How to make an infinite triangle out of paper. What you need to know about the Penrose triangle? History of impossible figures

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mathematic teacher

1.Introduction …………………………………………………. …… 3

2. Historical background ……………………………………… ..… 4

3. Main part ………………………………………………… .7

4. Proof of the impossibility of the Penrose triangle ... ... 9

5. Conclusions ……………………………………………… .. ………… 11

6. Literature ……………………………………………. …… 12

Relevance: Mathematics is a subject studied from the first to the final grade. Many students find it difficult, uninteresting, and unnecessary. But if you look behind the pages of a textbook, read additional literature, mathematical sophisms and paradoxes, then the idea of ​​mathematics will change, there will be a desire to study more than is studied in the school mathematics course.

Purpose of work:

show that the existence of impossible figures will broaden the horizons, develop spatial imagination, is used not only by mathematicians, but also by artists.

Tasks :

1. Study the literature on this topic.

2. Consider impossible figures, make a model of an impossible triangle, prove that an impossible triangle does not exist on a plane.

3. Make a sweep of the impossible triangle.

4. Consider examples of the use of the impossible triangle in the visual arts.

Introduction

Historically, mathematics has played an important role in the visual arts, in particular in the depiction of perspective, which implies a realistic depiction of a three-dimensional scene on a flat canvas or sheet of paper. According to modern views, math and art very distant disciplines, the first is analytical, the second is emotional. Mathematics does not play an obvious role in most work contemporary art and, in fact, many artists rarely or never even use perspective. However, there are many artists who focus on mathematics. Several significant figures in the visual arts have paved the way for these individuals.

Generally, there are no rules or restrictions on the use of various topics in the art of mathematics, such as impossible figures, Moebius strip, distortion or unusual perspective systems, and fractals.

History of impossible figures

Impossible figures are a certain kind of mathematical paradox, consisting of regular pieces connected in an irregular complex. If we try to formulate the definition of the term "impossible objects", it would probably sound something like this - physically possible figures assembled in an impossible form. But looking at them is much more pleasant, drawing up definitions.

The errors of spatial construction were encountered among artists a thousand years ago. But the first to build and analyze impossible objects is considered to be the Swedish artist Oscar Reutersvärd, who painted in 1934. the first impossible triangle of nine cubes.

Reuterswärd Triangle

Independently of Reutersward, the English mathematician and physicist Roger Penrose re-opens the impossible triangle and publishes its image in the British Journal of Psychology in 1958. The illusion uses a "false perspective". Sometimes this perspective is called Chinese, as this way of drawing, when the depth of the drawing is "ambiguous", is often found in the works of Chinese artists.

Escher Falls

In 1961. Dutchman M. Escher, inspired by the impossible Penrose triangle, creates the famous lithograph "Waterfall". The water in the picture flows endlessly, after the water wheel it goes further and gets back to the starting point. In fact, this is an image of a perpetual motion machine, but any attempt to build this structure in reality is doomed to failure.

Another example of impossible figures is shown in the figure "Moscow", which depicts an unusual scheme of the Moscow metro. At first, we perceive the whole image, but tracing the individual lines with a glance, we are convinced of the impossibility of their existence.

« Moscow ", graphics (ink, pencil), 50x70 cm, 2003

Drawing "Three snails" continues the tradition of the second famous impossible figure - an impossible cube (box).

"Three snails" Impossible cube

The combination of various objects can be found in the not-so-serious IQ (intelligence quotient) figure. It is interesting that some people do not perceive impossible objects due to the fact that their consciousness is not able to identify flat pictures with three-dimensional objects.

Donald Simanek argued that understanding visual paradoxes is one of the hallmarks of the kind of creativity that the best mathematicians, scientists and artists possess. Many works with paradoxical objects can be attributed to "intellectual mathematical games". Modern science speaks of a 7-dimensional or 26-dimensional model of the world. Such a world can be modeled only with the help of mathematical formulas, a person simply cannot imagine it. And this is where impossible figures come in handy.

The third popular impossible figure is Penrose's Incredible Staircase. You will continuously either ascend (counterclockwise) or descend (clockwise) along it. Penrose model formed the basis famous painting M. Escher "Up and Down" Incredible Penrose Staircase

Impossible trident

"Devil's Fork"

There is one more group of objects that cannot be implemented. The classic figure is the impossible trident, or "devil's fork." Upon closer examination of the picture, you will notice that the three teeth gradually turn into two on a single basis, which leads to a conflict. We compare the number of teeth above and below and come to the conclusion that the object is impossible. If you close it with your hand upper part trident, then we will see completely real picture- three round teeth. If we close the lower part of the trident, then we will also see a real picture - two rectangular teeth. But, if we consider the whole 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) crawls out forward. In addition to changing the foreground and background, this figure has another effect - the flat edges of the upper part of the trident become round in the lower part.

Main part.

Triangle- a figure consisting of 3 adjoining parts, which, with the help of unacceptable connections of these parts, creates an illusion from a mathematical point of view of an impossible structure. In another way, this three-bar is also called square Penrose

The graphic principle behind this illusion owes its formulation to a psychologist and his son Roger, a physicist. Penruz's square consists of 3 square bars located in 3 mutually perpendicular directions; each connects to the next at a right angle, all of which is placed in three-dimensional space. Here's a simple recipe for how to draw this isometric view of the Penruses' square:

· Cut off the corners of an equilateral triangle along lines parallel to the sides;

· Draw parallels to the sides inside the trimmed triangle;

· Cut the corners again;

· Once again, draw inside the parallel;

· Imagine one of the two possible cubes in one of the corners;

· Continue it with an L-shaped "piece";

· Run this structure in a circle.

· If we had chosen a different cube, then the square would have been “twisted” in the other direction .

Unfold an impossible triangle.

Fold line

Cut line

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

Place the first corner in the horizontal plane. We attach the second corner to it, directing one of its edges up. Finally, add the 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.

Now let's try to soapy look at the figure from different points in space (or make a real model out of wire). Imagine how it looks from one point, from another, from a third ... When you change the observation point (or - what is the same - when you rotate the structure in space), it will seem that the two "end" edges of our corners move relative to each other. It is not difficult to find such a position in which they will join (of course, in this case, the near corner will seem thicker to us than the longer one).

But if the distance between the edges is much less than the distance from the corners to the point from which we are considering our structure, then both edges will have the same thickness for us, and the idea will arise that these two edges are in fact an extension of each other.

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

And from the chosen point of observation, we see with our own eyes a miracle that has happened: there is a closed circuit of three corners. Just do not change the point of view, so that this illusion (in fact, it is an illusion!) Does not collapse. Now you can draw an object visible to you or place a camera lens on the found point and get a photograph of an impossible object.

The Penrose were the first to become interested in this phenomenon. They used the opportunities that arise when mapping three-dimensional space and three-dimensional objects on a two-dimensional plane (that is, during design) and drew attention to some design uncertainty - an open structure from three corners can be perceived as a closed circuit.

As already mentioned, wire can be easily made simplest model, in principle, explaining the observed effect. Take a straight piece of wire and divide it into three equal pieces. Then bend the outer parts so that they form a right angle with the middle part, and rotate about 900 relative to each other. Now turn this figure and observe it with one eye. At some position it will seem to be formed from a closed piece of wire. Turning on the table lamp, you can observe the shadow falling on the table, which also turns into a triangle at a certain position of the figure in space.

However, this design feature can be observed in another situation. If you make a ring of wire, and then spread it apart in different directions, you get one turn of a cylindrical spiral. This loop, of course, is open. But when projecting it onto a plane, you can get a closed line.

Once again, we made sure that the three-dimensional figure is reconstructed ambiguously from the projection onto the plane, from the drawing. That is, the projection contains some ambiguity, understatement, which give rise to the "impossible triangle".

And we can say that Penrose's "impossible triangle", like many other optical illusions, is on a par with logical paradoxes and puns.

Proof of the impossibility of the Penrose triangle

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.

It is extremely easy to prove that the impossible triangle does not exist, because each of its angles is a straight line, and their sum is equal to 2700 instead of the "prescribed" 1800.

Moreover, even if we consider an impossible triangle glued from corners less than 900, then in this case we can prove that an impossible triangle does not exist.

Consider another triangle, which consists of several parts. If the parts of which it consists are arranged differently, then you get exactly the same triangle, but with one small flaw. One square will not be enough. How is this possible? Or is it an illusion.

https://pandia.ru/text/80/021/images/image016_2.jpg "alt =" (! LANG: Impossible triangle" width="298" height="161">!}

Using the phenomenon of perception

Is there any way to enhance the effect of impossibility? Are some objects "more impossible" than others? And here features come to the rescue human perception... Psychologists have found that the eye begins to examine the object (picture) from the lower left corner, then the gaze slides to the right to the center and goes down to the lower right corner of the picture. Such a trajectory is possibly due to the fact that our ancestors, when meeting with the enemy, first looked at the most dangerous right hand, and then the gaze moved to the left, to the face and figure. Thus, artistic perception will significantly depend on how the composition of the picture is built. This feature in the Middle Ages was clearly manifested in the manufacture of tapestries: their drawing was a mirror image of the original, and the impression made by tapestries and originals differs.

This property can be successfully used when creating creations with impossible objects, increasing or decreasing the "degree of impossibility". The prospect also opens up of obtaining interesting compositions using computer technology or from several pictures rotated (maybe using of various kinds symmetries) one relative to the other, creating for the audience a different impression of the object and a deeper understanding of the essence of the concept, or from one that turns (constantly or in jerks) with the help of a simple mechanism at some angles.

This direction can be called polygonal (polygonal). The illustrations show images rotated relative to one another. The composition was created as follows: a drawing on paper, made in ink and pencil, was scanned, digitized and processed in a graphic editor. It is possible to note a regularity - the rotated picture has a greater "degree of impossibility" than the original one. This is easily explained: the artist in the process of work subconsciously seeks to create a "correct" image.

Conclusion

The use of various mathematical figures and laws is not limited to the above examples. Carefully studying all the given figures, you can find others that are not mentioned in this article, geometric bodies or a visual interpretation of mathematical laws.

Mathematical visual arts are thriving today, and many artists create paintings in Escher's style and in their own style. These artists work in a variety of fields, including sculpture, flat and three-dimensional painting, lithography, and computer graphics. And the most popular themes of mathematical art remain polyhedra, impossible figures, Mobius strips, distorted perspective systems and fractals.

Conclusions:

1. So, consideration of impossible figures develops our spatial imagination, helps to "get out" of the plane into three-dimensional space, which will help in the study of stereometry.

2. Models of impossible figures help to consider projections on a plane.

3. Consideration of mathematical sophisms and paradoxes instills interest in mathematics.

When doing this work

1. I learned how, when, where and by whom the impossible figures were first considered, that there are many such figures, these figures are constantly trying to depict artists.

2. Together with my dad I made a model of an impossible triangle, examined its projection onto a plane, saw the paradox of this figure.

3. Considered reproductions of artists, which depict these figures

4. My classmates were interested in my research.

In the future, I will use the knowledge gained in mathematics lessons and I was interested, but are there other paradoxes?

LITERATURE

1. Candidate of Technical Sciences D. RAKOV History of impossible figures

2. Impossible figures.- M .: Stroyizdat, 1990.

3. Alekseeva Illusions · 7 Comments

4. J. Timothy Anrach. - Amazing figures.
(LLC "Publishing house AST", LLC "Publishing house Astrel", 2002, 168 p.)

5. ... - Graphics.
(Art-Rodnik, 2001)

6. Douglas Hofstadter. - Gödel, Escher, Bach: this endless garland. (Publishing house "Bakhrakh-M", 2001)

7. A. Konenko - Secrets of impossible figures
(Omsk: Levsha, 199)

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

Discovery of the Penrose

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, composed of rectangular bars. But it's just optical illusion... It is impossible to build a real model of the impossible triangle.

The Penrose article contained several options for depicting an impossible triangle. - its "classic" presentation.

What are the elements of the 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 joining two identical rectangular bars at a right angle. Three such corners are required, and the bars, therefore, six pieces. These corners must be visually “connected” in a certain way to one another so that they form a closed circuit. What happens is the impossible triangle.

Place the first corner in the horizontal plane. We attach the second corner to it, directing one of its edges up. Finally, add the 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 the 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 how it looks from one point, from another, from a third. When you change the observation point, it will seem that the two "end" edges of our corners move relative to each other. It is not difficult to find such a position in which they will connect.

But if the distance between the edges is much less than the distance from the corners to the point from which we are considering our structure, then both edges will have the same thickness for us, and the idea will arise that these two edges are in fact an extension of each other. This situation is depicted in 4.

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

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

The Penrose were the first to become interested in this phenomenon. They used the opportunities that arise when mapping three-dimensional space and three-dimensional objects on a two-dimensional plane and drew attention to some design uncertainty - an open structure from three corners can be perceived as a closed circuit.

Proof of the impossibility of the Penrose triangle

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 also be interested in a purely mathematical proof.

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

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

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

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

But this is not the case. 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, in what form this idea will be presented, in what image it will appear to the general public. Will the embodiment be dry and difficult to perceive, or, on the contrary, the appearance of the idea will be bright, capturing our attention even against our will.

The impossible triangle has a happy fate. In 1961 g. Dutch artist Moritz Escher completed the lithograph, which he named "Waterfall". The artist has gone a long, but fast way from the very idea of ​​an impossible triangle to its stunning artistic embodiment. Recall that the Penrose article appeared in 1958.

At the heart of "Waterfall" are the two impossible triangles shown. One triangle is large, with another triangle inside it. It may seem that three identical impossible triangles are depicted. But this is not the point, the presented design is rather complicated.

At a cursory glance, its absurdity will not be visible to everyone and not immediately, 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 in order to understand this, we must expend certain intellectual and visual efforts.

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

And everything would be fine, as usual, if at the end of the path - namely at the point - we would not find that relative to the initial, initial point, we somehow mysteriously unthinkable climbed up the vertical!

To arrive at this paradoxical result, we must choose this very path, and even keep an eye on the level relative to the horizontal plane ... Not an easy task. In her decision, Escher came to the rescue ... water. Let us recall the song about movement from the wonderful vocal cycle of Franz Schubert "The Beautiful Miller":

And first in the imagination, and then at the hand of a wonderful master, naked and dry structures turn into aqueducts, along 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, together with the stream we break down, fall on the blades of a water mill, then again rush downstream ...

We go around this path once, twice, three times ... and only then we realize: moving in and out, we somehow fantastic way Let's rise to the top! Initial surprise develops into some kind of intellectual discomfort. It seems that we have become the victim of some kind of practical joke, the object of some kind of joke that we have not yet understood.

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

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

The artist specifically emphasizes, imposes on us the perception of his paintings as images of real three-dimensional objects. Volumetricness is emphasized by the image of quite real polyhedrons on the towers, brickwork with the most accurate representation of each brick in the walls of the aqueduct, 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 free from the captivity into 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 got one more - "physical" proof of the impossibility of the "impossible triangle": if such a triangle existed, then there would be Escher's "Waterfall", which is essentially a perpetual motion machine. But a perpetual motion machine is impossible, therefore, the "impossible triangle" is also impossible. And, perhaps, this "proof" is the most convincing.

What made Moritz Escher a phenomenon, a unique one that had no obvious predecessors in art and which cannot be imitated? This is a combination of planes and volumes, close attention to the bizarre forms of the microcosm - 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. These situations at first glance can both scare and bring a smile. You can joyfully look at the fun that the artist offers, or you can seriously immerse yourself in the depths of dialectics.

Moritz Escher showed that the world can be completely different from what we see and are accustomed to perceive - we just need to look at it from a different, new angle of view!

Moritz Escher

Moritz Escher was more fortunate as a scientist than as an artist. His prints and lithographs were seen as clues to theorem-proving 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, used basic patterns of mosaics, applying transformations to them. This is the great master of optical deception. Escher's engravings do not depict the world of formulas, but the beauty of the world. Their intellectual makeup is fundamentally the opposite of 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 now houses a museum.

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

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

In 1918-1919, Escher attended the Technical College in the Dutch town of Delft. He receives a reprieve from military service to continue his studies, but due to poor health, Moritz could not cope with curriculum, and was expelled. As a result, he never graduated. He studies at the School of Architecture and Ornament in the city of Haarlem, where he takes drawing lessons from Samuel Jeseren de Mesquite, who had a formative influence on the life and work of Escher.

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 drew many sketches mountain landscapes, which later formed the basis of his work. Later, he will constantly return to Italy, which will serve as a source of inspiration for him.

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

The end of the twenties proved 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 five solo exhibitions were held in Holland and Switzerland in one year. It was during this period that Escher's paintings were first called mechanical and "logical".

Escher travels a lot. Lives in Italy and Switzerland, Belgium. He studies Moorish mosaics, makes lithographs, prints. On the basis of travel sketches, he creates his first painting of an impossible reality Still Life with Street.

In the late thirties, Escher continued to experiment 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, where Escher and his family lived, fell into the zone of occupation. They find a house in Varna and move there in February 1941. Until the end of his days, Escher will live in this city.

In 1946, Escher became interested in intaglio printing technology. And although this technology was much more complicated than the one that Escher used before and took more time to create the picture, the results were impressive - thin lines and accurate shadow reproduction. One of the most famous works in gravure printing technique "Dew Drop" was completed in 1948.

In 1950, Moritz Escher gained popularity as a lecturer. At the same time, in 1950, his first solo exhibition was held in the United States and his work began to be bought. On April 27, 1955, Moritz Escher is knighted and becomes a nobleman.

In the mid-50s, Escher combines mosaics with figures that extend to infinity.

In the early 60s, the first book with Escher's works "Grafiek en Tekeningen" was published, in which the author himself commented on 76 works. The book helped gain understanding among mathematicians and crystallographers, including some from 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 new series operations Escher moved to new house in Laren, which had a studio, but poor health made it impossible to work hard.

In 1971, Moritz Escher passed away at the age of 73. Escher lived long enough to see the book "The World of M. K. 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 full version of the ones we looked at, in our opinion, is the site of Vlad Alekseev

This site presents not only widely famous paintings, including 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.

Dmitry Rakov

Our eyes do not know how
the nature of the objects.
Therefore, do not impose on them
delusions of reason.

Titus Lucretius Kar

The common expression "optical illusion" is inherently incorrect. The eyes cannot deceive us, since they are only an intermediate link between the object and the human brain. Optical illusion usually arises not due to what we see, but due to the fact that we unconsciously reason and involuntarily err: "through the eye, and not with the eye, the mind is able to look at the world."

One of the most spectacular trends in the artistic movement of optical art (op-art) is imp-art (impossible art), based on the depiction of impossible figures. Impossible objects are drawings on a plane (any plane is two-dimensional), depicting three-dimensional structures, the existence of which is impossible in the real three-dimensional world. The classical and one of the simplest shapes is the impossible triangle.

In an impossible triangle, every angle is itself possible, but a paradox arises when we consider it in its entirety. The sides of the triangle are directed simultaneously both towards the viewer and away from him, therefore its separate parts cannot form a real three-dimensional object.

In fact, our brain interprets a drawing on a plane as a three-dimensional model. Consciousness sets the "depth" at which each point of the image is. Our ideas about the real world are faced with a contradiction, with a certain inconsistency, and we have to make some assumptions:

• straight 2D lines are interpreted as straight 3D lines;
• 2D parallel lines are interpreted as 3D parallel lines;
• acute and obtuse angles are interpreted as right angles in perspective;
• the outer lines are considered the border of the form. This outer border is extremely important for the construction of a complete image.

Human consciousness first creates a general image of the object, and then examines the individual parts. Each angle is compatible with a spatial perspective, but when reunited, they form a spatial paradox. If you close any of the corners of the triangle, then the impossibility disappears.

History of impossible figures

The errors of spatial construction were encountered among artists a thousand years ago. But the first to build and analyze impossible objects is considered to be the Swedish artist Oscar Reutersvärd, who in 1934 drew the first impossible triangle, consisting of nine cubes.

		"Moscow", graphics (ink, pencil), 50x70 cm, 2003

Independently of Reutersward, the English mathematician and physicist Roger Penrose re-opens the impossible triangle and publishes its image in the British Journal of Psychology in 1958. The illusion uses a "false perspective". Sometimes this perspective is called Chinese, as this way of drawing, when the depth of the drawing is "ambiguous", is often found in the works of Chinese artists.

In the Three Snails drawing, the small and large cubes are not oriented in normal isometric projection. The smaller cube mates with the larger one on the front and back sides, which means, following the three-dimensional logic, it has the same dimensions on some sides as the large one. At first, the drawing seems to be a real representation of a rigid body, but as the analysis progresses, the logical contradictions of this object are revealed.

Drawing "Three snails" continues the tradition of the second famous impossible figure - an impossible cube (box).

"IQ", graphics (ink, pencil), 50x70 cm, 2001		"Up and down", M. Escher

The combination of various objects can also be found in the not-so-serious "IQ" (intelligence quotient) drawing. It is interesting that some people do not perceive impossible objects due to the fact that their consciousness is not able to identify flat pictures with three-dimensional objects.

Donald E. Simanek argued that understanding visual paradoxes is one of the hallmarks of the kind of creativity that the best mathematicians, scientists and artists possess. Many works with paradoxical objects can be classified as "intellectual mathematical games". Modern science speaks of a 7-dimensional or 26-dimensional model of the world. Such a world can be modeled only with the help of mathematical formulas, a person simply cannot imagine it. And this is where impossible figures come in handy. From a philosophical point of view, they serve as a reminder that any phenomena (in systems analysis, science, politics, economics, etc.) should be considered in all complex and non-obvious interrelationships.

Various impossible (and possible) objects are presented in the painting "Impossible Alphabet".

The third popular impossible figure is Penrose's Incredible Staircase. You will continuously either ascend (counterclockwise) or descend (clockwise) along it. Penrose's model formed the basis for the famous painting by M. Escher "Up and Down" ("Ascending and Descending").

There is one more group of objects that cannot be implemented. The classic figure is the impossible trident, or "devil's fork".

Upon closer examination of the picture, you will notice that the three teeth gradually turn into two on a single basis, which leads to a conflict. We compare the number of teeth above and below and come to the conclusion that the object is impossible.

Is there any more significant benefit to impossible drawings than mind games? In some hospitals, images of impossible objects are specially hung up, since their examination can take patients for a long time. It would be logical to hang such drawings at the cash desks, in the police and other places where waiting for their turn sometimes lasts for ages. The drawings could act as such "chronophages", i.e. eaters of time.

Also known as impossible triangle and tribar.

History

This figure gained wide popularity after the publication of an article about impossible figures v British Journal of Psychology English mathematician Roger Penrose v 1958 year... In this article, the impossible triangle was depicted in its most general form - in the form of three beams connected to each other at right angles. Influenced by this article in Dutch artist Maurits Escher created one of his famous lithographs « Waterfall ».

Sculptures

The 13-meter aluminum sculpture of an impossible triangle was erected in 1999 year in the town Perth (Australia)

Deutsches Technikmuseum Berlin February 2008 0004.JPG

The same sculpture when changing the viewpoint

Other figures

While it is quite possible to construct analogs of the Penrose triangle from regular polygons, the visual effect of them is not so impressive. As the number of sides increases, the object appears to be just curved or twisted.

see also

• Three hares (English Three hares )

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Excerpt from Penrose Triangle

Having said everything that was ordered to him, Balashev said that the Emperor Alexander wanted peace, but he would not start negotiations except on the condition that ... Then Balashev hesitated: he remembered the words that Emperor Alexander did not write in the letter, but which without fail ordered to insert into the rescript Saltykov and which he ordered Balashev to hand over to Napoleon. Balashev remembered these words: "until not a single armed enemy remains on Russian land," but some difficult feeling held him back. He could not say these words, although he wanted to do it. He hesitated and said: on condition that the French troops retreat beyond the Niemen.
Napoleon noticed Balashev's embarrassment when uttering the last words; his face trembled, the left calf of his leg began to tremble regularly. Without leaving his place, he began to speak in a voice higher and more hasty than before. During the subsequent speech, Balashev, more than once lowering his eyes, involuntarily observed the quivering of the calf in Napoleon's left leg, which intensified the more he raised his voice.
“I wish peace no less than Emperor Alexander,” he began. - Am I not for eighteen months doing everything to get it? I have been waiting for an explanation for eighteen months. But in order to start negotiations, what is required of me? He said, frowning and making an energetically questioning gesture with his small white and plump hand.
“The retreat of the troops beyond the Niemen, sir,” said Balashev.
- For the Neman? Repeated Napoleon. - So now you want to retreat beyond the Niemen - only the Niemen? - repeated Napoleon, looking directly at Balashev.
Balashev bowed his head respectfully.
Instead of the demand four months ago to retreat from Numbers, now they demanded to retreat only beyond the Niemen. Napoleon turned quickly and began to pace the room.
- You say that they require me to retreat beyond the Niemen to start negotiations; but they demanded of me in exactly the same way two months ago to retreat beyond the Oder and Vistula, and in spite of that, you agree to negotiate.
He silently walked from one corner of the room to another and again stopped opposite Balashev. His face seemed to be petrified in its stern expression, and his left leg trembled even faster than before. Napoleon knew this trembling of his left calf. La vibration de mon mollet gauche est un grand signe chez moi, [The trembling of my left calf is a great sign,] he said later.