The paradoxical world of impossible objects. Project "impossible figures" Impossible object as a paradox

Many people believe that impossible figures are truly impossible and they cannot be created in real world. However, we know from a school geometry course that a drawing depicted on a sheet of paper is a projection of a three-dimensional figure onto a plane. Therefore, any figure drawn on a piece of paper must exist in three-dimensional space. Moreover, three-dimensional objects, when projected onto a plane, produce a given flat figure of an infinite set. The same applies to impossible figures.

Of course, none of the impossible figures can be created by acting in a straight line. For example, if you take three identical pieces of wood, you will not be able to combine them to form an impossible triangle. However, when projecting a three-dimensional figure onto a plane, some lines may become invisible, overlap each other, join each other, etc. Based on this, we can take three different bars and make the triangle shown in the photo below (Fig. 1). This photograph was created by the famous popularizer of the works of M.K. Escher, the author of a large number of books by Bruno Ernst. In the foreground of the photograph we see the figure of an impossible triangle. There is a mirror in the background, which reflects the same figure from a different point of view. And we see that in fact the figure of an impossible triangle is not a closed, but an open figure. And only from the point from which we view the figure does it seem that the vertical bar of the figure goes beyond the horizontal bar, as a result of which the figure seems impossible. If we shifted the viewing angle a little, we would immediately see a gap in the figure, and it would lose its effect of impossibility. The fact that an impossible figure looks impossible from only one point of view is characteristic of all impossible figures.

Rice. 1. Photograph of an impossible triangle by Bruno Ernst.

As mentioned above, the number of figures corresponding to a given projection is infinite, so the above example is not the only way to construct an impossible triangle in reality. Belgian artist Mathieu Hamaekers created the sculpture shown in Fig. 2. The photo on the left shows a frontal view of the figure, making it look like an impossible triangle, the center photo shows the same figure rotated 45°, and the photo on the right shows the figure rotated 90°.

Rice. 2. Photograph of the impossible triangle figure by Mathieu Hemakerz.

As you can see, in this figure there is no straight lines, all elements of the figure are curved in a certain way. However, as in the previous case, the effect of impossibility is noticeable only at one viewing angle, when all curved lines are projected into straight lines, and, if you do not pay attention to some shadows, the figure looks impossible.

Another way to create an impossible triangle was proposed by the Russian artist and designer Vyacheslav Koleichuk and published in the journal “Technical Aesthetics” No. 9 (1974). All the edges of this design are straight lines, and the edges are curved, although this curvature is not visible in the frontal view of the figure. He created such a model of a triangle from wood.

Rice. 3. Model of the impossible triangle by Vyacheslav Koleichuk.

This model was later recreated by Gershon Elber, a member of the Computer Science Department at the Technion Institute in Israel. Its version (see Fig. 4) was first designed on a computer and then recreated in reality using a three-dimensional printer. If we slightly shift the viewing angle of the impossible triangle, we will see a figure similar to the second photograph in Fig. 4.

Rice. 4. A variant of constructing the impossible triangle by Elber Gershon.

It is worth noting that if we were now looking at the figures themselves, and not at their photographs, we would immediately see that none of the presented figures is impossible, and what is the secret of each of them. We simply would not be able to see these figures because we have stereoscopic vision. That is, our eyes, located at a certain distance from each other, see the same object from two close, but still different, points of view, and our brain, having received two images from our eyes, combines them into a single picture. It was said earlier that an impossible object looks impossible only from a single point of view, and since we view the object from two points of view, we immediately see the tricks with the help of which this or that object was created.

Does this mean that in reality it is still impossible to see an impossible object? No, you can. If you close one eye and look at the figure, it will look impossible. Therefore, in museums, when demonstrating impossible figures, visitors are forced to look at them through a small hole in the wall with one eye.

There is another way by which you can see an impossible figure, with both eyes at once. It consists of the following: it is necessary to create a huge figure the height of a multi-story building, place it in a vast open space and look at it from a very long distance. In this case, even looking at the figure with both eyes, you will perceive it as impossible due to the fact that both your eyes will receive images that are practically no different from each other. Such an impossible figure was created in the Australian city of Perth.

While an impossible triangle is relatively easy to construct in the real world, creating an impossible trident in three dimensions is not so easy. The peculiarity of this figure is the presence of a contradiction between the foreground and background of the figure, when the individual elements of the figure smoothly blend into the background on which the figure is located.

Rice. 5. The design is similar to an impossible trident.

The Institute of Ocular Optics in Aachen (Germany) was able to solve this problem by creating a special installation. The design consists of two parts. In front there are three round columns and a builder. This part is only illuminated at the bottom. Behind the columns there is a half-permeable mirror with a reflective layer located in front, that is, the viewer does not see what is behind the mirror, but sees only the reflection of the columns in it.

Rice. 6. Installation diagram reproducing the impossible trident.

Municipal budgetary educational institution

"Lyceum No. 1"

Research work on the topic

"Impossible Figures"

Completed by: Danil Slinchuk, 6B grade student

Head: mathematics teacher

Kazmenko Elena Alexandrovna

Introduction 3

1. Definition of impossible figures 4

2. Types of impossible figures 8

2.1. Amazing Triangle - Tribar 8

2.2. Endless staircase 9

2.3. Space fork 11

2.4. Impossible boxes 12

3. Application of impossible figures 13

3.1. Impossible figures in icon painting 13

3.2. Impossible figures in architecture and sculpture 15

3.3.Impossible figures in painting 16

3.4.Impossible figures in the philatelist 18

3.5.Impossible figures in design art 19

3.6.Impossible figures in animation 20

3.7.Impossible figures in logos and symbolism 21

4. Creating impossible figures 22

Conclusion 24

References 25

Introduction

Impossible figures have been known almost since the time of cave paintings; their systematic study began only in the middle of the 20th century, that is, almost before our eyes, and before that mathematicians dismissed them as an annoying misunderstanding.

In 1934, Oscar Reutersvard accidentally created his first impossible figure - a triangle made of nine cubes, but instead of correcting something, he began to create others impossible figures one after another.

Even such simple volumetric shapes as a cube, pyramid, parallelepiped can be represented as a combination of several figures located at different distances from the observer’s eye. There should always be a line along which the image of the individual parts combining into complete picture.

An “impossible figure” is a three-dimensional object made on paper that cannot exist in reality, but which, however, can be seen as a two-dimensional image.” This is one of the types of optical illusions, a figure that at first glance seems to be a projection of an ordinary three-dimensional object, upon careful examination of which contradictory connections of the elements of the figure become visible. An illusion is created of the impossibility of the existence of such a figure in three-dimensional space.

Despite a significant number of publications about impossible figures, their clear definition has not been formulated in essence. You can read that impossible figures include all optical illusions associated with the peculiarities of our perception of the world. On the other hand, a person can show you a figure of a green man or with ten arms and five heads and say that all these are impossible figures. At the same time, he will be right in his own way. After all, there are no green people with ten legs. By impossible figures we will understand flat images of figures perceived by a person unambiguously, as they are drawn without the person’s perception of any additional, actually not drawn images or distortions and which cannot be represented in three-dimensional form. The impossibility of representation in three-dimensional form is understood, of course, only directly, without taking into account the possibility of using special means in the manufacture of impossible figures, since an impossible figure can always be made by using an ingenious system of slots, additional supporting elements and bending the elements of the figure, and then photographing it under the right angle

I was faced with the question: “Do impossible figures exist in the real world?”

Project goal:

1. Find out how impossible figures are created and where they are used.

Project objectives:

1. Study literature on the topic “Impossible figures.”

2. Make a classification of impossible figures.

3. Consider ways to construct impossible figures.

4.Create an impossible figure.

The topic of my work is relevant because understanding paradoxes is one of the signs of the type of creative potential that the best mathematicians, scientists and artists possess. Many works with unreal objects can be classified as “intellectual mathematical games”. Such a world can only be modeled using mathematical formulas; humans simply cannot imagine it. And impossible figures are useful for the development of spatial imagination. A person tirelessly mentally creates around himself something that will be simple and understandable for him. He cannot even imagine that some objects around him may be “impossible.” In fact, the world is one, but it can be viewed from different angles.

Definition of impossible figures

There is still no clear definition of impossible figures. I found several different approaches to defining this concept.

An impossible figure is one of the types of optical illusions, a figure that at first glance seems to be a projection of an ordinary three-dimensional object, upon careful examination of which contradictory connections of the elements of the figure become visible.

Impossible figures are geometrically contradictory images of objects that do not exist in real three-dimensional space. Impossibility arises from the contradiction between the subconsciously perceived geometry of the depicted space and formal mathematical geometry.

Impossible figures are divided into two large classes: some have real three-dimensional models, while others cannot be created.

Typically, for a 3D model of an impossible figure to appear impossible, it must be viewed from a specific viewing angle to create the illusion of impossibility.

It is necessary to clarify the difference between the terms "impossible figure", "impossible object" and "three-dimensional model". A three-dimensional model is a physically representable object, when examined in space, all the cracks and bends become visible, which destroy the illusion of impossibility and this model loses its “magic”. When projecting this model onto a two-dimensional plane, an impossible figure is obtained. This impossible figure (as opposed to a three-dimensional model) creates the impression of an impossible object that can only exist in a person’s imagination, but not in space.

Impossible figures are quite often found in ancient engravings, paintings and icons - in some cases we have obvious errors in the transfer of perspective, in others - with deliberate distortions due to artistic design.

We are accustomed to believing photographs (and, to a lesser extent, drawings and drawings), naively believing that they always correspond to some kind of reality (real or fictional). An example of the first is a parallelepiped, the second is an elf or other fairy-tale animal. The absence of elves in the region of space/time we observe does not mean that they cannot exist. They still can (which is easy to verify with the help of plaster, plasticine or papier-mâché). But how to draw something that cannot exist at all?! What can’t be designed at all?!

There is a huge class of so-called “impossible figures”, mistakenly or deliberately drawn with errors in perspective, resulting in funny visual effects that help psychologists understand the principles of the (sub)conscious.

In medieval Japanese and Persian painting, impossible objects are an integral part of the oriental artistic style, which gives only a general outline of the picture, the details of which the viewer “has” to think out independently, in accordance with his preferences.

Paintings with distorted perspective can be found already at the beginning of the first millennium. A miniature from the book of Henry II, created before 1025 and kept in the Bavarian State Library in Munich, depicts a “Madonna and Child” (Fig. 1). The painting depicts a vault consisting of three columns, and the middle column, according to the laws of perspective, should be located in front of the Madonna, but is located behind her, which gives the painting the effect of unreality.

Figure 1. “Madonna and Child”

The article “Putting order in the impossible” (impossible.info/russian/articles/kulpa/putting-order.html) gives the following definition of impossible figures: “An impossible figure is a flat drawing that creates the impression of a three-dimensional object in such a way that the object, proposed by our spatial perception cannot exist, so that the attempt to create it leads to (geometric) contradictions clearly visible to the observer." The Penroses write approximately the same thing in their memorable article: “Each individual part of the figure looks like a normal three-dimensional object, but due to the incorrect connection of the parts of the figure, the perception of the figure completely leads to the illusory effect of impossibility,” but none of them answers the question: why is all this happening?

Meanwhile, everything is simple. Our perception is designed in such a way that when processing a two-dimensional figure that has signs of perspective (i.e. volumetric space), the brain perceives it as three-dimensional, choosing the simplest method of converting 2D to 3D, guided by life experience, and as was shown above, real prototypes of “impossible” figures are rather sophisticated designs with which our subconscious is unfamiliar, but even after becoming familiar with them, the brain still continues to choose the simplest (from its point of view) transformation option and only after Long-term training, the subconscious finally “enters the situation” and the apparent abnormality of the “impossible figures” disappears.

Consider a painting (yes, yes, a painting, not a computer-generated photorealistic drawing) drawn by a Flemish artist named Jos de Mey (Fig. 2). The question is - what physical reality could it correspond to?

At first glance architectural structure seems impossible, but after a moment’s hesitation the consciousness finds a saving option: the brickwork is in a plane perpendicular to the observer and rests on three columns, the tops of which seem to be located at an equal distance from the masonry, but in fact the empty space is simply “concealed” due to “successfully " of the selected projection. After consciousness has “deciphered” the picture, it (and all similar images) is perceived completely normally, and geometric contradictions disappear as imperceptibly as they appeared.

Figure 2. Impossible painting by Jos de Mey

Let's consider famous painting Maurits Escher/Maurits Escher “Waterfall” (Fig. 3) and its simplified computer model (Fig. 4), made in a photorealistic style. At first glance, there are no paradoxes; before us is an ordinary picture depicting... a drawing of a perpetual motion machine!!! But, as you know from a school physics course, a perpetual motion machine is impossible! How did Escher manage to depict in such detail something that could not exist in nature at all?!

Figure 3. Perpetual motion machine in Escher's "Waterfall" engraving.

Figure 4. Computer model of Escher's perpetual motion machine.

When trying to build an engine according to a drawing (or upon careful analysis of the latter), the “deception” immediately emerges - in three-dimensional space such designs are geometrically contradictory and can only exist on paper, that is, on a plane, and the illusion of “volume” is created only due to signs of perspective ( in this case - deliberately distorted) and in a drawing lesson we will easily get two points for such a masterpiece, pointing out errors in the projection.

Types of impossible figures

"Impossible figures" are divided into 4 groups:

An amazing triangle - tribar (Fig. 5).

Figure 5. Tribar

This figure is perhaps the first impossible object published in print. It appeared in 1958. Its authors, father and son Lionell and Roger Penrose, a geneticist and mathematician respectively, defined the object as a "three-dimensional rectangular structure." It was also called "tribar". At first glance, the tribar appears to be simply an image of an equilateral triangle. But the sides converging at the top of the picture appear perpendicular. At the same time, the left and right edges below also appear perpendicular. If you look at each detail separately, it seems real, but, in general, this figure cannot exist. It is not deformed, but the correct elements were incorrectly connected when drawing.

Here are some more examples of impossible figures based on the tribar (Fig. 6-9).

Figure 6. Triple deformed tribar Figure 7. Triangle of 12 cubes

Figure 8. Winged tribar Figure 9. Triple domino

The introduction to impossible figures (especially those performed by Escher) is, of course, stunning, but the fact that any of the impossible figures can be constructed in the real three-dimensional world is perplexing.

As you know, any two-dimensional image is a projection of a three-dimensional figure onto a plane (sheet of paper). There are quite a lot of projection methods, but within each of them the mapping is carried out uniquely, that is, there is a strict correspondence between a three-dimensional figure and its two-dimensional image. However, axonometric, isometric and other popular methods of projection are unidirectional transformations carried out with loss of information, and therefore the inverse transformation can be performed in an infinite number of ways, that is, a two-dimensional image corresponds to an infinite number of three-dimensional figures and any mathematician can easily prove that such a transformation is possible for any two-dimensional image. That is, in fact, there are no impossible figures!

Here's another display from Mathieu Hemakerz. There are many possible reverse mapping options (Fig. 10). Infinitely many!

Figure 10. Penrose triangle from different angles

Endless staircase

This figure is most often called the “Endless Staircase”, “Eternal Staircase” or “Penrose Staircase” - after its creator. It is also called the “continuously ascending and descending path” (Fig. 11).

Figure 11. Endless staircase

This figure was first published in 1958. A staircase appears before us, seemingly leading up or down, but at the same time, the person walking along it does not rise or fall. Having completed his visual route, he will find himself at the beginning of the path.

The “Endless Staircase” was successfully used by the artist Maurits K. Escher, this time in his lithograph “Ascent and Descend”, created in 1960.

Staircase with four or seven steps. The author could have been inspired by a pile of ordinary railroad sleepers to create this figure with a large number of steps. When you are about to climb this ladder, you will be faced with a choice: whether to climb four or seven steps.

The creators of this staircase took advantage of parallel lines to design the end pieces of the equally spaced blocks; Some blocks appear to be twisted to fit the illusion.

Space fork

The next group of figures is collectively called the “Space Fork”. With this figure we enter into the very core and essence of the impossible. Perhaps this is the most numerous class of impossible objects (Fig. 12).

Figure 12. Space fork

This notorious impossible object with three (or two?) teeth became popular with engineers and puzzle enthusiasts in 1964. The first publication dedicated to the unusual figure appeared in December 1964. The author called it a “Brace consisting of three elements.”

From a practical point of view, this strange trident or bracket-like mechanism is absolutely inapplicable. Some simply call it an "unfortunate mistake." One of the representatives of the aerospace industry proposed using its properties in the construction of an interdimensional space tuning fork.

Impossible boxes

Another impossible object appeared in 1966 in Chicago as a result of original experiments by photographer Dr. Charles F. Cochran. Many lovers of impossible figures have experimented with the Crazy Box. The author originally called it the "Free Box" and stated that it was "designed to send impossible objects in large numbers" (Fig. 14).

Figure 14. Impossible boxes

The “crazy box” is the frame of a cube turned inside out. The immediate predecessor of the “Crazy Box” was the “Impossible Box” (by Escher), and its predecessor, in turn, was the Necker Cube (Fig. 15).

Figure 15. Necker cube

It is not an impossible object, but it is a figure in which the depth parameter can be perceived ambiguously.

When we look at the Necker cube, we notice that the face with the dot is either in the foreground or in the background, it jumps from one position to another.

Application of impossible figures

Impossible figures sometimes find unexpected uses. Oscar Ruthersvard talks in his book "Omojliga figurer" about the use of imp art drawings for psychotherapy. He writes that the paintings, with their paradoxes, evoke surprise, focus attention and the desire to decipher. Psychologist Roger Shepard used the idea of a trident for his painting of the impossible elephant.

In Sweden, they are used in dental practice: by looking at pictures in the waiting room, patients are distracted from unpleasant thoughts in front of the dentist’s office.

3.1. Impossible figures in icon painting

Christianity very rarely used models of non-existent figures, but their images are often found in icons and frescoes. Not many models of impossible figures in temples have survived to this day. The most famous of them is the image of an impossible triangle located on the screen in front of the altar (Fig. 16). It is located in the Church of the Holy Trinity, built by Benedin monks from 1150 to 1550. Subsequently it was destroyed, and in 1869 it was restored and rebuilt.

Figure 16. Fresco in front of the altar

Images of impossible figures are found on icons and frescoes. This is usually an impossible colonnade. The base of the middle column is removed from the viewer. Until now, researchers have not come to the conclusion whether such a design is the artist’s intention or a mistake.

On the “Last Judgment” icon (early period) in the upper register on the left there is an image of Heavenly Jerusalem in the form of a walled city with many towers and gates (Fig. 17).

Figure 17. Icon “Last Judgment”

Inside it, behind eight thrones, the saints are represented by rank: apostles, martyrs, saints, hermits (fools), prophets, saints, martyrs and reverend women. Gradually this image became more and more stylized and simplified. By the middle of the 15th century, in the upper register of the icon there was already an arch with impossible ceilings.

These frescoes were created by Evgeny Matko in the Intercession Church in Voronezh region. On each of them you can see impossible constructions.

Decoration of the Chapel of the Nativity of the Virgin Mary near the village of Izhevtsy in the Chernivtsi region (Ukraine). The frescoes depict a large number of impossible figures, which is a characteristic technique of the artist. In most other examples of the use of impossible structures in icon painting, the appearance of impossible structures is associated more likely with the mistakes of the artists than with conscious intentions.

3.2.Impossible figures in architecture and sculpture

Abroad, on city streets, we can see architectural embodiments of impossible figures.

IN lately Several mini sculptures and three-dimensional models of impossible figures were created. They even erected a monument to them.

The Penrose Triangle is immortalized in the city of Petra in Australia. It was installed in 1999 and now everyone passing by can see the impossible figure (Fig. 18).

Figure 18. Perose Triangle in Australia

But as soon as you change the angle of view, the triangle turns from “impossible” into a real and aesthetically unattractive structure that has nothing to do with triangles (Fig. 19).

Figure 19. This is what the Penrose Triangle looks like from the other side

An example of impossible figures in architecture is the so-called Cube Houses. They were built in 1984 in Rotterdam (Netherlands) by architect Piet Blom. The houses are rotated at an angle of 45 degrees and arranged in a hexagonal grid. The design consists of 32 cubes connected to each other. Each cubic house consists of four floors. On the ground floor there is an entrance, on the second there is a kitchen and living room, on the third there is a bedroom and a bathroom, and on the fourth floor there is often a greenhouse. Roofs of houses painted white and gray colors, when viewed from the side, resemble mountain peaks covered with snow. This complex of buildings has another interesting property. From a bird's eye view, the buildings form a structure that looks like an impossible figure.

3.3.Impossible figures in painting

There is a whole direction in painting called impossibilism (“impossibility”) - the depiction of impossible figures and paradoxes. Interest in impossibilism flared up by 1980. The term was coined by Teddy Brunius, a professor of art history at the University of Copenhagen. This term precisely defines what is included in this new concept: the image of objects that seem real, but cannot exist in physical reality.

Fractal geometry studies the patterns manifested in the structure of natural objects, processes and phenomena that have a pronounced fragmentation, fracture and curvature.

Op art (English: Op-art - shortened version of optical art - optical art) - an artistic movement of the second half of the 20th century, using various visual illusions, based on the peculiarities of perception of flat and spatial figures. An independent direction in op art is the so-called imp-art, which uses the features of displaying three-dimensional objects on a plane to achieve optical illusions.

Most well-known representatives op art are Maurice Escher, Hungarian artist Istvan Orosz, Flemish artist Jos De Mey, Swiss artist Sandro del Pre. British artist Julian Beaver is one of the most famous artists of this movement, who depicts his masterpieces not on paper, but on the streets of the city, the walls of city houses, where everyone can admire them.

3.4.Impossible figures in philatelic work

In 1982, by order of the Swedish government, Oscar Reutersvärd made stamps with images of impossible figures (Fig. 20).

Figure 20. Swedish stamps with images of famous figures

The stamps were produced in limited editions; today they are very rare and in great demand among philatelists. Another edition is planned for the near future. The first of these stamps was dedicated to the mathematical congress in Innsbruck (Austria), held in 1981. The impossible Escher box is used as a basis (Fig. 21).

Figure 22. Stamp dedicated to mathematical progress

3.5.Impossible figures in design art

Impossible figures are often used to design magazine covers.

The cover of the first issue of 2008 of the magazine “Mathematics at School” depicts a collage of fragments of paintings by the Belgian artist Jos de Mey (Fig. 22).

Figure 22. Magazine “Mathematics at school”

Here you can see two frequent characters in the artist’s paintings - an owl and a man with a cube. For the Belgians, the owl is a symbol of theoretical knowledge, and at the same time a nickname for a stupid person. The man with the impossible cube is, in turn, one of the heroes of the lithography by M.K. Escher's "Belvedere", which de Mey borrowed for his paintings. It was de Mey who painted the clothes of this character in characteristic Dutch colors. You can also see other fragments from the paintings of the Belgian artist - a large impossible construction painted with mathematical formulas, as well as a tablet with Durer's magic square.

Impossible figures are traditionally used in the design of the covers of algebra textbooks for grade 7 (Fig. 23).

Figure 23. Algebra textbook

3.6.Impossible figures in animation

Interest in impossible figures was reflected in animation and cinema.

Who, as a child, did not watch the cartoon “In the Blue Sea, in the White Foam...”, filmed at the Armenfilm studio in 1984. The film tells the tale of how little boy frees the King of the Sea from the jug, after which he kidnaps the boy and drags him to the bottom of the sea (Fig. 24).

Figure 24. Still from the cartoon

At the beginning of the cartoon there is a scene in which there are perspective violations. In them, the King of the Sea operates with objects located at a great distance from him as if they were simply small in size and located next to him.

The modern popular American animated series Phineas and Ferb tells how two stepbrothers spend their summer holidays. Every day they start a new grandiose project (Fig. 25).

Figure 25. Still from the series

In episode 35 of the second season, "The Bottom Side of the Moon", the brothers build the tallest building in the world, which reaches the moon. One of the rooms of the building repeats Escher's Relativity.

3.7.Impossible figures in logos and symbols

Figure 26 shows the logo of the French automobile company Renault. In 1972, the impossible quadrangle became its symbol. The furniture store “Furniture Hallucinations” also uses an impossible triangle in its logo (Fig. 27).

Figure 26. Renault logo

Figure 27. Furniture store logo

Figure 28 shows the logo of the campaign for the production and sale of windows.

Figure 28. Logo of the “Russian Windows” campaign

Mathematicians claim that palaces in which you can go down the stairs leading up can exist. To do this, you just need to build such a structure not in three-dimensional, but, say, in four-dimensional space. But in the virtual world, which modern computer technology opens up for us, you can’t do anything like that. Nowadays, the ideas of a man who, at the dawn of the century, believed in the existence of impossible worlds are being realized.

Practical part

Creating impossible figures

As a survey of my classmates showed, most of the guys do not know about the existence of impossible figures (Appendix 1), although many automatically draw geometric shapes when talking on the phone, and easily depicted impossible figures. For example, you can draw five, six or seven parallel lines, end these lines at different ends in different ways - and the impossible figure is ready. If, for example, you draw five parallel lines, then they can be finished as two beams on one side and three on the other (Fig. 29).

Figure 29. Simple drawings of impossible figures

I created several impossible figures to better visualize how they could exist. To do this, I took scans for gluing from the Internet (Appendices 2,3 and 4). I printed out the development of an impossible triangle (tribar). The result is a figure that, at first glance, bears little resemblance to a tribar (Fig. 30).

Figure 30. Manufactured tribar

At first I thought that I had made a mistake in manufacturing, but after looking at it from a certain angle, everything turned out great. I note that to create a complete illusion, the correct angle of view and the correct lighting are necessary.

The following figures 31 and 32 show more complex figures, also made by me.

Figure 31. Impossible figure 1

Figure 32. Impossible figure 2

Conclusion

Impossible figures force our minds to first see what should not be, then look for the answer - what was done wrong, what is the hidden essence of the paradox. And sometimes the answer is not so easy to find - it is hidden in the optical, psychological, logical perception of the drawings.

The development of science, the need to think in a new way, the search for beauty - all these demands of modern life force us to look for new methods that can change spatial thinking and imagination.

Having studied the literature on the topic, you can answer the question “Are there impossible figures in the real world?” I realized that the impossible is possible and unreal figures can be made with your own hands. I created Ames models of the Impossible Triangle and two other figures. I was able to show that impossible figures can exist in the real world.

Impossible figures are widely used in modern advertising, industrial graphics, posters, design art and logos of various companies, there are many more areas in which impossible figures will be used.

Thus, we can say that the world of impossible figures is extremely interesting and diverse. The work can be used in mathematics classes to develop students' spatial thinking. For creative people prone to invention, impossible figures are a kind of lever for creating something new and unusual. All this allows us to talk about the relevance of the topic being studied.

References

Levitin Karl Geometrical Rhapsody. - M.: Knowledge, 1984, -176 p.

Penrose L., Penrose R. Impossible objects, Quantum, No. 5, 1971, p. 26

Reutersvard O. Impossible figures. - M.: Stroyizdat, 1990, 206 p.

Tkacheva M.V. Rotating cubes. - M.: Bustard, 2002. - 168 p.

Figure 1.

This is an impossible tri-bar. This drawing is not an illustration of a spatial object, since such an object cannot exist. Our EYE accepts this fact and the object itself without difficulty. We can come up with a number of arguments to defend the impossibility of an object. For example, face C lies in the horizontal plane, while face A is inclined towards us, and face B is inclined away from us, and if edges A and B diverge from each other, they do not can meet at the top of the figure, as we see in this case. We can note that the tribar forms a closed triangle, all three beams are perpendicular to each other, and the sum of its internal angles is equal to 270 degrees, which is impossible. We can use the basic principles of stereometry to help us, namely that three non-parallel planes always meet at the same point. However, in Figure 1 we see the following:

The dark gray plane C meets plane B; line of intersection - l;
The dark gray plane C meets the light gray plane A; line of intersection - m;
The white plane B meets the light gray plane A; line of intersection - n;
Intersection lines l, m, n intersect at three different points.

Thus, the figure in question does not satisfy one of the basic statements of stereometry, that three non-parallel planes (in this case A, B, C) must meet at one point.

To summarize: no matter how complex or simple our reasoning may be, the EYE signals us about contradictions without any explanation on its part.

The impossible tribar is paradoxical in several respects. It takes a split second for the eye to convey the message: “This is a closed object consisting of three bars.” A moment later follows: “This object cannot exist...”. The third message can be read as: "...and thus the first impression was wrong." In theory, such an object should break up into many lines that have no significant relationship with each other and no longer assemble into the form of a tribar. However, this does not happen, and the EYE signals again: “This is an object, a tribar.” In short, the conclusion is that it is both an object and not an object, and this is the first paradox. Both interpretations have equal force, as if the EYE left the final verdict to a higher authority.

The second paradoxical feature of the impossible tribar arises from considerations about its construction. If block A is directed towards us, and block B is directed away from us, and yet they are joined, then the angle they form must lie in two places at the same time, one closer to the observer, and the other farther away. (The same applies to the other two angles, since the object remains identically shaped when turned up at the other angle.)

Figure 2. Bruno Ernst, photograph of an impossible tribar, 1985
Figure 3. Gerard Traarbach, "Perfect timing", oil on canvas, 100x140 cm, 1985, printed backwards
Figure 4. Dirk Huiser, "Cube", irisated screenprint, 48x48 cm, 1984

The reality of impossible objects

One of the most difficult questions about impossible figures concerns their reality: do they really exist or not? Naturally, the picture of an impossible tribar exists, and this is not in doubt. However, at the same time, there is no doubt that the three-dimensional form presented to us by the EYE, as such, does not exist in the surrounding world. For this reason, we decided to talk about the impossible objects, not about the impossible figures(although they are better known by that name in English). This seems to be a satisfactory solution to this dilemma. And yet, when we, for example, carefully examine the impossible tribar, its spatial reality continues to confuse us.

Faced with an object disassembled into separate parts, it is almost impossible to believe that simply by connecting bars and cubes with each other, you can get the desired impossible tribar.

Figure 3 is especially attractive to crystallography specialists. The object appears to be a slowly growing crystal; cubes are inserted into the existing crystal lattice without disturbing the overall structure.

The photograph in Figure 2 is real, although the tri-bar made from cigar boxes and photographed from a certain angle is not real. This is a visual joke created by Roger Penrose, co-author of the first article and the Impossible Tribar.

Figure 5.

Figure 5 shows a tribar made up of numbered blocks measuring 1x1x1 dm. By simply counting the blocks, we can find out that the volume of the figure is 12 dm 3, and the area is 48 dm 2.

Figure 6.
Figure 7.

In a similar way, we can calculate the distance that a ladybug will travel along the tribar (Figure 7). The center point of each block is numbered and the direction of movement is indicated by arrows. Thus, the surface of the tribar appears as a long continuous road. Ladybug must complete four full circles before returning to the starting point.

Figure 8.

You may begin to suspect that the impossible tribar has some secrets on its invisible side. But you can easily draw a transparent impossible tribar (Fig. 8). In this case, all four sides are visible. However, the object continues to look quite real.

Let's ask the question again: what exactly makes the tri-bar a figure that can be interpreted in so many ways. We must remember that the EYE processes the image of an impossible object from the retina in the same way as images ordinary items- chair or home. The result is a "spatial image". At this stage there is no difference between an impossible tri-bar and a regular chair. Thus, the impossible tribar exists in the depths of our brain at the same level as all other objects around us. The eye's refusal to confirm the three-dimensional "viability" of a tribar in reality in no way diminishes the fact that an impossible tribar is present in our heads.

In Chapter 1, we encountered an impossible object whose body disappeared into nothingness. In the pencil drawing "Passenger Train" (Fig. 11), Fons de Vogelaere subtly used the same principle with a reinforced column on the left side of the picture. If we follow the column from top to bottom, or close the lower part of the picture, we will see a column that is supported by four supports (of which only two are visible). However, if you look at the same column from below, you will see a fairly wide opening through which a train can pass. Solid stone blocks at the same time turn out to be... thinner than air!

This object is simple enough to categorize, but turns out to be quite complex when we begin to analyze it. Researchers such as Broydrick Thro have shown that the very description of this phenomenon leads to contradictions. Conflict in one of the borders. The EYE first calculates the contours and then assembles shapes from them. Confusion occurs when contours have two purposes in two different shapes or parts of a shape, as in Figure 11.

Figure 9.

A similar situation arises in Figure 9. In this figure, the contour line l appears both as the boundary of form A and as the boundary of form B. However, it is not the boundary of both forms at the same time. If your eyes look first at the top of the drawing, then, looking down, the line l will be perceived as the boundary of shape A and will remain so until it is discovered that A is an open shape. At this point the EYE offers a second interpretation for the line l, namely, that it is the boundary of shape B. If we follow our gaze back up the line l, then we will return again to the first interpretation.

If this were the only ambiguity, then we could talk about a pictographic dual figure. But the conclusion is complicated by additional factors, such as the phenomenon of the figure disappearing from the background, and, in particular, the spatial representation of the figure by the EYE. In this regard, you can take a different look at Figures 7, 8 and 9 from Chapter 1. Although these types of shapes manifest themselves as real spatial objects, we can temporarily call them impossible objects and describe them (but not explain them) in the following general terms: The EYE calculates from these objects two different mutually exclusive three-dimensional shapes that nevertheless exist simultaneously. This can be seen in Figure 11 in what appears to be a monolithic column. However, upon re-examination, it appears to be open, with a wide gap in the middle through which, as shown in the picture, a train could pass.

Figure 10. Arthur Stibbe, "In front and behind", cardboard/acrylic, 50x50 cm, 1986
Figure 11. Fons de Vogelaere, "Passenger Train", pencil drawing, 80x98 cm, 1984

Impossible object as a paradox

Figure 12. Oscar Reutersvärd, "Perspective japonaise n° 274 dda", colored ink drawing, 74x54 cm

At the beginning of this chapter we saw the impossible object as a three-dimensional paradox, that is, an image whose stereographic elements contradict each other. Before exploring this paradox further, it is necessary to understand whether there is such a thing as a pictoraphic paradox. In fact, it exists - think of mermaids, sphinxes and other fairy-tale creatures often found in the visual arts of the Middle Ages and early Renaissance. But in this case, it is not the work of the EYE that is disrupted by such a pictographic equation as woman + fish = mermaid, but our knowledge (in particular, knowledge of biology), according to which such a combination is unacceptable. Only where the spatial data in the retinal image contradict each other does the EYE's "automatic" processing fail. The EYE is not ready to process such strange material, and we are witnessing a visual experience that is new to us.

Figure 13a. Harry Turner, drawing from the series "Paradoxical patterns", mixed media, 1973-78
Figure 13b. Harry Turner, "Corner", mixed media, 1978

We can divide the spatial information contained in the retinal image (when looking with only one eye) into two classes - natural and cultural. The first class contains information that is not influenced by a person's cultural environment, and which is also found in paintings. This true "uncorrupted nature" includes the following:

Objects of the same size appear smaller the further away they are. This is the basic principle of linear perspective, which has played a major role in the visual arts since the Renaissance;
An object that partially blocks another object is closer to us;
Objects or parts of an object connected to each other are at the same distance from us;
Objects located relatively far from us will be less distinguishable and will be hidden by the blue haze of spatial perspective;
The side of the object on which the light falls is brighter than the opposite side, and shadows point in the direction opposite to the light source.

Figure 14. Zenon Kulpa, “Impossible Figures”, ink/paper, 30x21 cm, 1980

In a cultural setting, the following two factors play an important role in our assessment of space. People have created their living space in such a way that right angles predominate in it. Our architecture, furniture and many tools are essentially made up of rectangles. We can say that we have packaged our world into a rectangular coordinate system, into a world of straight lines and angles.

Figure 15. Mitsumasa Anno, "Cube Section"
Figure 16. Mitsumasa Anno, "Intricate Wooden Puzzle"
Figure 17. Monika Buch, "Blue Cube", acrylic/wood, 80x80 cm, 1976

Thus, our second class of spatial information - cultural, is clear and understandable:

A surface is a plane that continues until other details tell us that it has not ended;
The angles at which the three planes meet define the three cardinal directions, so zigzag lines can indicate expansion or contraction.

Figure 18. Tamas Farcas, "Crystal", irisated print, 40x29 cm, 1980
Figure 19. Frans Erens, watercolor, 1985

In our context, the distinction between natural and cultural environments is very useful. Our visual sense evolved in natural environments, and it also has an amazing ability to accurately and accurately process spatial information from cultural categories.

Impossible objects (at least most of them) exist due to the presence of mutually contradictory spatial statements. For example, in the painting by Jos de Mey “Double-guarded gateway to the wintery Arcadia” (Fig. 20), the flat surface forming the upper part of the wall breaks down into several planes at the bottom, located at different distances from the observer. The impression of different distances is also formed by the overlapping parts of the figure in Arthur Stibbe's painting "In front and behind" (Fig. 10), which contradict the rule of a flat surface. In the watercolor drawing by Frans Erens (Fig. 19), the shelf, shown in perspective, with its decreasing end tells us that it is located horizontally, moving away from us, and it is also attached to the supports in such a way as to be vertical. In the painting "The five bearers" by Fons de Vogelaere (Fig. 21), we will be stunned by the number of stereographic paradoxes. Although the painting does not contain paradoxical overlapping objects, it does contain many paradoxical connections. Of interest is the way in which the central figure is connected to the ceiling. The five figures supporting the ceiling connect the parapet and the ceiling with so many paradoxical connections that the EYE goes on an endless search for the point from which it is best to view them.

Figure 20. Jos de Mey, "Double-guarded gateway to the wintery Arcadia", canvas/acrylic, 60x70 cm, 1983
Figure 21. Fons de Vogelaere, "The five bearers", pencil drawing, 80x98 cm, 1985

You might think that with each possible type of stereographic element that appears in a painting, it would be relatively easy to create a systematic overview of the impossible figures:

Those that contain elements of perspective that are in mutual conflict;
Those in which perspective elements are in conflict with spatial information indicated by overlapping elements;
etc.

However, we will soon discover that we will not be able to find existing examples for many such conflicts, while some impossible objects will be difficult to fit into such a system. However, such a classification will allow us to discover many more hitherto unknown types of impossible objects.

Figure 22. Shigeo Fukuda, "Images of illusion", screenprint, 102x73 cm, 1984

Definitions

To conclude this chapter, let's try to define impossible objects.

In my first publication about paintings with impossible objects, M.K. Escher, which appeared around 1960, I came to the following formulation: a possible object can always be considered as a projection - a representation of a three-dimensional object. However, in the case of impossible objects, there is no three-dimensional object of which this projection is a representation, and in this case we can call the impossible object an illusory representation. This definition is not only incomplete, but also incorrect (we will return to this in Chapter 7), since it relates only to the mathematical side of impossible objects.

Figure 23. Oscar Reutersvärd, "Cubic organization of space", colored ink drawing, 29x20.6 cm.
In reality, this space is not filled because the larger cubes are not connected to the smaller cubes.

Zeno Kulpa offers the following definition: an image of an impossible object is a two-dimensional figure that creates the impression of an existing three-dimensional object, and this figure cannot exist in the way we spatially interpret it; thus, any attempt to create it leads to (spatial) contradictions that are clearly visible to the viewer.

Kulpa's last point suggests one practical way to find out whether an object is impossible or not: just try to create it yourself. You will soon see, perhaps even before you begin construction, that you cannot do this.

I would prefer a definition that emphasizes that the EYE, when analyzing an impossible object, comes to two contradictory conclusions. I prefer this definition because it captures the reason for these mutually conflicting conclusions, and also clarifies the fact that impossibility is not a mathematical property of a figure, but a property of the viewer's interpretation of the figure.

Based on this, I propose the following definition:

An impossible object has a two-dimensional representation, which the EYE interprets as a three-dimensional object, and at the same time, the EYE determines that this object cannot be three-dimensional, since the spatial information contained in the figure is contradictory.

Figure 24. Oscar Reutersväird, “Impossible four-bar with Crossbars”
Figure 25. Bruno Ernst, "Mixed illusions", photography, 1985

Exists big class images about which you can say: “What are we seeing? Something strange.” These include drawings with a distorted perspective, objects that are impossible in our three-dimensional world, and unimaginable combinations of very real objects. Appearing at the beginning of the 11th century, such “strange” drawings and photographs have today become a whole movement of art called imp art.

A little history

Paintings with distorted perspective can be found already at the beginning of the first millennium. A miniature from the book of Henry II, created before 1025 and kept in the Bavarian State Library in Munich, depicts a Madonna and Child. The painting depicts a vault consisting of three columns, and the middle column, according to the laws of perspective, should be located in front of the Madonna, but is behind her, which gives the painting a surreal effect. Unfortunately, we will never know whether this technique was a conscious act of the artist or his mistake.

Images of impossible figures, not as a conscious direction in painting, but as techniques that enhance the effect of the perception of the image, are found among a number of painters of the Middle Ages. Pieter Bruegel's painting "The Magpie on the Gallows", created in 1568, shows a gallows of an impossible design, which gives the effect to the whole picture. The well-known engraving by the 18th century English artist William Hogarth, “False Perspective,” shows the absurdity to which an artist’s ignorance of the laws of perspective can lead.

At the beginning of the 20th century, the artist Marcel Duchamp painted an advertising painting "Apolinere enameled" (1916-1917), stored in the Philadelphia Museum of Art. In the design of the bed on the canvas you can see impossible three- and quadrangles.

The founder of the direction of impossible art - imp-art (impossible art) is rightly called the Swedish artist Oscar Rutesvard (Oscar Reutersvard). The first impossible figure "Opus 1" (N 293aa) was drawn by the master in 1934. The triangle is made up of nine cubes. The artist continued his experiments with unusual objects and in 1940 created the figure “Opus 2B”, which is a reduced impossible triangle consisting of only three cubes. All cubes are real, but their location in three-dimensional space is impossible.

The same artist also created the prototype of the “impossible staircase” (1950). The most famous classical figure, the Impossible Triangle, was created by the English mathematician Roger Penrose in 1954. He used linear perspective, and not parallel, like Rootesward, which gave the picture depth and expressiveness and, therefore, a greater degree of impossibility.

Most famous artist Imp art became M. C. Escher. Among his most famous works are the paintings “Waterfall” (1961) and “Ascending and Descending”. The artist used the “endless staircase” effect, discovered by Rootesward and later expanded by Penrose. The canvas depicts two rows of men: when moving clockwise, the men constantly rise, and when moving counterclockwise, they descend.

A bit of geometry

There are many ways to create optical illusions (from the Latin word “iliusio” - error, delusion - inadequate perception of an object and its properties). One of the most spectacular is the direction of imp art, based on images of impossible figures. Impossible objects are drawings on a plane (two-dimensional images), executed in such a way that the viewer gets the impression that such a structure cannot exist in our real three-dimensional world. Classic, as already mentioned, and one of the simplest such figures is the impossible triangle. Each part of the figure (the corners of the triangle) exists separately in our world, but their combination in three-dimensional space is impossible. Perceiving the entire figure as a composition of irregular connections between its real parts leads to the deceptive effect of an impossible structure. The gaze glides along the edges of an impossible figure and is unable to perceive it as a logical whole. In reality, the view tries to reconstruct the real three-dimensional structure (see figure), but encounters a discrepancy.

From a geometric point of view, the impossibility of a triangle lies in the fact that three beams connected in pairs to one another, but along three different axes of the Cartesian coordinate system, form a closed figure!

The process of perceiving impossible objects is divided into two stages: recognizing the figure as a three-dimensional object and realizing the “irregularity” of the object and the impossibility of its existence in the three-dimensional world.

The existence of impossible figures

Many people believe that impossible figures are truly impossible and cannot be created in the real world. But we must remember that any drawing on a sheet of paper is a projection of a three-dimensional figure. Therefore, any figure drawn on a piece of paper must exist in three-dimensional space. Impossible objects in paintings are projections of three-dimensional objects, which means that the objects can be realized in the form of sculptural compositions (three-dimensional objects). There are many ways to create them. One of them is the use of curved lines as the sides of an impossible triangle. The created sculpture looks impossible only from a single point. From this point, the curved sides look straight, and the goal will be achieved - a real "impossible" object will be created.

About the benefits of imp art

Oscar Rootesvaard talks in the book “Omojliga figurer” (there is a Russian translation) about the use of imp art drawings for psychotherapy. He writes that the paintings, with their paradoxes, evoke surprise, focus attention and the desire to decipher. In Sweden, they are used in dental practice: by looking at pictures in the waiting room, patients are distracted from unpleasant thoughts in front of the dentist’s office. Remembering how long one has to wait for an appointment in various Russian bureaucratic and other institutions, one can assume that impossible pictures on the walls of reception areas can brighten up the waiting time, calming visitors and thereby reducing social aggression. Another option would be to install in reception areas slot machines or, for example, mannequins with corresponding faces as dart targets, but, unfortunately, this kind of innovation was never encouraged in Russia.

Using the phenomenon of perception

Is there any way to enhance the effect of impossibility? Are some objects more "impossible" than others? And this is where features come to the rescue. human perception. 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 drops 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 a mirror image of the original, and the impression produced by the tapestries and the originals differs.

This property can be successfully used when creating creations with impossible objects, increasing or decreasing the “degree of impossibility”. There is also the prospect of obtaining interesting compositions using computer technology, either from several paintings rotated (perhaps using different types of symmetries) one relative to the other, giving viewers a different impression of the object and a deeper understanding of the essence of the design, or from one rotated ( 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 a graphics 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.

Combinations, combinations

There is a group of impossible objects, the sculptural implementation of which is impossible. Perhaps the most famous of them is the “impossible trident”, or “devil’s fork” (P3-1). If you look closely at the object, you will notice that three teeth gradually turn into two on a common basis, leading to a conflict of perception. We compare the number of teeth above and below and come to the conclusion that the object is impossible. Based on the “fork,” a great many impossible objects have been created, including those where a part that is cylindrical at one end becomes square at the other.

In addition to this illusion, there are many other types of optical illusions (illusions of size, movement, color, etc.). The illusion of depth perception is one of the oldest and most famous optical illusions. The Necker cube (1832) belongs to this group, and in 1895 Armand Thiery published an article about a special type of impossible figures. In this article, for the first time, an object was drawn that later received the name Thierry and was used countless times by op art artists. The object consists of five identical rhombuses with sides of 60 and 120 degrees. In the figure you can see two cubes connected along one surface. If you look from the bottom up, you can clearly see the lower cube with two walls at the top, and if you look from the top down, you can clearly see the upper cube with the walls below.

The most simple figure of the Thierry-like ones, this is apparently a “pyramid-opening” illusion, which is a regular rhombus with a line in the middle. It is impossible to say exactly what we see - a pyramid rising above the surface, or an opening (depression) on it. This effect was used in the graphic "Labyrinth (Pyramid Plan)" of 2003. The painting received a diploma at the international mathematical conference and exhibition in Budapest in 2003 "Ars(Dis)Symmetrica" 03. The work uses a combination of the illusion of depth perception and impossible figures.

In conclusion, we can say that the imp art direction is like component Optical art is actively developing, and in the near future we will undoubtedly expect new discoveries in this area.

Candidate of Technical Sciences D. RAKOV (Institute of Mechanical Science named after A. A. Blagonravov RAS).

LITERATURE

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

Under this name, the magazine has been publishing drawings of all sorts of impossible figures and objects for almost forty years. See "Science and Life" No. 5, 8, 1969; No. 2, 1970; No. 1, 1979; No. 10, 1986; No. 11 1989; No. 8, 1994

GU Osmeryzhskaya main secondary school

Impossible figures

Direction: physics and mathematics

Performer of the work : Dippel Sergey, 6th grade student of the Osmeryzhsk secondary school, Pavlodar region, Kachira district, Osmeryzhsk village

Head of work: Dovzhenko Natalya Vladimirovna mathematics teacher at Osmeryzhskaya secondary school

2013

Resume/abstract/………………………………………………………………2

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

1. A little bit of history……………………………………………..………….5

2. Types of impossible figures…………….…………………………………….9

3. Oscar Ruthersward – father of the impossible figure……….………………..16

4. Impossible figures are possible!……………………………………...18 5. Application of impossible figures……………………………………..……19

Conclusion……………………………………………………………………………….....21

References……………………………………………………………22

Resume /abstract/

Project stages:

Stage 1.

Statement of the problem, setting goals, objectives of information and research work;

Conducting conversations about impossible figures;

Staging problematic issue, motivation to implement the project;

Conducting preliminary work on the topic “Impossible figures”;

Discussion and drawing up a step-by-step work plan, creating a bank of ideas and proposals. Selection of information sources.

Stage 2. Project implementation activities.

Information and educational conversations;

Information retrieval work;

Experimental work;

Literature review

Achieving goals

Introduction

For some time now I have been interested in figures that at first glance seem ordinary, but upon closer inspection you can see that something is wrong with them. The main interest for me was the so-called impossible figures, looking at which one gets the impression that they cannot exist in the real world. I wanted to know more about them.

Despite the fact that impossible figures have been known almost since the time of cave paintings, their systematic study began only in the middle of the 20th century, that is, almost before our eyes, and before that mathematicians dismissed them as an annoying misunderstanding.

In 1934, Oscar Reutersvard accidentally created his first impossible figure, a triangle made of nine cubes, but instead of correcting anything, he began creating other impossible figures one after another.

Even such simple volumetric shapes as a cube, pyramid, parallelepiped can be represented as a combination of several figures located at different distances from the observer’s eye. There should always be a line along which the images of individual parts are combined into a complete picture.

“An impossible figure is a three-dimensional object made on paper that cannot exist in reality, but which, however, can be seen as a two-dimensional image.” This is one of the types optical illusions, a figure that at first glance seems to be a projection of an ordinary three-dimensional object, upon careful examination of which contradictory connections of the elements of the figure become visible. An illusion is created of the impossibility of the existence of such a figure in three-dimensional space.

Despite a significant number of publications about impossible figures, their clear definition has not been formulated in essence. You can read that impossible figures include all optical illusions associated with the peculiarities of our perception of the world. On the other hand, a person can show you a figure of a green man or with ten arms and five heads and say that all these are impossible figures. At the same time, he will be right in his own way. After all, there are no green people with ten legs. Therefore, by impossible figures we will understand flat images of figures perceived by a person unambiguously, as they are drawn without human perception of any additional, actually not drawn images or distortions and which cannot be represented in three-dimensional form. The impossibility of representation in three-dimensional form is understood, of course, only directly, without taking into account the possibility of using special means in the manufacture of impossible figures, since an impossible figure can always be made by using an ingenious system of slots, additional supporting elements and bending the elements of the figure, and then photographing it under the right angle

I was faced with the question: “Do impossible figures exist in the real world?”

Project goals:

1. Find out how unreal figures are created.

2. Find areas of application of impossible figures.

Project objectives:

1. Study literature on the topic “Impossible figures.”

2. Make a classification of impossible figures.

3. Consider ways to construct impossible figures.

4.Create an impossible figure.

Impossible figures

A little bit of history

In medieval Japanese and Persian painting, impossible objects are an integral part of the eastern artistic style, which gives only a general outline of the picture, the details of which the viewer “has” to think out independently, in accordance with his preferences. Here is the school in front of us. Our attention is drawn to the architectural structure in the background, the geometric inconsistency of which is obvious. It can be interpreted as either the inner wall of a room or the outer wall of a building, but both of these interpretations are incorrect, since we are dealing with a plane that is both an outer and an outer wall, that is, the picture depicts a typical impossible object.

Paintings with distorted perspective can be found already at the beginning of the first millennium. A miniature from the book of Henry II, created before 1025 and kept in the Bavarian State Library in Munich, depicts a Madonna and Child. The painting depicts a vault consisting of three columns, and the middle column, according to the laws of perspective, should be located in front of the Madonna, but is located behind her, which gives the painting the effect of unreality.

In the article "Bringing order to the impossible" ( impossible.info/russian/articles/kulpa/putting-order.html) the following definition of impossible figures is given: " An impossible figure is a flat drawing that gives the impression of a three-dimensional object in such a way that the object suggested by our spatial perception cannot exist, so that the attempt to create it leads to (geometric) contradictions clearly visible to the observer". The Penroses write approximately the same thing in their memorable article: " Each individual part of the figure appears to be a normal three-dimensional object, but due to the incorrect connection of the parts of the figure, the perception of the figure completely leads to the illusory effect of impossibility", but none of them answers the question: why is all this happening?

Meanwhile, everything is simple. Our perception is designed in such a way that when processing a two-dimensional figure that has signs of perspective (i.e. volumetric space), the brain perceives it as three-dimensional, choosing the simplest method of converting 2D to 3D, guided by life experience, and as shown above, real prototypes “impossible” figures are rather sophisticated designs with which our subconscious is unfamiliar, but even after becoming familiar with them, the brain still continues to choose the simplest (from its point of view) transformation option, and only after lengthy training does the subconscious finally “enter the situation” and the apparent abnormality of the “impossible figures” disappears.

Let's start with the easy one. Consider a painting (yes, a painting, not a computer-generated photorealistic drawing) by a Flemish artist named Jos de Mey. The question is - what physical reality could it correspond to?

At first glance, the architectural structure seems impossible, but after a moment’s hesitation, the consciousness finds a saving option: the brickwork is in a plane perpendicular to the observer and rests on three columns, the tops of which seem to be located at an equal distance from the masonry, but in fact the empty space is simply “hidden” “due to the “successfully” chosen projection. After consciousness has “deciphered” the picture, it (and all similar images) is perceived completely normally and geometric contradictions disappear as imperceptibly as they appeared.

Impossible painting by Jos de May

Consider the famous painting by Maurits Escher “Waterfall” and its simplified computer model, made in a photorealistic style. At first glance, there are no paradoxes; before us is an ordinary picture depicting... a drawing of a perpetual motion machine!!! But, as you know from a school physics course, a perpetual motion machine is impossible! How did Escher manage to depict in such detail something that could not exist in nature at all?!

Perpetual motion machine in the engraving "Waterfall" by Escher.

Computer model of Escher's perpetual motion machine.

Types of impossible figures.

"Impossible figures" are divided into 4 groups. So, the first one:

An amazing triangle - tribar.

Here are some more examples of impossible figures based on the tribar.

Triple Warped Tribar 12 Cube Triangle

Winged Tribar Triple Domino

Let's return to the Penrose Triangle and try to construct a three-dimensional figure, the projection of which onto a two-dimensional plane would look like the indicated image. Naturally, it will not be possible to solve such a problem directly, but if you think carefully and choose the right angle, then... one of the possible options is shown in the figure.

Possible impossible Penrose Triangle.

Here's another display from Mathieu Hemakerz. There are many possible reverse mapping options. So many. Infinitely many!

The same Penrose Triangle from different angles.

By the way, the Penrose Triangle is immortalized in the form of a statue in Perth (Australia). Created by artist Brian McKay and architect Ahmad Abas, it was erected in Claisebrook Park in 1999 and now everyone passing by can see the next "impossible" figure.

Perose Triangle in Australia

But as soon as you change the angle of view, the triangle turns from “impossible” into a real and aesthetically unattractive structure that has nothing to do with triangles.

This is what the Penrose Triangle actually looks like.

Endless staircase

This figure is most often called the “Endless Staircase”, “Eternal Staircase” or “Penrose Staircase” - after its creator. It is also called the "continuously ascending and descending path."

The “Endless Staircase” was successfully used by the artist Maurits K. Escher, this time in his lithograph “Ascent and Descend”, created in 1960.

The creators of this staircase took advantage of parallel lines to design the end pieces of the equally spaced blocks; Some blocks appear to be twisted to fit the illusion.

Space fork.

The next group of figures is collectively called the “Space Fork”. With this figure we enter into the very core and essence of the impossible. This may be the largest class of impossible objects.

Impossible boxes

The “crazy box” is the frame of a cube turned inside out. The immediate predecessor of the Crazy Box was the Impossible Box (by Escher), and its predecessor in turn was the Necker Cube.

It is not an impossible object, but it is a figure in which the depth parameter can be perceived ambiguously.

When we look at the Necker cube, we notice that the face with the dot is either in the foreground or in the background, it jumps from one position to another.

Oscar Ruthersward - father of the impossible figure.

The “father” of impossible figures is the Swedish artist Oscar Rutersvard. Swedish artist Oscar Ruthersvard, a specialist in creating images of impossible figures, claimed that he was poorly versed in mathematics, but, nevertheless, elevated his art to the rank of science, creating a whole theory of creating impossible figures according to a certain number of patterns.

A pair of impossible figures from Oscar Reutersvärd.

He divided the figures into two main groups. He called one of them “true impossible figures.” These are two-dimensional images of three-dimensional bodies that can be colored and shadowed on paper, but they do not have a monolithic and stable depth.

Another type is dubious impossible figures. These figures do not represent single solid bodies. They are a combination of two or more figures. They cannot be painted, nor can light and shadow be applied to them.

A true impossible figure consists of a fixed number of possible elements, while a doubtful one “loses” a certain number of elements if you follow them with your eyes.

One version of these impossible figures is very easy to do, and many of those who mechanically draw geometric figures when talking on the phone have done this more than once. You need to draw five, six or seven parallel lines, finish these lines at different ends in different ways - and the impossible figure is ready. If, for example, you draw five parallel lines, then they can end up as two beams on one side and three on the other.

In the figure we see three options for dubious impossible figures. On the left is a three-seven beam structure, built from seven lines, in which three beams turn into seven. The figure in the middle, built from three lines, in which one beam turns into two round beams. The figure on the right, constructed from four lines, in which two round beams turn into two beams

During his life, Ruthersvard painted about 2,500 figures. Ruthersvard's books have been published in many languages, including Russian.

Impossible figures are possible!

Many people believe that impossible figures are truly impossible and cannot be created in the real world. But we must remember that any drawing on a sheet of paper is a projection of a three-dimensional figure. Therefore, any figure drawn on a piece of paper must exist in three-dimensional space. Impossible objects in paintings are projections of three-dimensional objects, which means that the objects can be realized in the form of sculptural compositions. There are many ways to create them. One of them is the use of curved lines as the sides of an impossible triangle. The created sculpture looks impossible only from a single point. From this point, the curved sides look straight, and the goal will be achieved - a real "impossible" object will be created.

Russian artist Anatoly Konenko, our contemporary, divided impossible figures into 2 classes: some can be simulated in reality, while others cannot. Models of impossible figures are called Ames models.

I made my own impossible figure. I took forty-two cubes and glued them together to form a cube with part of the edge missing. I note that to create a complete illusion, the correct angle of view and the correct lighting are necessary.

I create my impossible figures using O. Ruthersward's advice. I drew seven parallel lines on paper. I connected them from below with a broken line, and from above I gave them the shape of parallelepipeds. Look at it first from above then from below. You can come up with an infinite number of such figures.

Application of impossible figures

In Sweden, they are used in dental practice: by looking at pictures in the waiting room, patients are distracted from unpleasant thoughts in front of the dentist’s office.

Impossible figures inspired artists to create a whole new movement in painting called impossibilism. Impossibilists include Dutch artist Escher. He is the author of the famous lithographs “Waterfall”, “Ascent and Descent” and “Belvedere”. The artist used the “endless staircase” effect discovered by Rootesward.

Abroad, on city streets, we can see architectural embodiments of impossible figures.

The most famous use of impossible figures is in popular culture- logo of the automobile concern "Renault"

Mathematicians claim that palaces in which you can go down the stairs leading up can exist. To do this, you just need to build such a structure not in three-dimensional, but, say, in four-dimensional space. But in the virtual world, which modern computer technology opens up for us, you can’t do anything like that. This is how these days the ideas of a man who, at the dawn of the century, believed in the existence of impossible worlds are being realized.

Conclusion.

The development of science, the need to think in a new way, the search for beauty - all these demands of modern life force us to look for new methods that can change spatial thinking and imagination.

After studying the literature on the topic, I was able to answer the question “Are there impossible figures in the real world?” I realized that the impossible is possible and unreal figures can be made with your own hands. I created the Ames model of the Impossible Cube. After looking at ways to construct impossible figures, I was able to draw my own impossible figures. I was able to show that

Conclusion: All impossible figures can exist in the real world.

There are many more areas where impossible figures will be used.

Thus, we can say that the world of impossible figures is extremely interesting and diverse. The study of impossible figures is quite important from a geometry point of view. The work can be used in mathematics classes to develop students' spatial thinking. For creative people prone to invention, impossible figures are a kind of lever for creating something new and unusual.

References

Levitin Karl Geometrical Rhapsody. - M.: Knowledge, 1984, -176 p.

Penrose L., Penrose R. Impossible objects, Quantum, No. 5, 1971, p. 26

Reutersvard O. Impossible figures. – M.: Stroyizdat, 1990, 206 p.

Tkacheva M.V. Rotating cubes. – M.: Bustard, 2002. – 168 p.

Internet resources:

http://wikipedia.tomsk.ru

http://www.konenko.net/imp.htm

http://www.im-possible.info/russian/articles/reut_imp/