This work by Escher depicts a paradox - the falling water of a waterfall drives a wheel that directs the water to the top of the waterfall. The waterfall has the structure of an “impossible” Penrose triangle: the lithograph was created based on an article in the British Journal of Psychology.
The structure is made up of three crossbars stacked on top of each other at right angles. The waterfall in the lithograph works like a perpetual motion machine. Depending on the movement of the eye, it alternately appears that both towers are identical and that the tower on the right is one floor lower than the left tower.
A regular park (or garden; also French or geometric park; sometimes also “garden in a regular style”) is a park that has a geometrically regular layout, usually with pronounced symmetry and regularity of composition. It is characterized by straight alleys, which are axes of symmetry, flower beds, parterres and pools of regular shape, pruning of trees and shrubs giving the plantings a variety of geometric shapes.
“Two Pines and a Flat Distance” (Chinese: 雙松平遠) is a handwritten scroll created around 1310 by the Chinese artist Zhao Mengfu. The scroll depicts a landscape with pine trees, part of which is filled with calligraphy. The work is currently in the collection of the Metropolitan Museum of Art, where the drawing was transferred in 1973.
The game of Chinese chess (French: Le jeu d'échets chinois) - etching by the British engraver John Ingram (English: John Ingram, 1721-1771?, active until 1763) based on a drawing by the French artist Francois Boucher. Depicts supposedly the Chinese national game of Xiangqi (Chinese 象棋, pinyin xiàngqí), in fact a fantasy game (all the pieces in real Xiangqi are checker-shaped).
Diorama (Ancient Greek διά (dia) - “through”, “through” and ὅραμα (horama) - “view”, “spectacle”) - a ribbon-shaped, semicircularly curved pictorial picture with a foreground subject (structures, real and fake items). Diorama is classified as a mass entertainment art, in which the illusion of the viewer’s presence in natural space is achieved through a synthesis of artistic and technical means. If the artist performs a full all-round view, then they speak of a “panorama”.
A snow globe, also called a “glass ball with snow,” is a popular Christmas souvenir in the form of a glass ball containing a certain model (for example, a house decorated for the holiday). When such a ball is shaken, artificial “snow” begins to fall on the model. Modern snow globes are very beautifully decorated; many have a winding mechanism and even a built-in mechanism (similar to that used in music boxes) that plays a New Year's tune.
Constellations are a series of 23 small gouaches by Joan Miró, begun in 1939 in Varengeville-sur-Mer and completed in 1941, between Mallorca and Mont-roig del Camp. The Morning Star, one of the most important works in the series, is preserved by the Joan Miró Foundation. The works were a gift from the artist to his wife; she later donated them to the Foundation.
Astrarium, also called Planetarium, is an ancient astronomical clock created in the 14th century by the Italian Giovanni de Dondi. The appearance of this instrument marked the development in Europe of technologies related to the manufacture of mechanical watch instruments. The Astrarium modeled the Solar System and, in addition to counting time and presenting calendar dates and holidays, showed how the planets moved across the celestial sphere. This was his main task, in comparison with the astronomical clock, the main...
“Regular division of the plane” is a series of woodcuts by the Dutch artist Escher, which he began in 1936. These works are based on the principle of tessellation, in which space is divided into parts that completely cover the plane, without intersecting or overlapping each other.
Kinetic architecture is a branch of architecture in which buildings are designed in such a way that their parts can move relative to each other without disturbing the overall integrity of the structure. In another way, kinetic architecture is called dynamic, and is referred to as the direction of architecture of the future.
Crop circles (English crop circles), or agroglyphs (Port. agroglifos; French agroglyphes; “agro” + “glyphs”) - geoglyphs; geometric patterns in the form of rings, circles and other shapes, formed in the fields with the help of fallen plants. They can be both small and very large, completely visible only from a bird's eye view or from an airplane. They attracted public attention starting in the 1970s and 1980s, when they began to be discovered in large numbers in the south of Great Britain.
Imaginary Prisons, Fantastic Images of Prisons, or Dungeons, are a series of etchings by Giovanni Battista Piranesi, begun in 1745, which become the author's best-known work. Around 1749-1750, 14 sheets were published, and in 1761 the series of engravings was reprinted in 16 sheets. In both editions, the engravings had no titles, but in the second, in addition to reworking, the works received serial numbers. The last edition was published in 1780.
Dance with the Veil (French: Danser avec un voile) is a sculpture by Antoine Emile Bourdelle. Is on permanent exhibition at the Pushkin Museum of Fine Arts. A. S. Pushkin in Moscow. Made of bronze in 1909, size - 69.5 x 26 x 51 cm.
The Bollingen Tower is a structure created by the Swiss psychiatrist and psychologist Carl Gustav Jung. It is a small castle with several towers, located in the town of Bollingen on the shores of Lake Zurich at the mouth of the Obersee River.
Landscape style, unlike the regular one, is as close to nature as possible. It was created in the East and gradually spread throughout the world. China and Japan have always admired the natural beauty of nature, believed that when creating landscapes, it is necessary to proceed from the laws of nature. Only in this case can harmony and balance be achieved. Designing a site in a landscape style requires much less effort compared to the regular style. It does not require special changes to the terrain to create a cascade of waterfalls. You can take advantage of the natural topography of your site and organize a small pond of free outlines in its lowland, surrounding it with a flower bed of unpretentious ornamental plants, and on a hill, arrange an alpine slide, covered with moss and surrounded by river pebbles.
Baroque, as we know, sought to introduce movement into architecture, to create the illusion of movement (“illusory” is typical of Baroque). In the gardening art of the Baroque there was a clear opportunity to move from illusion to real implementation movements in art. Therefore, fountains cascades and waterfalls are a typical phenomenon of Baroque gardens. The water shoots up and, as it were, overcomes the laws of nature. Stump swaying in the wind is also an element of movement in Baroque gardens.
The Japanese have always considered nature to be a divine creation. Since ancient times, they have worshiped its beauty, worshiped mountain peaks, rocks and stones, mighty ancient trees, picturesque ponds and waterfalls. According to the Japanese, the most beautiful areas of the natural landscape are the home of spirits and gods. In the VI-VII centuries. the first artificially created Japanese appear gardens that are miniature imitation of the sea coast, later Chinese-style gardens using stone fountains and bridges became popular. During the Heian era, the shape of the ponds in the palace parks changed. It becomes more whimsical: waterfalls, streams, and fishing pavilions decorate parks and gardens.
The second stage of restoration work lasted from 1945 to 1951. During this time, fountains were restored, the lost decorative sculpture. Finally, on August 26, 1946, it was introduced The Alley of Fountains, Terrace and Italian (“Bowls”) fountains, water cannons and waterfalls of the Grand Cascade are in action. And on September 14, 1947, a fountain with a bronze group “Samson Tearing the Lion’s Mouth” began operating. From 1947 to 1950, decorative parts were made for the Grand Cascade to replace stolen ones: bas-reliefs, herms, mascarons, brackets, monumental statues “Tritons”, “Volkhov”, “Neva”. At the same time, the largest fountains of the Lower Park began to function: “Adam”, “Eve”, Menagernye, Roman, “Nymph”, “Danaida”, the “Golden Mountain” cascade, and the “Umbrella” joker fountain. As a result of the second stage of restoration, the seven fountains of the Monplaisir Garden resumed operation.
In addition, in the park “Golden Gate" there are many other interesting areas: Chalet Park, Shakespeare Garden, Bible Garden, the tallest man-made waterfall in the western United States, the Young Museum of Fine Arts, the magnificent Stribing Arbotherium botanical garden and others.
Landowners of the early 19th century saw the ideal in natural beauty, and therefore decisively changed ponds to lakes, smooth alleys to winding paths, evenly trimmed lawns to lawns, where instead of individual trees with crowns-balls or squares, miniature groves of greenery appeared. Man-made nature was complemented by “almost like real" waterfalls, "medieval" towers,“shepherd’s huts and ruins” are buildings stylized to resemble dilapidation and neglect, made up of assorted (old and new, large and small) parts, covered with creeping greenery for added effect.
Switzerland in literature. Albrecht von Haller (1708-1777) wrote the epic poem "The Alps", the story "The Magic" by Thomas Mann mountain" made Davos famous, and Jean-Jacques Rousseau glorified the beauty of Lake Geneva in his novel “Julia, or the New Heloise.” Thanks to The Notes of Sherlock Holmes, Reichenbach Falls is like the grave of Professor Moriarty.
The book describes the highest mountains and the deepest ocean trenches, the driest deserts and the largest seas, the highest volcanoes and geysers, the deepest abysses and the longest caves, the tallest waterfalls in general the most, the most, the most.
The attractiveness of the trail is associated with the picturesque landscape, the harmonious combination of living and inanimate nature, and the diversity of plant and animal life. world, the originality of particularly attractive objects and natural phenomena (lakes, beautiful streams, rocks, canyons, waterfalls, caves, etc.).
An impossible figure is one of the types of optical illusions, a figure that at first glance seems to be a projection of an ordinary three-dimensional object,
upon careful examination, contradictory connections of the elements of the figure become visible. An illusion is created of the impossibility of the existence of such a figure in three-dimensional space.
The most famous impossible figures are the impossible triangle, the endless staircase and the impossible trident.
Impossible Perrose Triangle
The Reutersvard Illusion (Reutersvard, 1934)
Note also that the change in figure-ground organization made it possible to perceive a centrally located “star.”
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Escher's impossible cube
In fact, all impossible figures can exist in the real world. Thus, all objects drawn on paper are projections of three-dimensional objects, therefore, it is possible to create a three-dimensional object that, when projected onto a plane, will look impossible. When looking at such an object from a certain point, it will also look impossible, but when viewed from any other point, the effect of impossibility will be lost.
A 13-meter sculpture of an impossible triangle made of aluminum was erected in 1999 in Perth (Australia). Here the impossible triangle was depicted in its most general form - in the form of three beams connected to each other at right angles.
Devil's fork
Among all the impossible figures, the impossible trident (“devil’s fork”) occupies a special place. 
If we close the right side of the trident with our hand, we will see a very real picture - three round teeth. If we close the lower part of the trident, we will also see the real picture - two rectangular teeth. But, if we consider the entire figure as a whole, it turns out that three round teeth gradually turn into two rectangular ones.
Thus, you can see that the foreground and background of this drawing are in conflict. That is, what was originally in the foreground goes back, and the background (middle tooth) comes forward. In addition to the change in foreground and background, there is another effect in this drawing - the flat edges of the right side of the trident become round on the left.
The effect of impossibility is achieved due to the fact that our brain analyzes the contour of the figure and tries to count the number of teeth. The brain compares the number of teeth in the figure on the left and right sides of the picture, which gives rise to the feeling that the figure is impossible. If the number of teeth in the figure were significantly greater (for example, 7 or 8), then this paradox would be less pronounced.
Some books claim that the impossible trident belongs to a class of impossible figures that cannot be recreated in the real world. Actually this is not true. ALL impossible figures can be seen in the real world, but they will only look impossible from one single point of view.
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Impossible elephant
How many legs does an elephant have?
Stanford psychologist Roger Shepard used the idea of a trident for his picture of the impossible elephant.
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Penrose staircase(endless staircase, impossible staircase)
The Endless Staircase is one of the most famous classical impossibilities.
It is a design of a staircase in which, if moving along it in one direction (counterclockwise in the picture to the article), a person will endlessly ascend, and if moving in the opposite direction, he will constantly descend.
In other words, we are presented with a staircase that seems to lead up or down, but the person walking along it does not rise or fall. Having completed his visual route, he will find himself at the beginning of the path. If you actually had to walk up those stairs, you would walk up and down them aimlessly an infinite number of times. You can call it an endless Sisyphean task!
Since the Penroses published this figure, it has appeared in print more often than any other impossible object. The “Endless Staircase” can be found in books about games, puzzles, illusions, in textbooks on psychology and other subjects.
"Rise and Descend"
The "Endless Forest" was successfully used by the artist Maurits K. Escher, this time in his enchanting lithograph "Ascent and Descend", created in 1960.
In this drawing, reflecting all the possibilities of the Penrose figure, the very recognizable Endless Staircase is neatly inscribed in the roof of the monastery. Hooded monks continuously move up the stairs in a clockwise and counterclockwise direction. They go towards each other along an impossible path. They never manage to go up or down.
Accordingly, The Endless Staircase has become more often associated with Escher, who redrew it, than with the Penroses, who invented it.
How many shelves are there?
Where is the door open?
Outward or inward?
Impossible figures occasionally appeared on the canvases of past masters, for example, such is the gallows in the painting of Pieter Bruegel (the Elder)
"The Magpie on the Gallows" (1568)
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Impossible Arch
Jos de Mey is a Flemish artist who trained at the Royal Academy of Fine Arts in Ghent (Belgium) and then taught interior design and color to students for 39 years. Since 1968, his focus has been on drawing. He is best known for his meticulous and realistic execution of impossible structures.
The most famous are the impossible figures in the works of the artist Maurice Escher. When examining such drawings, each individual detail seems quite plausible, but when you try to trace the line, it turns out that this line is no longer, for example, the outer corner of the wall, but the inner one.
"Relativity"
This lithograph by the Dutch artist Escher was first printed in 1953.
The lithograph depicts a paradoxical world in which the laws of reality do not apply. Three realities are united in one world, three forces of gravity are directed perpendicular to one another.
An architectural structure has been created, the realities are united by stairs. For people living in this world, but in different planes of reality, the same staircase will be directed either up or down.
"Waterfall"
This lithograph by the Dutch artist Escher was first printed in October 1961.
This work by Escher depicts a paradox - the falling water of a waterfall drives a wheel that directs the water to the top of the waterfall. The waterfall has the structure of an “impossible” Penrose triangle: the lithograph was created based on an article in the British Journal of Psychology.
The structure is made up of three crossbars stacked on top of each other at right angles. The waterfall in the lithograph works like a perpetual motion machine. It also seems that both towers are the same; in fact, the one on the right is one floor below the left tower.
Well, more modern works :o)
Endless photography
Amazing construction site
Chess board
What do you see: a huge crow with prey or a fisherman in a boat, fish and an island with trees?
Rasputin and Stalin
Youth and old age
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Nobleman and Queen
Illusionary works of art have a certain charm. They are a triumph of fine art over reality. Why are illusions so interesting? Why do so many artists use them in their works? Perhaps because they do not show what is actually drawn. Everyone celebrates the lithograph "Waterfall" by Maurits C. Escher. The water circulates here endlessly; after the wheel rotates, it flows further and ends up back to the starting point. If such a structure could be built, then there would be a perpetual motion machine! But upon closer examination of the picture, we see that the artist is deceiving us, and any attempt to build this structure is doomed to failure.
To convey the illusion of three-dimensional reality, two-dimensional drawings (drawings on a flat surface) are used. Usually the deception consists of drawing projections of solid figures, which the person tries to imagine as three-dimensional objects in accordance with his personal experience.
Classical perspective is effective at simulating reality in the form of a “photographic” image. This view is incomplete for several reasons. It does not allow us to see the scene from different points of view, get closer to it, or view the object from all sides. It doesn't give us the effect of depth that a real object would have. The depth effect occurs because our eyes look at an object from two different perspectives, and our brain combines them into one image. A flat drawing represents a scene from only one specific point of view. An example of such a drawing would be a photograph taken using a conventional monocular camera.
When using this class of illusions, the drawing appears at first glance to be an ordinary representation of a solid body in perspective. But upon closer examination, the internal contradictions of such an object become visible. And it becomes clear that such an object cannot exist in reality.
Escher's Falls is based on the Penrose illusion, sometimes called the impossible triangle illusion. Here this illusion is illustrated in its simplest form. 
It seems that we see three square bars connected into a triangle. If you close any corner of this figure, you will see that all three bars are connected correctly. But when you remove your hand from the closed corner, the deception becomes obvious. Those two bars that will connect in this corner should not be even close to each other.
The Penrose illusion uses "false perspective". "False perspective" is also used when constructing isometric images. Sometimes this perspective is called Chinese (translator's note: Reutersvard called this perspective Japanese). This method of painting was often used in Chinese fine arts. With this method of drawing, the depth of the drawing is ambiguous.
In isometric drawings, all parallel lines appear parallel, even if they are inclined with respect to the observer. An object that is tilted at an angle away from the observer appears exactly the same as if it were tilted toward the observer at the same angle. A rectangle bent in half (the figure of Mach) clearly shows such ambiguity. This figure may appear to you to be an open book, as if you are looking at the pages of a book, or it may appear to be a book with its binding turned towards you and you are looking at the cover of a book. This figure may also appear to be two parallelograms superimposed, but very few people will see this figure as parallelograms.
The figure of Thiery illustrates the same duality 
Consider the Schroeder staircase illusion, a "pure" example of isometric depth ambiguity. This figure can be perceived as a staircase that could be climbed from right to left, or as a view of the staircase from below. Any attempt to change the position of the figure's lines will destroy the illusion.
This simple drawing resembles a line of cubes, shown from the outside to the inside. On the other hand, this drawing resembles a line of cubes, shown above and below. But it is very difficult to perceive this drawing as just a series of parallelograms.
Let's paint some areas black. Black parallelograms can look as if we are looking at them either from below or from above. Try, if you can, to see this picture differently, as if we were looking at one parallelogram from below, and at the other from above, alternating them. Most people cannot perceive this picture in this way. Why are we unable to perceive a picture in this way? I believe this is the most complex of the simple illusions.
The picture on the right uses the illusion of an impossible triangle in an isometric style. This is one of the "shading" samples from the AutoCAD(TM) drafting software. This sample is called "Escher".
An isometric drawing of a wire cube structure shows isometric ambiguity. This figure is sometimes called the Necker cube. If the black dot is in the center of one side of the cube, is that side the front side or the back side? You can also imagine that the point is near the bottom right corner of the side, but you still won't be able to tell whether that side is the front side or not. You also have no reason to assume that the point is on the surface of the cube or inside it; it could just as well be in front of the cube or behind it, since we have no information about the actual dimensions of the point.
If you imagine the faces of a cube as wooden planks, you can get unexpected results. Here we used an ambiguous connection of horizontal planks, which will be discussed below. This version of the figure is called the impossible box. It is the basis for many similar illusions.
An impossible box cannot be made of wood. And yet we see here a photograph of an impossible box made of wood. This is a lie. One of the drawer slats that appears to run behind the other is actually two separate slats with a gap, one closer and one further away than the intersecting slats. Such a figure is visible only from a single point of view. If we were looking at a real structure, then with our stereoscopic vision we would see a trick that makes the figure impossible. If we changed our point of view, this trick would become even more noticeable. This is why when impossible figures are shown at exhibitions and museums, you are forced to look at them through a small hole with one eye.
What is this illusion based on? Is it a variation of Much's book?
In fact, it is a combination of the Much illusion and the ambiguous connection of lines. The two books share a common middle surface of the figure. This makes the slant of the book cover ambiguous.
The Poggendorf illusion, or "crossed rectangle", misleads us into which line A or B is a continuation of line C. A definite answer can only be given by applying a ruler to line C and seeing which line coincides with it.
Illusions of shape are closely related to illusions of position, but here the very structure of the design forces us to change our judgment about the geometric shape of the design. In the example below, the short slanted lines create the illusion that the two horizontal lines are curved. In fact, these are straight parallel lines.
These illusions take advantage of our brain's ability to process visual information, including cross-hatched surfaces. One pattern of shading may dominate so much that other elements of the design appear distorted.
A classic example is a set of concentric circles with a square superimposed on them. Although the sides of the square are perfectly straight, they appear to be curved. You can verify that the sides of the square are straight by applying a ruler to them. Most shape illusions are based on this effect.
The following example works on the same principle. Although both circles are the same size, one of them looks smaller than the other. This is one of many size illusions.
An explanation for this effect can be our perception of perspective in photographs and paintings. In the real world, we see two parallel lines converge as the distance increases, so we perceive that the circle touching the lines is further away from us and therefore must be larger.
If the circles and areas bounded by the lines are painted black, the illusion will be weaker.
The width of the brim and the height of the hat are the same, although it does not seem so at first glance. Try rotating the image 90 degrees. Has the effect persisted? This is an illusion of relative sizes within a painting.
Tilted circles are projected onto the plane by ellipses, and these ellipses have depth ambiguity. If the figure (above) is a tilted circle, then there is no way to know whether the upper arc is closer to us or further away from us than the lower arc.
The ambiguous connection of lines is an essential element in the ambiguous ring illusion:
Ambiguous Ring, © Donald E. Simanek, 1996.
If you cover half of the picture, the rest will resemble half of an ordinary ring.
When I came up with this figure, I thought it could be an original illusion. But later I saw an advertisement with the logo of the fiber optic corporation Canstar. Although the Canstar emblem is mine, they can be classified as the same class of illusions. Thus, I and the corporation independently developed the figure of the impossible wheel. I think if you dig deeper, you can probably find earlier examples of the impossible wheel.
Another of Penrose's classic illusions is the impossible staircase. It is most often depicted as an isometric drawing (even in the work of Penrose). Our version of the endless staircase is identical to the Penrose version (except for the shading).
It can also be depicted in perspective, as is done in the lithograph by M. C. Escher.
The deception in the lithograph “Ascent and Descent” is constructed in a slightly different way. Escher placed a staircase on the roof of a building and depicted the building below in such a way as to convey the impression of perspective.
The artist depicted an endless staircase with a shadow. Like shading, a shadow could destroy the illusion. But the artist placed the light source in such a place that the shadow blends well with other parts of the painting. Perhaps the shadow of the stairs is an illusion in itself.
Some people are not at all intrigued by illusory pictures. “It’s just a wrong picture,” they say. Some people, perhaps less than 1% of the population, do not perceive them because their brains are unable to convert flat pictures into three-dimensional images. These people tend to have difficulty understanding technical drawings and illustrations of three-dimensional figures in books.
Others may see that there is “something wrong” with the picture, but they will not think to ask how the deception is achieved. These people never have the need to understand how nature works; they cannot focus on details due to a lack of basic intellectual curiosity.
Perhaps understanding visual paradoxes is one of the hallmarks of the kind of creativity that the best mathematicians, scientists and artists possess. Among the works of M.C. Escher there are many illusion paintings, as well as complex geometric paintings, which can be classified more as “intellectual mathematical games” than art. However, they make an impression on mathematicians and scientists.
It is said that people living on some Pacific island or deep in the Amazon jungle, where they have never seen a photograph, will not be able to initially understand what the photograph represents when shown it to them. Interpreting this particular type of image is an acquired skill. Some people are better at this skill, others worse.
Artists began to use geometric perspective in their works much earlier than the invention of photography. But they could not study it without help from science. Lenses became generally available only in the 14th century. At that time they were used in experiments with darkened chambers. A large lens was placed in a hole in the wall of a darkened chamber so that the inverted image was displayed on the opposite wall. The addition of a mirror allowed the image to be cast from the floor to the ceiling of the chamber. This device was often used by artists who were experimenting with the new "European" perspective style in art. By that time, mathematics was already sophisticated enough to provide a theoretical basis for perspective, and these theoretical principles were published in books for artists.
Only by trying to draw illusory pictures yourself can you appreciate all the subtleties necessary to create such deceptions. Very often the nature of illusion imposes its own limitations, imposing its “logic” on the artist. As a result, the creation of a painting becomes a battle between the artist's wit and the strangeness of an illogical illusion.
Now that we've discussed the nature of some illusions, you can use them to create your own illusions, as well as categorize any illusions you come across. After a while you will have a large collection of illusions, and you will need to demonstrate them in some way. I designed a glass display case for this.
Showcase of illusions. © Donald E. Simanek 1996.
You can check the convergence of the lines in perspective and other aspects of the geometry of this drawing. By analyzing such pictures and trying to draw them, you can find out the essence of the deceptions used in the picture. M. C. Escher used similar tricks in his painting Belvedere (below).
Donald E. Simanek, December 1996. Translation from English
Do science and art have common points of intersection? Can one of these worlds complement and enrich the other with discoveries? The great creators of the Renaissance would not even have seen a contradiction in this formulation of the question. For them, the ways of understanding the world and expressing themselves were not divided as strictly as for us. The works of the Dutch graphic artist Maurits (Maurice) Escher usually have a hypnotic effect on people, because they blur the rigid boundaries in our minds between the logical and the impossible, between the constant and the changing.
In fact, each of the paintings is a scientific and artistic study of the patterns of space and the characteristics of our perception. Experts consider his work in the context of the theory of relativity and psychoanalysis. But you can simply distract yourself for a few minutes and immerse yourself in a world where the clear logic that reigns inside the drawing suddenly turns out to be distorted in relation to our world.
Surreal watercolors created by Spanish artist Borge Sanchez,
