supervisor

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 first to senior grades. Many students find it difficult, uninteresting and unnecessary. But if you look beyond the pages of the textbook, read additional literature, mathematical sophisms and paradoxes, your idea of ​​mathematics will change, and you will have a desire to study more than is studied in the school mathematics course.

Goal of the work:

show that the existence of impossible figures expands horizons, develops spatial imagination, and 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 impossible triangle does not exist on the plane.

3. Make a development of an impossible triangle.

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

Introduction

Historically, mathematics has played important role in the visual arts, particularly in perspective painting, which involves realistically depicting a three-dimensional scene on a flat canvas or sheet of paper. According to modern views, mathematics and art disciplines very distant from each other, the first is analytical, the second is emotional. Mathematics does not play an obvious role in most jobs contemporary art, and, in fact, many artists rarely or never even use perspective. However, there are many artists whose focus is on mathematics. Several significant figures in the visual arts paved the way for these individuals.

In fact, there are no rules or restrictions on use various topics in mathematical art, such as impossible figures, Möbius strips, distortion or unusual perspective systems, and fractals.

History of impossible figures

Impossible figures are a certain type of mathematical paradox, consisting of regular parts connected in an irregular complex. If we try to formulate a 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.

Errors in spatial construction were encountered by artists even a thousand years ago. But the Swedish artist Oscar Reutersvärd, who painted in 1934, is rightfully considered to be the first to construct and analyze impossible objects. the first impossible triangle, consisting of nine cubes.

Reutersvaerd's triangle

Independent of Reuters, English mathematician and physicist Roger Penrose rediscovers the impossible triangle and publishes its image in a British psychology journal in 1958. The illusion uses “false perspective.” Sometimes this perspective is called Chinese, since a similar method of drawing, when the depth of the drawing is “ambiguous,” was 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 passes further and ends up back at the starting point. Essentially, this is an image of a perpetual motion machine, but any attempt to actually build this structure is doomed to failure.

Another example of impossible figures is presented in the drawing “Moscow”, which depicts an unusual diagram of the Moscow metro. At first we perceive the image as a whole, but when we trace the individual lines with our gaze, we become convinced of the impossibility of their existence.

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

The “Three Snails” drawing continues the tradition of the second famous impossible figure - the impossible cube (box).

"Three Snails" Impossible Cube

A combination of various objects can also be found in the not entirely serious drawing “IQ” (intelligence quotient). Interestingly, some people do not perceive impossible objects because their minds are unable to identify flat pictures with three-dimensional objects.

Donald Simanek has suggested 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” math games». Modern science speaks of a 7-dimensional or 26-dimensional model of the world. Such a world can only be modeled using mathematical formulas; humans simply cannot imagine it. This is where impossible figures come in handy.

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

Impossible trident

"Devil's Fork"

There is another group of objects that cannot be implemented. The classic figure is the impossible trident, or "devil's fork". If you carefully study the picture, you will notice that three teeth gradually turn into two on a single base, 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 we close the upper part of the trident with our hand, we will see completely 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, it can be seen that the front and background of this picture conflict. That is, what was originally on foreground goes back, and the back (middle tooth) comes forward. In addition to the change of foreground and background, there is another effect in this drawing - the flat edges of the upper part of the trident become round at the bottom.

Main part.

Triangle- a figure consisting of 3 adjacent parts, which, through unacceptable connections of these parts, creates the illusion of a mathematically impossible structure. This three-beam structure is also called differently square Penroses

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

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

· Draw parallels to the sides inside the trimmed triangle;

· Trim the corners again;

· Draw parallels inside again;

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

· Continue it with an L-shaped “thing”;

· Run this design in a circle.

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

Development of an impossible triangle.

Inflection line

Cut line

What elements are used to construct an impossible triangle? More precisely, from what elements does it seem to us (precisely it seems!) 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.

Now let’s try to look at the figure from different points in space (or make a real wire model). Imagine what it looks like from one point, from another, from a third... When the observation point changes (or - which is the same thing - when the structure is rotated in space), it will seem that the two “end” edges of our corners are moving relative to each other. It is not difficult to choose a position in which they will connect (of course, the near corner will seem thicker to us than the longer one).

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.

By the way, if we simultaneously look at the display 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 the point of observation so that this illusion (in fact, it is an 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 (that is, design) and drew attention to some of the uncertainty of design - an open structure of three corners can be perceived as a closed circuit.

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

We were once again convinced that from a projection onto a plane, from a drawing, a three-dimensional figure is reconstructed ambiguously. That is, the projection contains some ambiguity, understatement, which gives rise to the “impossible triangle.”

And we can say that the “impossible triangle” of the Penroses, like many other optical illusions, is on a par with logical paradoxes and puns.

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.

It is extremely easy to prove that an impossible triangle does not exist, because each of its angles is right, and their sum is 2700 instead of the “positioned” 1800.

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

Let's consider another triangle, which consists of several parts. If the parts of which it consists are arranged differently, you will get exactly the same triangle, but with one small flaw. One square will be missing. How is this possible? Or is it still an illusion?

https://pandia.ru/text/80/021/images/image016_2.jpg" alt="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 the peculiarities of human perception come to the rescue. Psychologists have found that the eye begins to examine an object (picture) from the lower left corner, then the gaze slides to the right to the center and descends to the lower right corner of the picture. This trajectory may be due to the fact that our ancestors, when meeting an 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 constructed. This feature was clearly manifested in the Middle Ages in the manufacture of tapestries: their design was mirror image original, and the impression produced 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 of receiving interesting compositions using computer technology or from several pictures rotated (maybe using various types symmetries) one relative to the other, creating in viewers a different impression of the object and a deeper understanding of the essence of the design, or from one that rotates (constantly or jerkily) using a simple mechanism at certain angles.

This direction can be called polygonal (polygonal). The illustrations show images rotated relative to each other. The composition was created as follows: a drawing on paper, made in ink and pencil, was scanned, converted into digital form and processed in graphic editor. A regularity can be noted - 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 strives to create the “correct” image.

Conclusion

Using various mathematical figures and laws is not limited to the above examples. By carefully studying all the given figures, you can find others not mentioned in this article. geometric bodies or visual interpretation mathematical laws.

Mathematical fine arts are flourishing today, and many artists create paintings in Escher's style and in their own style. These artists work in various directions, including sculpture, painting on flat and three-dimensional surfaces, lithography and computer graphics. And the most popular topics in mathematical art remain polyhedra, impossible figures, Möbius strips, distorted perspective systems and fractals.

Conclusions:

1. So, consideration of impossible figures develops our spatial imagination, helps us “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 performing this work

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

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

3. Examined reproductions of artists depicting these figures

4. My classmates were interested in my research.

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

LITERATURE

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

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

3. Website of V. Alekseev Illusions · 7 Comments

4. J. Timothy Unrach. – Amazing figures.
(AST Publishing House LLC, Astrel Publishing House LLC, 2002, 168 p.)

5. . - Graphic arts.
(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)

Several impossible figures have been invented - a ladder, a triangle and an x-prong. These figures are actually quite real in a three-dimensional image. But when an artist projects volume onto paper, the objects seem impossible. The triangle, which is also called the “tribar,” has become a wonderful example of how the impossible becomes possible when you put in the effort.

All these figures are beautiful illusions. The achievements of human genius are used by artists who paint in the imp art style.

Nothing is impossible. This can be said about the Penrose triangle. This is a geometrically impossible figure, the elements of which cannot be connected. After all, the impossible triangle became possible. Swedish painter Oscar Reutersvärd introduced the world to the impossible triangle made of cubes in 1934. O. Reutersvard is considered the discoverer of this visual illusion. In honor of this event on postage stamp Sweden later published this drawing.

And in 1958, mathematician Roger Penrose published a publication in an English magazine about impossible figures. It was he who created the scientific model of illusion. Roger Penrose was an incredible scientist. He conducted research in the theory of relativity, as well as the fascinating quantum theory. He was awarded the Wolf Prize together with S. Hawking.

It is known that the artist Maurits Escher, under the impression of this article, painted his amazing work - the lithograph “Waterfall”. But is it possible to make a Penrose triangle? How to do it, if possible?

Tribar and reality

Although the figure is considered impossible, making a Penrose triangle with your own hands is as easy as shelling pears. It can be made from paper. Origami lovers simply could not ignore the tribar and nevertheless found a way to create and hold in their hands a thing that previously seemed beyond the imagination of a scientist.

However, we are deceived by our own eyes when we look at the projection of a three-dimensional object from three perpendicular lines. The observer thinks he sees a triangle, although in fact he does not.

Geometry crafts

The tribar triangle, as stated, is not actually a triangle. The Penrose triangle is an illusion. Only at a certain angle does an object look like an equilateral triangle. However, the object in in kind- these are 3 faces of the cube. In such an isometric projection, 2 angles coincide on the plane: the one closest to the viewer and the farthest.

The optical illusion, of course, quickly reveals itself as soon as you pick up this object. The shadow also reveals the illusion, since the shadow of the tribar clearly shows that the angles do not coincide in reality.

Tribar made of paper. Scheme

How to make a Penrose triangle with your own hands from paper? Are there any schematics for this model? Today, 2 layouts have been invented in order to fold such an impossible triangle. Basic geometry tells you exactly how to fold an object.

To fold a Penrose triangle with your own hands, you will need to allocate only 10-20 minutes. You need to prepare glue, scissors for several cuts and paper on which the diagram is printed.

From such a blank the most popular impossible triangle is obtained. The origami craft is not too difficult to make. Therefore, it will definitely work out the first time, even for a schoolchild who has just started studying geometry.

As you can see, it turns out to be a very nice craft. The second piece looks different and folds differently, but the Penrose triangle itself ends up looking the same.

Steps to create a Penrose triangle from paper.

Choose one of 2 blanks convenient for you, copy the file and print. Here we give an example of the second layout model, which is a little simpler.

The “Tribar” origami blank itself already contains all the necessary tips. In fact, instructions for the circuit are not required. It is enough just to download it onto a thick paper medium, otherwise it will be inconvenient to work and the figure will not work out. If you cannot immediately print on cardboard, then you need to attach the sketch to the new material and cut out the drawing along the contour. For convenience, you can fasten with paper clips.

What to do next? How to fold a Penrose triangle with your own hands step by step? You need to follow this action plan:

1. Let's direct reverse side scissors those lines where you need to bend, according to the instructions. Bend all the lines
2. We make cuts where necessary.
3. Using PVA, we glue together those scraps that are intended to hold the part together into a single whole.

The finished model can be repainted in any color, or you can take colored cardboard for work in advance. But even if the object is made of white paper, all the same, everyone who enters your living room for the first time will certainly be discouraged by such a craft.

Triangle drawing

How to draw a Penrose triangle? Not everyone likes to do origami, but many people love to draw.

To begin with, draw a regular square of any size. Then a triangle is drawn inside, the base of which is the bottom side of the square. A small rectangle fits into each corner, all sides of which are erased; Only those sides that are adjacent to the triangle remain. This is necessary to ensure that the lines are straight. The result is a triangle with truncated corners.

The next stage is the image of the second dimension. A strictly straight line is drawn from the left side of the upper lower corner. The same line is drawn starting from the lower left corner, and is slightly not brought to the first line of the 2nd dimension. Another line is drawn from the right corner parallel to the bottom side of the main figure.

The final stage is to draw the third one inside the second dimension using three more small lines. Small lines start from the lines of the second dimension and complete the image of a three-dimensional volume.

Other Penrose figures

Using the same analogy, you can draw other shapes - a square or a hexagon. The illusion will be maintained. But still, these figures are no longer so amazing. Such polygons simply appear to be very twisted. Modern graphics allows you to make more interesting versions of the famous triangle.

In addition to the triangle, the Penrose Staircase is also world famous. The idea is to trick the eye, making it appear that a person is continuously rising upward when moving clockwise, and downwards when moving counterclockwise.

The continuous staircase is best known for its association with M. Escher’s painting “Ascent and Descend”. It is interesting that when a person walks all 4 flights of this illusory staircase, he invariably ends up back where he started.

There are also other objects known that mislead the human mind, such as the impossible block. Or a box made according to the same laws of illusion with intersecting edges. But all these objects have already been invented based on an article by a remarkable scientist - Roger Penrose.

Impossible triangle in Perth

The figure named after the mathematician is honored. A monument was erected to her. In 1999, in one of the cities of Australia (Perth), a large Penrose triangle made of aluminum was installed, which is 13 meters in height. Tourists enjoy taking pictures next to the aluminum giant. But if you choose a different angle for photography, the deception becomes obvious.

Dmitry Rakov

Our eyes cannot know
the nature of objects.
So don’t force it on them
delusions of reason.

Titus Lucretius Carus

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 occurs not because of what we see, but because we unconsciously reason and involuntarily get mistaken: “the mind can look at the world through the eye, and not with the eye.”

One of the most spectacular areas of 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 that are impossible to exist in the real three-dimensional world. The classic and one of the simplest figures is the impossible triangle.

In an impossible triangle, each angle is itself possible, but a paradox arises when we consider it as a whole. The sides of the triangle are directed both towards and away from the viewer, so its individual parts cannot form a real three-dimensional object.

Strictly speaking, 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 located. Our ideas about the real world face a contradiction, some 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;
• external lines are considered as the boundary of the form. This outer boundary is extremely important for constructing a complete image.

Human consciousness first creates a general image of an object, and then examines individual parts. Each angle is compatible with 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

Errors in spatial construction were encountered by artists even a thousand years ago. But the first to construct 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 (mascara, pencil), 50x70 cm, 2003

Independent of Reuters, English mathematician and physicist Roger Penrose rediscovers the impossible triangle and publishes an image of it in a British psychology journal in 1958. The illusion uses “false perspective.” Sometimes this perspective is called Chinese, since a similar method of drawing, when the depth of the drawing is “ambiguous,” was often found in the works of Chinese artists.

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

The "Three Snails" drawing continues the tradition of the second famous impossible figure - the impossible cube (box).

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

A combination of various objects can also be found in the not entirely serious drawing “IQ” (intelligence quotient). Interestingly, some people do not perceive impossible objects because their minds are unable to identify flat pictures with three-dimensional objects.

Donald E. Simanek has suggested 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 only be modeled using mathematical formulas; humans simply cannot imagine it. This is where impossible figures come in handy. From a philosophical point of view, they serve as a reminder that any phenomena (in system analysis, science, politics, economics, etc.) should be considered in all complex and non-obvious relationships.

A variety of impossible (and possible) objects are presented in the painting "Impossible Alphabet".

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

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

If you carefully study the picture, you will notice that three teeth gradually turn into two on a single base, 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 greater benefit from impossible drawings than mind games? Some hospitals specially hang images impossible objects, since examining them can occupy patients for a long time. It would be logical to hang such drawings at ticket offices, police stations and other places where waiting in line sometimes lasts an eternity. The drawings could act as sort of “chronophages”, i.e. time wasters.

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 new type an optical illusion 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 that's just 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 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 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's remember the song about movement from the wonderful vocal cycle Franz Schubert's "The Beautiful Miller's Wife":

And first in the imagination, and then at hand wonderful master bare and dry structures are transformed into aqueducts through which clean and fast streams of water flow. 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 great 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 Great master 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, studying at 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 performed 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 was unable to complete his studies. 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 drew many sketches mountain landscapes, which later formed the basis of his work. 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 complicated than what Escher had used before and required more time to create a picture, the results were impressive - fine lines and accurate shadow rendering. One of the most famous works using the intaglio printing technique "Dew Drop" was completed in 1948.

In 1950, Moritz Escher gained popularity as a lecturer. At the same time, in 1950, its first personal exhibition in the United States and people are starting to buy his work. 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 new series Escher's operations moved to new house in Laren, which had a studio, but poor health made it impossible to work 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 language and was very pleased with it.

Various impossible pictures can be found on the websites of mathematicians and programmers. Most full version of the ones we looked at, in our opinion, is the site of Vlad Alekseev

This site presents not only a wide range of 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.