How to make an infinite triangle out of paper. Graphic illusions: Impossible and inverted figures

The impossible triangle is one of the amazing mathematical paradoxes. When you first look at it, you cannot doubt for a second its real existence. However, this is only an illusion, a deception. And the very possibility of such an illusion will be explained to us by mathematics!

Opening of the Penroses

In 1958, the British Journal of Psychology published an article by L. Penrose and R. Penrose, in which they introduced a new type of optical illusion, which they called the “impossible triangle.”

A visually impossible triangle is perceived as a structure that actually exists in three-dimensional space, made up of rectangular bars. But this is just an optical illusion. It is impossible to build a real model of an impossible triangle.

The Penroses' article contained several options for depicting an impossible triangle. - his “classic” presentation.

What elements are used to construct an impossible triangle?

More precisely, from what elements does it seem to us to be built? The design is based on a rectangular corner, which is obtained by connecting two identical rectangular bars at right angles. Three such corners are required, and therefore six pieces of bars. These corners must be visually “connected” to one another in a certain way so that they form a closed chain. What happens is an impossible triangle.

Place the first corner in the horizontal plane. We will attach a second corner to it, directing one of its edges upward. Finally, we attach a third corner to this second corner so that its edge is parallel to the original horizontal plane. In this case, the two edges of the first and third corners will be parallel and directed in different directions.

If we consider a bar to be a segment of unit length, then the ends of the bars of the first corner have coordinates, and, the second corner - , and, the third - , and. We got a “twisted” structure that actually exists in three-dimensional space.

Now let’s try to mentally look at it from different points in space. Imagine what it looks like from one point, from another, from a third. As the viewing point changes, the two “end” edges of our corners will appear to move relative to each other. It is not difficult to find a position in which they will connect.

But if the distance between the ribs is much less than the distance from the corners to the point from which we view our structure, then both ribs will have the same thickness for us, and the idea will arise that these two ribs are actually a continuation of one another. This situation is depicted 4.

By the way, if we simultaneously look at the reflection of the structure in the mirror, we will not see a closed circuit there.

And from the chosen observation point we see with our own eyes the miracle that has happened: there is a closed chain of three corners. Just do not change your point of observation so that this illusion does not collapse. Now you can draw an object that you can see or place a camera lens at the found point and get a photograph of an impossible object.

The Penroses were the first to become interested in this phenomenon. They took advantage of the possibilities that arise when mapping three-dimensional space and three-dimensional objects onto a two-dimensional plane and drew attention to some of the design uncertainty - an open structure of three corners can be perceived as a closed circuit.

Proof of the impossibility of the Penrose triangle

By analyzing the features of a two-dimensional image of three-dimensional objects on a plane, we understood how the features of this display lead to an impossible triangle. Perhaps someone will be interested in a purely mathematical proof.

It is extremely easy to prove that an impossible triangle does not exist, because each of its angles is right, and their sum is equal to 270 degrees instead of the “positioned” 180 degrees.

Moreover, even if we consider an impossible triangle glued together from angles less than 90 degrees, then in this case we can prove that an impossible triangle does not exist.

We see three flat edges. They intersect in pairs along straight lines. The planes containing these faces are orthogonal in pairs, so they intersect at one point.

In addition, the lines of mutual intersection of the planes must pass through this point. Therefore, straight lines 1, 2, 3 must intersect at one point.

But that's not true. Therefore, the presented design is impossible.

"Impossible" art

The fate of this or that idea - scientific, technical, political - depends on many circumstances. And not least of all, it depends on the exact form in which this idea will be presented, in what form it will appear to the general public. Will the embodiment be dry and difficult to perceive, or, conversely, the manifestation of the idea will be bright, capturing our attention even against our will.

The impossible triangle has a happy fate. In 1961, the Dutch artist Moritz Escher completed a lithograph he called Waterfall. The artist has come a long but fast way from the very idea of ​​an impossible triangle to its stunning artistic embodiment. Let us remember that the Penroses' article appeared in 1958.

"Waterfall" is based on the two impossible triangles shown. One triangle is large, with another triangle located inside it. It may seem that three identical impossible triangles are depicted. But this is not the point; the presented design is quite complex.

At a quick glance, its absurdity will not be immediately visible to everyone, since every connection presented is possible. as they say, locally, that is, in a small area of ​​the drawing, such a design is feasible... But in general it is impossible! Its individual pieces do not fit together, do not agree with each other.

And to understand this, we must expend certain intellectual and visual efforts.

Let's take a journey through the facets of the structure. This path is remarkable in that along it, as it seems to us, the level relative to the horizontal plane remains unchanged. Moving along this path, we neither go up nor go down.

And everything would be fine, familiar, if at the end of the path - namely at the point - we would not discover that, relative to the initial, starting point, we had somehow risen up vertically in some mysterious, inconceivable way!

To arrive at this paradoxical result, we must choose exactly this path, and also monitor the level relative to the horizontal plane... Not an easy task. In her decision, Escher came to the aid of...water. Let us recall the song about movement from Franz Schubert’s wonderful vocal cycle “The Beautiful Miller’s Wife”:

And first in the imagination, and then under the hand of a wonderful master, bare and dry structures turn into aqueducts through which clean and fast streams of water run. Their movement captures our gaze, and now, against our will, we rush downstream, following all the turns and bends of the path, fall down with the flow, fall onto the blades of a water mill, then rush downstream again...

We go around this path once, twice, three times... and only then do we realize: moving down, we are somehow fantastically rising to the top! The initial surprise develops into a kind of intellectual discomfort. It seems that we have become the victim of some kind of practical joke, the object of some joke that we have not yet understood.

And again we repeat this path along a strange conduit, now slowly, with caution, as if fearing a trick from the paradoxical picture, critically perceiving everything that happens on this mysterious path.

We are trying to unravel the mystery that amazed us, and we cannot escape from its captivity until we find the hidden spring that lies at its basis and brings the unthinkable whirlwind into non-stop motion.

The artist specifically emphasizes and imposes on us the perception of his painting as an image of real three-dimensional objects. The volumetricity is emphasized by the image of very real polyhedrons on the towers, brickwork with the most accurate representation of each brick in the walls of the aqueduct, and rising terraces with gardens in the background. Everything is designed to convince the viewer of the reality of what is happening. And thanks to art and excellent technology, this goal has been achieved.

When we break out of the captivity in which our consciousness falls, we begin to compare, contrast, analyze, we find that the basis, the source of this picture is hidden in the design features.

And we received one more - “physical” proof of the impossibility of the “impossible triangle”: if such a triangle existed, then Escher’s “Waterfall”, which is essentially a perpetual motion machine, would also exist. But a perpetual motion machine is impossible, therefore, the “impossible triangle” is also impossible. And perhaps this “evidence” is the most convincing.

What made Moritz Escher a phenomenon, a unique one who had no obvious predecessors in art and who cannot be imitated? This is a combination of planes and volumes, close attention to the bizarre forms of the microworld - living and inanimate, to unusual points of view on ordinary things. The main effect of his compositions is the effect of the appearance of impossible relationships between familiar objects. At first glance, these situations can both frighten and make you smile. You can joyfully look at the fun that the artist offers, or you can seriously plunge into the depths of dialectics.

Moritz Escher showed that the world may be completely different from how we see it and are used to perceiving it - we just need to look at it from a different, new angle!

Moritz Escher

Moritz Escher was luckier as a scientist than as an artist. His engravings and lithographs were seen as keys to the proof of theorems or original counterexamples that defied common sense. At worst, they were perceived as excellent illustrations for scientific treatises on crystallography, group theory, cognitive psychology or computer graphics. Moritz Escher worked in the field of relationships between space, time and their identity, using basic mosaic patterns and applying transformations to them. This is a great master of optical illusions. Escher's engravings depict not the world of formulas, but the beauty of the world. Their intellectual makeup is radically opposed to the illogical creations of the surrealists.

Dutch artist Moritz Cornelius Escher was born on June 17, 1898 in the province of Holland. The house where Escher was born is now a museum.

Since 1907, Moritz has been studying carpentry and playing the piano and studying in high school. Moritz's grades in all subjects were poor, with the exception of drawing. The art teacher noticed the boy's talent and taught him to make wood engravings.

In 1916, Escher completed his first graphic work, an engraving on purple linoleum - a portrait of his father G. A. Escher. He visits the studio of the artist Gert Stiegemann, who had a printing press. Escher's first engravings were printed on this press.

In 1918-1919, Escher attended the Technical College in the Dutch town of Delft. He receives a deferment from military service to continue his studies, but due to poor health, Moritz failed to cope with the curriculum and was expelled. As a result, he never received higher education. He studies at the School of Architecture and Ornament in the city of Haarlem. There he takes drawing lessons from Samuel Geserin de Mesquite, who had a formative influence on Escher's life and work.

In 1921, the Escher family visited the Riviera and Italy. Fascinated by the vegetation and flowers of the Mediterranean climate, Moritz made detailed drawings of cacti and olive trees. He sketched many sketches of mountain landscapes, which later formed the basis of his works. Later he would constantly return to Italy, which would serve as a source of inspiration for him.

Escher begins to experiment in a new direction for himself; even then, mirror images, crystalline figures and spheres are found in his works.

The end of the twenties turned out to be a very fruitful period for Moritz. His work was shown at many exhibitions in Holland, and by 1929 his popularity had reached such a level that in one year five solo exhibitions were held in Holland and Switzerland. It was during this period that Escher's paintings were first called mechanical and "logical".

Asher travels a lot. Lives in Italy and Switzerland, Belgium. He studies Moorish mosaics, makes lithographs and engravings. Based on travel sketches, he creates his first picture of the impossible reality, Still Life with Street.

At the end of the thirties, Escher continued experiments with mosaics and transformations. He creates a mosaic in the form of two birds flying towards each other, which formed the basis of the painting “Day and Night”.

In May 1940, the Nazis occupied Holland and Belgium, and on May 17, Brussels entered the occupation zone, where Escher and his family lived at that time. They find a house in Varna and move there in February 1941. Asher will live in this city until the end of his days.

In 1946, Escher began to become interested in intaglio printing technology. And although this technology was much more complex than what Escher had used before and required more time to create a picture, the results were impressive - fine lines and accurate rendering of shadows. One of the most famous works in the intaglio printing technique, “Dew Drop,” was completed in 1948.

In 1950, Moritz Escher gained popularity as a lecturer. Then, in 1950, his first personal exhibition took place in the United States and his works began to be bought. On April 27, 1955, Moritz Escher was knighted and became a nobleman.

In the mid-50s, Escher combined mosaics with figures extending into infinity.

In the early 60s, the first book with Escher’s works, Grafiek en Tekeningen, was published, in which 76 works were commented on by the author himself. The book helped gain understanding among mathematicians and crystallographers, including some in Russia and Canada.

In August 1960 Escher gave a lecture on crystallography at Cambridge. The mathematical and crystallographic aspects of Escher's work are becoming very popular.

In 1970, after a new series of operations, Escher moved to a new house in Laren, which included a studio, but poor health prevented him from working much.

In 1971, Moritz Escher died at the age of 73. Escher lived long enough to see The World of M. C. Escher translated into English and was very pleased with it.

Various impossible pictures can be found on the websites of mathematicians and programmers. The most complete version of the ones we have looked at, in our opinion, is Vlad Alekseev’s website

This site presents not only well-known paintings, including those by M. Escher, but also animated images, funny drawings of impossible animals, coins, stamps, etc. This site is alive, it is periodically updated and replenished with amazing drawings.