warmed up, stolen from someone else's work, that is, aesthetics. No, please, ”I said,“ take away the poetry of any phenomena, even machine ones, and merge with modernity - although I do not think of myself in isolation from it - of course, it is necessary!

Friends, - I said, - I will go to the scandal with you, but even then, indeed, in order to start with nakedness, with a touch, as if we were born for the first time ...

Where is it, my conspirators, top hats, tailcoats, face make-up assert as a new form; the dances of cafes are not taken as a scabies of overwhelmed townspeople, but as a new gesture, the damn dreadnoughts are elevating values ​​to train the body ... No, my friends, maybe I have not grown up to all this, but then it’s better long live the geometry of Euclid, I’ll go better in its space, as long as you compose poetry with dreadnoughts ...

We will not go to holiznu, - others shout. - We are people of the twentieth century, we are the future, to start with the primitiveness of holism!

Then the hunter offered in my honor a drink of the hottest drinks in the famous old Roman tavern, somewhere near the tower of St. Angela.

I remember toasts, fun. I remember, but already vaguely, myself on the table, speaking in Russian, because the meaning of the speech was very complex. Then everything became dim in my memory. We were walking or driving somewhere, it was as if I was trying to swim in the Tiber, and then everything calmed down ...

There is a dream when you wake up in a dream and decipher what you have just seen and talk about the absurdity of the vision, analyze it and console yourself that now I will soberly do what I have failed to do in a dream, but the absurdities begin to be even more pretentious, but their imprint in memory, after a real awakening , much stronger than the first dream ...

Who does not know nightmares that plunge us into a hopeless situation, into final death, or when you commit an irreparable crime in a dream: murder, betrayal, violence against the defenseless. And suddenly, like a shutter, it clicks in consciousness and as if you wake up.

It's nice to know that the torment turned out to be nonsense, but that was not the case: from somewhere, usually from a gap of some kind, a muzzle will appear, and this muzzle will beckon you into the gap ... braided legs, for which there is no support - they are like whips. And in no way let out of the larynx a cry for help and to give yourself strength. And after every effort you extract from your throat, pressing on it from your belly, a wheeze, which both wakes you up and saves you from a nightmare ...

The damned questions that produced the Roman crisis in me were, of course, deeper than Futurism and my stalled still life. These periodic approaches to the chasms of life in youth often end tragically. If the young man is ill with this process, it will be followed by a violent growth of the body refreshed by the fracture.

In the process of breaking, everything becomes confusing and unclear. People become strangers, bothering about trifles, and boredom squeezes desires in a loop. You will not catch on to anything, the romance of the landscape fades away, and the person grows dim even in his best in feeling and sincerity positions. It seems that all this complex, piled up life, with the culture of cities, with cells of relationships, is made by people out of nothing to do, a game of "as if", but deprived true meaning games: lightness, fantasy, and most importantly, the usual, as in children, consciousness that this is a game, that it will not cause misfortune and grief, and if its system is screwed up tightly, the game is changed ... built on submission to their own lusts to the desires of others, and all this is serious and eternal; besides, the life itself, of course, of the other and the right to it are at stake.

In times of difficulty, I was often saved by the synthesis of all the phenomena accessible to my observation and imagination: you will bring the way of life to the rotation of the worlds and the earth among them, and it will become clear that there is no violence and tediousness in world building: play and a free agreement turn the whole this bulk, - they argue, trick things-giants and miniatures, but no one is irreparable from this death. Dress-up game: today's form - tomorrow is a nebula, a large organism disintegrates into microbes; decompose, crystallize, change weight and elasticity, pass through temperature systems and microbes, again - an annular nebula, vortex, concentrating in a solar and planetary organism. Cunning, grow your form, strengthen ties with those like you, listen to the air, earthly and fiery comrades from which your cells are woven, establish a natural relationship with them ... , nasty things that are not beneficial to either the tyrant or the victim.

The muzzles rose to me from all the cracks of life, like ancient horrors. Instead of living people, I was surrounded by "acting duties" - there was no one to play with. Art, like a glass ball, hung over the earthly officials. Everything in him became closed, isolated ...

I felt myself thrown out of the people's tub, and I had nothing to do with myself ...

Sleep in some way this blackout, or ...

This "or" is well known and generally available: to break your form, to release motor and combustible energy for other forms.

I will not describe the rather banal choice of the method of destroying myself in these cases, but the danger to my life rescued me from the deplorable ending, threatening it from the outside, in addition to my desire ...

Kuzma Sergeevich Petrov-Vodkin

Euclid space

Chapter one

DEPARTURE FROM THE NEST

By the time of the final exams, we schoolchildren, imperceptibly for ourselves, matured. Everyone's breasts swelled and calmed down. The faces became more concerned. Our voices hardened and spoke more boldly about girls.

They started smoking, however, still on the sly from their parents. Having overcome the nausea and disgust from tobacco smoke, we developed the gestures of puffs, holding a cigarette between our fingers, in the corner of the mouth, and chanting words released simultaneously with the smoke.

Conversations have become smarter. We moved on to questions that we hadn't thought about a year ago. The pressing issue there were - the advantages and meaningfulness of this or that profession: each planned his own path or whom his parents had planned, but few of us decided to irrevocably cross over to Khlynovsk, and few realized the scarcity of our educational baggage, and not many had the need to replenish it.

We are sitting in the courtyard of the school, - Pyotr Antonovich is unwell, - we know the culprit of ill health - the buffet on the Suvorov, we sit and discuss our assumptions.

I will sweep the streets in Moscow, but I will not stay in Khlynovsk! - declares Pozdnukhov - our poet, romantic. He is an orphan; the uncle, at whom Pozdnukhov was orphaned, was also a boar, he made furniture out of twigs; so the uncle decided that since he had brought his nephew to "high science", so now give him food.

I’ll go on foot, ”Pozdnukhov continues,“ I have my luggage ready: Pushkin’s book, forty kopecks and dried biscuits over the winter ...

And we knew, it was clear from the person that he would hold back what he was talking about.

Petya Sibiryakov is not cheerful, he also has an explosive inside, but he is too soft: he is sent to Saratov to a trading enterprise. Kuznetsov, the son of a postman, will continue his father's profession in telegraph officials. Kira Tutin must master the "heights of mechanics" - this is his decision. The most calm of all for his fate is Vasya Serov, he will go through life in a straight line, his mind is rosary and tenacious, whoever does not step aside in his path will fall himself; With the logic of naked truths, Vasya will conquer all his infirmities, love, pity, and interplanetary mysteries, close the valves of reason to the past and the future in order to align the present in length and breadth.

Three students were supposed to be kept in a railway school.

And you? - they turn to me. And I don’t know, or, on the contrary, I know my inclination too well, but I don’t have a definite form of action, I sense the roundabout paths that lie ahead of me, I don’t even know if there is a suitable school for me, and what to call the vehicle. what I would like to do - after all, I was a pioneer in Khlynovsk, who opened a new occupation.

Our circle is calloused, tell him a clear lesson. He will not be afraid of the blackness of the work, he will not laugh at it, only so that there is no distant transmission in the work and so that its usefulness is justified. How was I to justify the artist's occupation?

I'll also go to the train! - I express to my comrades the decision that has just ripened in me, - I had to start life somewhere and not interrupt the teachings.

Of the graduates, we had two horse breeders - Serov and Tutin.

Serov went through the whole school with grades. He had no fantasy of play and pranks. The whole educational material he knew from now and to this day. He did not refuse his comrades who came running to him for help, but it seemed to him so unnatural not to know anything that his help seemed arrogant and always slightly pricked the pride of the one who came to her.

Vasya's big head with black eyes, which, combined with a grimace at the corner of his mouth, seemed mocking and unkind, this head, which threw out certain, school truths, was the object of many observations for me. I was cold to him, but I could not help but admire his brain box, in which the formulas of mathematics, the vocation of the Varangians, and the catechism were so tightly packed. Answering the lesson in a low, sonorous voice, Serov, as it were, ordered the square of the hypotenuse to be built with legs, Rurik, Sineus and Truvor unquestioningly came to own Russia, members of the symbol of faith typed inevitability with stone slabs.

Within me there was a disagreement with such tyrannical unconditionality, but I could not resist his conclusions.

Well, this smart Vaska, - said the laughing Grisha Yurkin, - good, by God, he will get through to the police station!

I don’t know if extremes or straightforwardness converge, but our teacher of the law could not get enough of Serov. He used to scold him a lesson, and the protopope gently smoothes the folds of his robes and dak in his beard and sighs, and dissolves in Vasya's eloquence from his own tongue-tied tongue.

That would be a bishop, but one of his own, - apparently, the priest dreamed.

And it happened that after the lesson, the teacher of the law called Vasya into solitude and convinced the young man to choose an appropriate career.

Again he called to the clergy, - Serov answered our questions.

Grisha Yurkin was asleep and saw himself as a priest.

Oh you, there you are, oh you! .. - Yurkin gasped in earnest over his dream. - Why is this long-maned not inviting me? After all, Vaska by heart, and I know the church by heart ... Wait, I'll tell him a lesson ... Serov's fancy is useless, but I'm trying to talk about my rootless orphan ... Oh, I would have fed my mother with spies in sour cream! ..

It must be said that the absurd Grisha knew the holy calendar and the catechism very well, but his misfortune was the everlasting speculation; he confused words by consonance - his spies ripened in the manure, the mushrooms betrayed their homeland. And so, when, at the very first lesson, the one who dreamed of a clerical title suggested answering the service, we listened with pleasure to Grishino's presentation. He even managed to avoid words that confused him. The archpriest also got on his guard in an amicable way and asked the last question about "the proskomedia of previously consecrated gifts," and Yurkin began to answer him clearly, without hesitation, that your Serov was:

Microscopy of lethargy, of sated gifts is performed ...

The teacher of law with fists rushed at the poor young man:

Devilish balda! Shut the darkened throat! Laughing Grisha was about to snort at the shaking of Protopopov's beard, but then he became very indignant.

So, to spite you, I’ll prove it, a kind of propag-pop! .. - he threatened after the departing one.

And what, Yurkin nevertheless became a priest in the village of Levitin. The peasants, they say, loved the cheerful, simple priest, and if not for the vodka, to which Grisha indulged himself immensely, perhaps he would have stepped behind the archpriest. But one day, during mass, the green serpent seemed to him coming from the kliros. Yurkin threw the censer into the snake's mouth and swore obscenely at the horrified crowd of parishioners.

Grisha died in an insane asylum in the province.

The second horse breeder was Tutin. If Serov accumulated knowledge, then Kira passed it through himself - others learned easily from him. I have already talked about him in Khlynovsk and outlined his influence on me.