The impossible triangle of cubes in the real world. Impossible reality

Many believe that impossible figures are really impossible, and they cannot be created in the real world. However, from the school year of geometry, we know that the drawing shown on a sheet of paper is a projection of a three-dimensional figure on a plane. Consequently, any figure drawn on a sheet of paper should exist in three-dimensional space. Moreover, three-dimensional objects, when projection on the plane of which, the predetermined flat figure is the endless set. The same applies to the impossible figures.

Of course, none of the impossible figures can not be created, acting straightforward. For example, if you take three identical wooden bars, you will not be able to combine them so that it is an impossible triangle. However, when projecting a three-dimensional figure on a plane, some lines may become invisible, overlapping each other, to stick together with each other, etc. Based on this, we can take three different bars and make a triangle, presented in the photo below (Fig. 1). This photo is created by the famous popularizer of works M.K. Escher, the author of a large number of books Bruno Ernst. In the foreground of the photo, we see the figure of the impossible triangle. In the background there is a mirror, which reflects the same figure from another point of view. And we see that in fact the figure of the impossible triangle is not closed, but an open figure. And only from the point with which we overlook the figure it seems that the vertical lump of the figure goes beyond the horizontal bar, as a result of which the figure seems impossible. If we had shifted the viewing angle a bit, you would immediately be visible to the gap in the figure, and she would lose their effect of impossibility. The fact that the impossible figure looks impossible only from one point of view is characteristic of all impossible figures.

Fig. one. Photo of an impossible triangle made by Bruno Ernst.

As mentioned above, the number of figures corresponding to a given projection, an infinite set, therefore the above example is not the only way to build an impossible triangle in reality. The Belgian artist Mathieu Hemachers (Mathieu Hamaekers) created a sculpture presented in Fig. 2. Photography on the left shows the frontal view of the figure, in which it looks like an impossible triangle, the central photo shows the same figure, turned 45 °, and the photo on the right is a figure, turned 90 °.

Fig. 2. A photograph of the shape of the impossible triangle Mathie Chemacherz.

As you can see, there are no straight lines in this figure, all the elements of the shape are curved in a certain way. However, as in the previous case, the effect of the impossibility is noticeable only at one corner of the review, when all curved lines are projected into direct, 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 magazine "Technical Aesthetics" No. 9 (1974). All edges of this design are straight lines, and the verges are bent, although the figure of this bent is not visible on the frontal form. He created such a tree triangle model.

Fig. 3. Model of the impossible triangle Vyacheslav Koleichuk.

Later, this model was recreated by an employee of the Faculty of Computer Sciences Institute of Technion in Israel Elber Gershon. Its option (see Fig. 4) was first designed on the computer, and then recreated in reality using a three-dimensional printer. If you move a little overview angle of the impossible triangle, then we will see a figure similar to the second photo in Fig. four.

Fig. four. An option to build an impossible triangle of Elbera Gershon.

It is worth noting that if we looked at the figures now, and not in their photos, we would immediately see that none of the figures presented is impossible, and what is the secret of each of them. We would simply be able to see these figures impossible, since we have stereoscopic vision. That is, our eyes located at a certain distance from each other see the same object from two relatives, but still different, points of view, and our brain, having received two images from our eyes combines them into a single picture. It has previously been said that the impossible object looks impossible only from a single point of view, and since we overlook the object from two points of view, then we immediately see those tricks, with which one or another object is created.

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

There is another way, with which you can see the impossible figure, with two eyes at once. It is as follows: it is necessary to create a huge figure with a high-storey house, arrange it on an extensive open space and look at it from a very long distance. In this case, even looking at the figure with two eyes, you will perceive it as an impossible due to the fact that both of your eyes will receive images practically no different friend. Such an impossible figure was created in the Australian city of Perth.

If the impossible triangle is relatively easy to construct in the real world, then it is not so easy to create an impossible trident in three-dimensional space simply. A feature of this figure is the presence of a contradiction between the front and the background of the figure when the individual elements of the shapes are smoothly transhibited into the background on which the figure is located.

Fig. five. Design similar to the impossible trident.

At the Eye Optics Institute in the city of Aachen (Germany), this task was able to solve this task by creating a special installation. The design consists of two parts. Three round columns and builder are located in front of the front. This part is illuminated only at the bottom. The columns are the semipermeable (Half-Permeable) mirror with a reflective layer, located in front, that is, the viewer does not see what is behind the mirror, and sees only the reflection of the columns.

Fig. 6.Installation circuit, reproducing impossible trident.

Municipal budgetary educational institution

"Lyceum №1"

Research work on the topic

"Impossible figures"

Performed: Slichuk Danil Student 6B Class

Leader: Mathematics Teacher

Kazmenko Elena Alexandrovna

Introduction 3.

1. Determination of the impossible figures 4

2. Types of impossible figures 8

2.1. Amazing Triangle - Trybar 8

2.2. Infinite staircase 9.

2.3. Space fork 11.

2.4. Impossible boxes 12.

3. Application of impossible figures 13

3.1. Impossible figures in Iconopy 13

3.2. Impossible figures in architecture and sculpture 15

3.3.Nogening figures in painting 16

3.4.Nogening figures in philatelist 18

3.5. Clear figures in investigative art 19

3.6. Clear figures in animation 20

3.7.Nogening figures in logos and symbols 21

4. Creating impossible figures 22

Conclusion 24.

References 25.

Introduction

Impossible figures are known almost since the times of rocking painting, their systematic study began only in the middle of the 20th century, that is, almost before our eyes, and before that mathematics, they shook them from the annoying misunderstanding.

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

Even such simple bulk forms, like a cube, pyramid, parallelepiped can be represented as a combination of several figures located at different distances from the eye of the observer. There should always be a line in which the image of individual parts of combining in a holistic picture.

"Impossible Figure" is a three-dimensional object made on paper, which cannot actually exist, but which, however, can be seen as a two-dimensional image. " This is one of the types of optical illusions, a figure that seems at first glance the projection of the usual three-dimensional object, with attentive consideration of which contradictory connections of the elements of the figure become visible. The illusion of the impossibility of the existence of such a figure in three-dimensional space is created.

Despite the significant number of publications about the impossible figures of their clear definition on the merits are not formulated. It is possible to read that the 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 ten hands and five heads and say that all this is impossible figures. At the same time, he will be in his right. After all, there are no green people with ten legs. Under impossible figures, we understand the flat images of the figures perceived by a person definitely, as they are drawn without perception by a person, no additional, in fact, 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 immediately without taking into account the use of special means in the manufacture of impossible figures, since it is always impossible to make a figure, applying a hitromic system of slots, additional supporting elements and bending elements of the figure, and then photographed it The right angle

The question arose in front of me: "Do impossible figures exist in the real world?"

Objective of the project:

1. To consider how impossible figures are created and where they are used.

Project tasks:

1. Subject literature on the topic "Impossible Figures".

2. Create the classification of impossible figures.

3. Discussion ways to build impossible figures.

4. Create an impossible figure.

The topic of my work is relevant because the understanding of paradoxes is one of the signs of the type of creative potential that the best mathematics, scientists and artists possess. Many work with unreal objects can be attributed to "intellectual mathematical games". You can simulate a similar world with the help of mathematical formulas, a person is simply not able to present it. And for the development of spatial imagination, impossible shapes are useful. A man tirelessly mentally creates around him that it will be simple and understandable for him. He can not even imagine that some objects surrounding it can be "impossible." In fact, the world is one, but you can consider it from different sides.

1. Determining the impossible figures

Until now, there is no clear definition of impossible figures. I found several different approaches to the definition of this concept.

The impossible figure is one of the types of optical illusions, the figure that seems at first glance the projection of the usual three-dimensional object, with attentive consideration of which contradictory connections of the elements of the figure become visible.

Impossible figures are geometrically controversial images of objects that do not exist in real three-dimensional space. Inability arises from a 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, and for others it is impossible to create.

As a rule, the three-dimensional model of the impossible figure looks impossible, it should be considered from a certain viewing angle, so that the illusion of the inability arose.

It is necessary to clarify the difference between the terms "impossible figure", "impossible object" and "three-dimensional model". The three-dimensional model is a physically representative object, when considering which in space, all the slots and bends become visible, which destroy the illusion of the impossibility and this model loses its "magic". When projection of this model, the two-dimensional plane turns out the impossible figure. This impossible figure (in contrast to the three-dimensional model), creates the impression of an impossible object, which can exist only in the imagination of a person, but not in space.

Impossible figures are often found quite often on ancient engravings, paintings and icons - in some cases we have with obvious errors of the transmission of the prospects, in others - with intentional distortions due to the artistic intent.

We are accustomed to believeing \u200b\u200bphotographs (and somewhat less - 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 - an elf or another fabulous beast. The absence of elves in the space / time area observed by us does not mean that they cannot exist. Still as they can (what is easy to make sure the help of plaster, plasticine or papier-mache). But how to draw something that can not be at all?! What can not be constructed at all?!

There is a huge class of so-called "impossible figures", erroneously or deliberately drawn with errors of transmitting prospects, as a result of what fun visual effects arise, helping psychologists to deal with the principles of work (under) consciousness.

In medieval Japanese and Persian painting, the impossible objects are an integral part of the eastern artistic style, which gives only a common painting sketch, the details of which "have to" think about the viewer on their own, in accordance with its preferences.

Pictures with a distorted perspective are found already at the beginning of the first millennium. At miniature from the book of Henry II, created until 1025 and stored in the Bavarian State Library in Munich, Madonna with a Baby was drawn (Fig. 1). The picture shows a set consisting of three columns, and the average column according to the laws of the prospects should be located ahead of Madonna, but is behind it, which gives the picture the effect of unreality.

Figure 1. "Madonna with Baby"

In the article "Guidance of the order in the impossible" (impossible.info/russian/articles/kulpa/peutting-order.html) is given the following definition of the impossible figures: "The impossible figure is a flat pattern that creates the impression of a three-dimensional object in such a way that the object, The proposed by our spatial perception cannot exist, so that an attempt to create it leads to (geometric) contradictions, clearly visible observer. " Approximately the same writer and penis 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 pieces of the figure, the perception of the figure fully leads to the illusory effect of the impossibility," but none of them answer the question: why all this happens?

Meanwhile, everything is simple. Our perception is designed so that when processing a two-dimensional figure, having signs of perspective (i.e. volumetric space), the brain perceives it as three-dimensional, choosing the simplest way to transform 2D in 3D, guided by life experience, and as shown above, real prototypes "Impossible" figures are quite trimmed structures with which our subconscious is unfamiliar, but even after acquaintance with them, the brain still continues to choose the simplest (from his point of view) the transformation option and only after long-term workouts, the subconsciously finally "enters the situation" And the apparent abnormality of "impossible figures" disappears.

Consider a picture (yes, yes, it is the picture, and not a photorealistic drawing generated by a computer), drawn by the Flemish artist named Jos de Mea / Jos de Mey (Fig. 2). The question is - what physical reality she could fit?

At first glance, the architectural structure seems impossible, but after the second zaminka consciousness finds a rescue option: the brickwork is in the plane perpendicular to the observer and relies on three columns, the vertices of which seem to be located at an equal distance from the masonry, but in fact the empty space simply "closes "Due to the" successful "of the selected projection. After the consciousness "deciphered" the picture, it (and all similar images) is perceived perfectly normal, and geometric contradictions are also unnoticed, as they appear.

Figure 2. Impossible Picture of Josa de Maya

Consider the famous picture of Maurice Escher / Maurits Escher "Waterfall" / "Waterfall" (Fig. 3) and its simplified computer model (Fig. 4), performed in the photorealistic style. At first glance there are no paradoxes, we have an ordinary picture, depicting ... Drawing of the Eternal Motor !!! But after all, as you know from the school year of physics, the eternal engine is impossible! How did Eschru managed to portray what in nature can not be in nature?!

Figure 3. Eternal engine on the engraving "waterfall" of the Escher.

Figure 4. Computer model of the Escher Eternal Engine.

When trying to build the engine according to the drawing (or with attentive analysis of the latter), "deception" pops up immediately - in three-dimensional space, such structures are geometrically contradictory and can exist only on paper, that is, on the plane, and the illusion of "volume" is created only due to the signs of perspective ( In this case, it is deliberately distorted) and at the lesson of the drawing for such a masterpiece we will easily win two points, pointing out the projection errors.

Types of impossible figures

"Impossible figures" are divided into 4 groups:

1. An amazing triangle is tribar (Fig. 5).

Figure 5. Tribar

This - figure is possible the first impossible object published in the print. She appeared in 1958. Its authors, father and son Lionell and Roger Penrouse, genetic and mathematician, respectively, determined this object as a "three-dimensional rectangular structure". She also got the name "Tribar". At first glance, the tribar seems simply an image of an equilateral triangle. But the sides converging at the top of the drawing seem perpendicular. At the same time, the left and right faces at thenime also seem 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.

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

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

Figure 8. Winged tribar Figure 9. Triple domino Acquaintance with impossible figures (especially in the execution of Escher), of course, stunning, but the fact that any of the impossible figures can be constructed in the real three-dimensional world, leads to beware. As you know, any two-dimensional image is a projection of a three-dimensional figure on a plane (sheet of paper). There are quite a lot of projection methods, but within each of them the mapping is definitely performed, that is, there is a strict correspondence between the three-dimensional figure and its two-dimensional image. However, axonometric, isometric and other popular projection methods are unidirectional transformations carried out with loss of information and therefore the inverse transformation can be performed by an infinite set of ways, that is, a two-dimensional image corresponds to an infinite multiple three-dimensional figures and any mathematician will easily prove that such a conversion is possible for any Two-dimensional image. That is, in fact no impossible figures! But another map from Mathieu Hemacherza. Possible reverse display options (Fig.10). Infinitely a lot! Figure 10. Triangle Penrose in various angles

1. Infinite staircase

This figure is most often called the "endless staircase", the "eternal staircase" or "Penrose Ladder" - by the name of its creator. It is also called "continuously ascending and downward trail" (Fig.11).

Figure 11. Infinite staircase

For the first time this figure was published in 1958. We have a staircase leading, seemingly up, up or down, but at the same time, a person walking through it does not rise and does not fall. After completing his visual route, it will be at the beginning of the way.

The "endless staircase" artist Mauritz K. Escher was successfully used, this time in his lithography "climbing and descent" created in 1960.

Staircase with four or family steps. To create this figure with a large number of steps the author could inspire a bunch of ordinary railway sleepers. Having gathered to climb on this staircase, you will stand before choosing: Whether to rise in four or seven steps.

The creators of this staircase used parallel lines in the development of finite parts of blocks located at the same distance; It seems that some blocks are twisted to match the illusion.

1. Space fork

The next group of figures under the general name "Space fork". With this figure we enter the very core and the essence of the impossible. Maybe this is the most numerous class of impossible objects (Fig. 12).

Figure 12. Space fork

This notorious impossible object with three (or two-time) teeth became popular with engineers and puzzle lovers in 1964. The first publication dedicated to an unusual figure appeared in December 1964. The author called her "bracket consisting of three elements."

From a practical point of view, this strange trident or a mechanism in the form of a bracket is absolutely not applicable. Some call it just a "annoying mistake." One of the representatives of the aerospace industry proposed to use its properties when constructing interdimal spacecaps.

1. Impossible boxes

Another impossible object appeared in 1966 in Chicago as a result of the original experiments of the photographer Dr. Charles F. Kokrane. Many lovers of impossible figures conducted experiments with a "crazy box." Initially, the author called it a "free box" and stated that it was "designed to send impossible objects in large quantities" (Fig.14).

Figure 14. Impossible Boxes

"Crazy Box" is an inside out of a cube frame inside out. The immediate predecessor of the "crazy box" was the "impossible box" (author Esher), and its predecessor, in turn, became the cube of Necker (Fig. 15).

Figure 15. Cube Necker

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

When we look into the cube of Necker, we notice that the face with a point is on the front, then in the background, it jumps out of one position to another.

Application of impossible figures

Impossible figures sometimes find unexpected use. Oscar Rutherrsvard tells in the book "Omojliga Figurer" on the use of imp-art drawings for psychotherapy. He writes that paintings with their paradoxes are surprising, sharpen attention and desire to decipher. Psychologist Roger Shepard used the idea of \u200b\u200ba trident for his picture of the impossible elephant.

In Sweden, they are used in dental practice: considering paintings in the reception, patients are distracted from unpleasant thoughts in front of the dentist's cabinet.

3.1. Impossible figures in icon stock

Christianity very rarely used the models of non-existent figures, but their images are often found on icons and frescoes. Until our time, there are not so many models of impossible figures in the temples. The most famous of them is an 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, placed by the Benongensky monks from 1150 to 1550. Subsequently, it was destroyed, in 1869 - restored and rebuilt.

Figure 16. Fresco in front of the altar

The images of the impossible figures occurs on icons and frescoes. This is usually the impossible colonnade. The base of the middle column is removed from the viewer. Until now, researchers have not concluded that such a design of the artist or a mistake.

On the Icon "Terrible Court" (early period) in the upper case, the image of the heavenly Jerusalem in the form of a city, discharged with walls with a multitude of towers and a gate (Fig. 17).

Figure 17. Icon "Scary Court"

Inside him, behind the eight thrones, the saints are presented by the ranks: the apostles, martyrs, reverend, hermits (yarodovy), prophets, saint, martyrs and reverend wives. Gradually, this image was increasingly stylized and simplified. By the middle of the XV century in the upper case, the icons had already been an arc with impossible overlaps.

These frescoes were created by Evgeny Madko in the Pokrovsky temple in the Voronezh region. Each of them can see the impossible designs.

The decoration of the chapel of the Nativity of the Virgin Near Village Izhevsk in Chernovetsky region (Ukraine). The frescoes depicted a large number of impossible figures, which is the characteristic technique of artist. In most other examples of using impossible designs in icon painting, the emergence of impossible designs is connected, rather, with errors of artists than conscious intentions.

3.2. Almost figures in architecture and sculpture

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

Recently, several mini sculptures and volumetric models of impossible figures were created. They even put a monument.

Penrose's triangle is immortalized in the city of Peter in Australia. It was installed in 1999 and now everything passing by, can see the impossible figure (Fig. 18).

Figure 18. Triangle perose in Australia

But it is worth a change angle of view, as a triangle from the "impossible" turns into a real and aesthetically unattractive structure that has no relation to the triangles (Fig. 19).

Figure 19. It looks like a triangle of Penrose on the other hand

As an example of the impossible figures in the architecture, the so-called cubic houses can be given. They were built in 1984 in Rotterdam (Netherlands) by architect in bat blomot. At home are deployed at an angle of 45 degrees and are located on a hexagonal grid. The design consists of 32 cubes connected to each other. Each cubic house consists of four floors. On the first floor - entrance, on the second is a kitchen and a living room, on the third - a bedroom and a bathroom, on the fourth floor, often arrange a greenhouse. Roofs of houses painted in white and gray colors, when viewed, resemble mountain peaks covered with snow. This complex of buildings has another interesting feature. From the height of a bird's eyelet of the building form a design looking like an impossible figure.

3.3. Clear figures in painting

In painting, there is a whole direction that is called impossilism ("inability") - an image of impossible figures, paradoxes. Interest in the impusilism broke out by 1980. This term was introduced into the appeal of Teddy Brunius, a professor of art history of Copenhagen University. This term accurately determines what enters this new concept: an image of objects that seem real, but cannot exist in physical reality.

Fractal geometry studies the patterns that manifest themselves in the structure of natural objects, processes and phenomena with explicitly pronounced fragmentation, brokenness and curvature.

OP-ART (eng. Op-Art - Abbreviated OPTICAL ART version - optical art) - the artistic course of the second half of the 20th century, using various visual illusions based on the peculiarities of the perception of flat and spatial figures. An independent direction in OP-Art is the so-called imp-art (IMP-ART), which is used to achieve optical illusions, the features of the display of three-dimensional objects on the plane.

The most famous representatives of OP-Art are Maurice Escher, Hungarian artist Ishthan Oros, Flemish artist Jos de Mea, Swiss artist Sandro Del Pre. British artist Julian Beaver is one of the most famous artists of this area, which depicts its masterpieces not on paper, but on the streets of the city, the walls of urban homes, where they can admire everything.

3.4.Nogening figures in philatelist

In 1982, according to the order of the Government of Sweden, Oscar Reuturasvard made stamps with images of impossible figures (Fig.20).

Figure 20. Swedish brands with images of famous shapes

The brands were released by limited edition, today they are greatly rare and are in great demand among philatelists. In the near future, they are planned another circulation. The first of these brands was devoted to the Mathematical Congress in Innsbruck (Austria), held in 1981. The inscription of the Escher drawer is taken as the basis (Fig.21).

Figure 22. Mark dedicated to Mathematical Curgress

3.5.Nogening figures in investigative art

Not rare impossible shapes are used to design log covers.

On the cover of the first issue of 2008, the magazine "Mathematics at school" depicts a collage of fragments of the pictures of the Belgian artist Zhosa de Maya (Fig.22).

Figure 22. Magazine "Mathematics in School"

Here you can see two frequent characters of the artist's paintings - owl and man with a cube. Owl for Belgians is a symbol of theoretical knowledge, and at the same time nicknamed a stupid man. A person with an impossible cube is in turn one of the heroes of lithography M.K. Escher Belvedere, who borrowed de Mea for his paintings. It was de Mea who painted the clothes of this character in characteristic Dutch colors. You can also see other fragments from the pictures of the Belgian artist - a large impossible design, painted by mathematical formulas, as well as a sign with a magic square of Durera.

In the design of textbooks for algebra for grade 7, impossible shapes are traditionally used (Fig.23).

Figure 23. Tutorial Algebra

3.6. Clear figures in animation

Interest in impossible figures was reflected in animation and cinema.

Who in childhood did not watch the cartoon "in the blue sea, in white foam ...", shot at the Armenfilm studio in 1984. The film tells a fairy tale about how little boy frees from the jug of the king of the sea, after which he abducts the boy and pulls it into the bottom of the sea (Fig.24).

Figure 24. Frame from the cartoon

At the beginning of the cartoon there is a scene in which there are violations of the prospects. In them, the king of the sea operates with objects from him at a high distance as if just a small size and are located next to it.

In the modern popular American animation series Phineas and Ferb, it is described about how two consolidated brothers spend the summer holidays. Every day, they inspire a new grand project (Fig.25).

Figure 25. Frame from the series

In the 35 episode of the second season "Fuffstit side of the Moon" brothers build the highest building in the world, which reaches the moon. One of the rooms of the building repeats the relativity of the Escher.

3.7. Clear figures in logos and symbols

Figure 26 shows the logo of the French automotive company Renault. In 1972, its symbol was the impossible quadricle. Also, the impossible triangle in his logo uses the furniture store "Furniture hallucinations" (Fig.27).

Figure 26. Renault logo

Figure 27. Logo Furniture Store

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

Figure 28. Logo campaign "Russian windows"

Mathematics argue that both the palaces in which can be descended down the stairs leading up can exist. For this you only need to build such a structure not in three-dimensional, but, say, in four-dimensional space. And in the virtual world, which opens up a modern computer technology, and it can not be done. Nowadays, man's ideas are carried out, who still believed at the dawn in the existence of impossible worlds.

Practical part

Creating impossible figures

As the survey of my classmates showed, most of the guys do not know about the existence of impossible figures (Appendix 1), although many mechanically draw geometric shapes when they speak the phone, and easily depicted impossible shapes. For example, you can spend five, six or seven parallel lines, finish these lines in different ends in different ways - and the impossible figure is ready. If, for example, spend 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 more clearly imagine how they can exist. To do this, I took a scanner for gluing on the Internet (Appendix 2.3 and 4). The scan of the impossible triangle (tribara) printed on the printer. As a result, a figure was turned out, at first glance, a little similar to the tribar (Fig. 30).

Figure 30. Made Tribar

At first I thought that I was mistaken in the manufacture, but looking at her at a certain angle, everything turned out perfectly. I note that the correct angle of view and correct lighting are needed to create a complete illusion.

The following figures 31 and 32 show more complex figures, as well as MOT.

Figure 31. Impossible Figure 1

Figure 32. Impossible Figure 2

Conclusion

Impossible figures make our mind first see what should not be, then look for an answer - what is done not as the raisin of the paradox is hidden. And sometimes the answer is sometimes not so simple - it is hidden in the optical, psychological, logical perception of drawings.

The development of science, the need to think in a new way, the search for a beautiful - all these requirements of modern life makes looking for new methods that are able to change spatial thinking, imagination.

After having studied literature on the topic, you can answer the question "Do impossible figures exist in the real world?" I realized that the impossible is possible and unrealistic figures can be made with their own hands. I created the AMEM models of the "impossible triangle" and two more figures. I managed to show that impossible figures can exist in the real world.

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

Thus, it can be said that the world of impossible figures is extremely interesting and diverse. Work can be used in mathematics classes for the development of spatial thinking of students. For creative people prone to invention, impossible figures are a kind of lever to create something new, unusual. All this allows us to talk about the relevance of the topic under study.

Bibliography

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

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

Reethersward O. Impossible Figures. - M.: Stroyzdat, 1990, 206 p.

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

The impossible figure is one of the types of optical illusions, the figure that seems at first glance the projection of the usual three-dimensional object,

With a careful consideration of which the contradictory compounds of the elements of the figure become visible. The illusion of the impossibility of the existence of such a figure in three-dimensional space is created.

Impossible figures

The most famous impossible figures: the impossible triangle, an infinite staircase and an impossible trident.

Impossible triangle Perrose

Reutersvard Illusion (Reutersvard, 1934)

Pay attention also to the fact that the change in the organization "Figure-background" made possible perception located in the center of "Stars".
_________

Impossible cube Escher

In fact, all impossible figures can exist in the real world. Thus, all objects drawn on paper are projections of three-dimensional objects, therefore, you can create such a three-dimensional object, which when projection to the plane will look impossible. When looking at such an object from a specific point, it will also look impossible, but when a review from any other point, the effect of impossibility will be lost.

The 13-meter sculpture of the impossible triangle of aluminum was erected in 1999 in Perth (Australia). Here, the impossible triangle was depicted in the most general form - in the form of three beams connected to each other under the right corners.

Chestova fork
Among all the impossible figures, the impossible trident occupies a special place ("Damn Fork").

If you close the right side of the troll, we will see a completely real picture - three round teeth. If you close the lower part of the trident, we will also see the real picture - two rectangular teeth. But, if we consider the whole figure of the whole, it turns out that three round teeth are gradually turning into two rectangular.

Thus, it can be seen that the front and rear plans of this picture conflict. That is, what was originally in the foreground goes back, and the back plan (medium tooth) gets out forward. In addition to the change of the front and rear plans in this picture there is another effect - the flat faces of the right side of the trident become round in the left.

The effect of the inability is achieved due to the fact that our brain analyzes the contour of the figure and tries to calculate the number of teeth. The brain compares the number of teeth of the figure in the left and right part of the figure, because of which there is a feeling of the impossibility of the figure. If the number of teeth in the figure was significantly larger (for example, 7 or 8), then this paradox would be less pronounced.

Some books argue that the impossible trident belongs to the class of impossible figures that cannot be recreated in the real world. In fact, it is not. All impossible figures can be seen in the real world, but they will be impossible to look only from one single point of view.

______________

Impossible elephant


How many legs of the elephant?

Psychologist from Stepford Roger Shepard (Roger Shepard) used the idea of \u200b\u200ba trident for his picture of the impossible elephant.

______________

Penrose staircase (Infinite staircase, impossible staircase)

The infinite staircase "is one of the most famous classical impossibility.

It is the design of the staircase at which in the event of a movement along it in one direction (in the figure to the article counterclockwise), a person will raise an infinitely, and when moving in the opposite - constantly descend.

In other words, we appear the staircase leading, seemingly up or down, but at the same time a person walking along it does not rise and does not fall. After completing his visual route, it will be at the beginning of the way. If you really had to go through this stairs, you would be aimlessly climbed and descended the infinite number of times. You can call it an endless sympathetic work!

Since Penrouse published this figure, it appeared in print more often than any other impossible object. The "endless staircase" can be found in books about games, puzzles, illusions, in textbooks on psychology and other subjects.

"Climbing and descent"

The "Infinite Forestry" "was successfully used by the artist Mauritz K. Escher, this time in his charming lithography" climbing and descent "created in 1960.
In this picture, reflecting all the features of the Figure Figure, a completely recognizable endless staircase is neatly inscribed in the roof of the monastery. Monks in the hoods are continuously moving along the stairs in the direction clockwise and against it. They go towards each other by the impossible path. They are never able to go upstairs, nor go down.

Accordingly, the "endless staircase" became more likely to be associated with Escher, who resulted in it than with Penrose, who came up with it.

How many shelves are there?

Where is the door open?

Outside or inward?

Impossible figures occasionally appeared on the canvases of the masters of the past, for example, such a gallows on the picture of Peter Bruegel (senior)
"Forty on the gallows" (1568)

__________

Impossible Arch

Jos De Mey - Flemish artist, studied at the Royal Academy of Fine Arts in Ghent (Belgium), and then taught students design interiors and color for 39 years. Since 1968, the center has become drawing. It is most famous for the careful and realistic performance of impossible structures.

The most famous impossible figures in the works of the artist Maurice Escher. When viewed by such drawings, each individual item seems quite plausible, but when trying to trace the line, it turns out that this line is already, for example, not an outer angle of the wall, but internal.

"Relativity"

This lithography of the Dutch artist Escher was first printed in 1953.

On lithographs, a paradoxical world is depicted, in which the laws of reality do not apply. In one world, three reality are combined, three gravity are directed perpendicular to one other.

A architectural structure has been created, reality combined with stairs. For people living in this world, but in different planes of reality, the same staircase will be directed or up or down.

"Waterfall"

This lithography of the Dutch artist Escher was first printed in October 1961.

In this work, Escher depicts a paradox - the waterfall falling water controls the wheel, which directs water to the vertex of the waterfall. The waterfall has the structure of the "impossible" triangle of Penrose: lithography was created based on the article in the British magazine of psychology.

The design is made up of three crossbars, put on each other at right angles. Waterfall on lithography works as an eternal engine. It also seems that both towers are the same; In fact, the one is the right, on the floor below the left tower.

Well, more modern work: O)
Infinite photography

Amazing construction

Chess board

Inverted pictures

What do you see: a huge crow with prey or fisherman in a boat, fish and island with trees?

Rasputin and Stalin

Youth and old age

_________________

Welject and Queen

___________________

Angry and merry

Picture 1.

This is the impossible tribar. This drawing is not an illustration of a spatial object, since such an object cannot exist. Our eye takes this fact and the object itself without difficulty. We can come up with a number of arguments in defense of the impossibility of object for example, the face C lies in the horizontal plane, while the face A is tilted to us, and the face b, tilted from us, and, if the faces A and B divert each other, they are not May 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 inner corners is obtained equal to 270 degrees, which is impossible. We can attract the basic basic principles of stereometry, namely, that three non-parallel planes are always found at one point. However, in Figure 1, we see the following:

• The dark gray plane C is found with the plane B; Line of intersection - l.;
• A dark gray plane C occurs with a light gray plane A; Line of intersection - m.;
• White plane B occurs with light gray plane A; Line of intersection - n.;
• Lines intersection l., m., n. intersect in three different points.

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

We summarize: no matter how difficult or simple, our reasoning, the eye signals us about contradictions without any explanation on his part.

The impossible tribar is paradoxal in several ways. The eye requires a split second to transfer 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 decay into a variety of lines that do not have significant relationships with each other and no longer assembled in the form of tribara. However, this does not happen, and the eye signals again: "This is an object, tribar". In short, the conclusion is that it is the object and not an object, and this is the first paradox. Both interpretations have the same force as if the eye left the final verdict of the superior instance.

The second paradoxical feature of the impossible tribara arises from reasoning about its design. If the bar A is aimed to us, and the bar b - from us, and yet they are joined, then the angle that they form should lie in two places at the same time, one closer to the observer, and the other further. (The same applies to two other corners, since the object remains an identical form when they repel another angle up.)

Figure 2. Bruno Ernst, photo of the impossible tribara, 1985
Figure 3. Gerard Traarbach, "Perfect Timing", canvas / oil, 100x140 cm, 1985, printed on the contrary
Figure 4. Dirk Huizer, "Cube", Irisated ScreenPrint, 48x48 cm, 1984

Reality of impossible objects

One of the most difficult issues about impossible figures concerns their reality: Do they really exist or not? Naturally, the drawing of the impossible tribar exists, and this is not questioned. However, at the same time, there is no doubt that a three-dimensional form presented by the eye for us is not in the world around. For this reason, we decided to talk about impossible objects, not about impossible figures (Although, under such name in English, they are more known). It seems that this is a satisfactory solution to this dilemma. And yet, when we, for example, are examining attentively impossible tribar, its spatial reality continues to confuse us.

Faced with an object in the disassembled on some parts of the form, it is almost impossible to believe that, simply connecting bars and cubes with each other, you can get the desired impossible tribar.

Figure 3 is especially attractive for crystallography professionals. The object seems to be slowly growing crystal, Cubes are inserted into the existing crystal lattice without a violation of the overall structure.

A picture in Figure 2 is real, although the tribar composed of cigars and photographed at a certain angle is unreal. This is a visual joke, invented by Roger Penrose, the co-author of the first article and the impossible tribar.

Figure 5.

Figure 5 shows the tribar compiled from the numbered blocks of 1x1x1 dm. We can figure out the simple calculation of the blocks that the volume of the figure is 12 dm 3, and for goodbye - 48 dm 2.

Figure 6.
Figure 7.

In the same way, we can calculate the distance that the ladybug in the tribar will pass (Figure 7). The central point of each bar is numbered, and the direction of movement is marked by arrows. Thus, the surface of the tribar is represented as a long continuous road. Ladybug must make four complete circles before returning to the starting point.

Figure 8.

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

Let's set the question again: what actually makes the tribar figure, which can be interpreted with such a variety of ways. It should be remembered that the eye processes the image of the impossible object from the retina as well as images of ordinary items - stool or at home. The result is the "Spatial Image". At this stage there is no difference between the impossible tribar and the usual chair. Thus, the impossible tribar exists in the depths of our brain at the same level as all other objects surrounding us. The refusal of the eye to confirm the three-dimensional "viability" of the tribara in the reality in no way reduces the fact of the presence of the impossible tribar in our head.

In Chapter 1, we met with an impossible object, whose body disappeared to nowhere. In the pencil figure "Passenger train" (Fig. 11) FONS de Vogelaere finely used the same principle with a reinforced column in the left side of the picture. If we get a look at 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 (from which only two are visible). However, if you look at the same column from below, we will see a fairly wide opening, through which the train can drive. Solid stone blocks at the same time turn out to be ... Thinned air!

This object is easy enough to categorize, but it turns out to be quite complicated when we start it to analyze. Researchers, such as Broydrick Thro, showed that the very description of this phenomenon leads to contradictions. Conflict in one of the borders. The eye first calculates the contours, and then collects the shapes of them. The confusion occurs when the contours have two assignments at once in two different figures or pieces of the figure, as in Figure 11.

Figure 9.

A similar situation occurs in Figure 9. In this figure, the line contour l. It is also manifested both as a border of form A and as a border of the form B. However, it is not the boundary of both forms at the same time. If your eyes look first on the top of the drawing, then dropping down, line l. It will be perceived as the border of form A and will remain so as long as it is found that A is an open figure. At this point, the eye offers second interpretation for the line l., namely, that it is the border of form B. If you follow the view back up the line l., we will return again to the first interpretation.

If it were the only ambiguity, we could talk about a pictographic dual figure. But the conclusion is complicated by additional factors, such as the phenomenon of the disappearance of the shape against the background of the background, and, in particular, the spatial representation of the eye shape. In this regard, you can already take pictures of 7.8 and 9 from chapter 1 in a different way. Although these types of figures exhibit themselves as real spatial objects, we can temporarily call them impossible objects and describe them (but not explain) in the following common concepts: the eye calculates on the basis of these objects two different mutually exclusive three-dimensional forms, which, however, exist. At the same time. This can be seen in Figure 11 in the fact that it seems to us is a monolithic column. However, when re-examined, it appears to be open, with a spacious interval in the middle through which, as shown in the figure, a train can drive.

Figure 10. Arthur Stibbe, "In Front and Behind", cardboard / acrylic, 50x50 cm, 1986
Figure 11. FONS de Vogelaere, "Passenger Train", Picture Picture, 80x98 cm, 1984

Impossible object as a paradox

Figure 12. Oscar Reutersvärd, "Perspective Japonaise N ° 274 DDA", painted drawing of mascara, 74x54 cm

At the beginning of this chapter, we saw an impossible object as a three-dimensional paradox, that is, the image, whose stereographic elements are contradiction with each other. Before investigating this paradox deeper, it is necessary to understand whether such a phenomenon exists as a pictophic paradox. In fact, it exists - think about mermaids, sphinxes and other fabulous beings, often found in the visual arts of the Middle Ages and the early Renaissance. But in this case, the eye does not work out with such a pictographic equation as a woman + fish \u003d mermaid, and our knowledge (in particular, knowledge of biology), according to which such a combination is unacceptable. Only where the spatial data on the image from the retina mutually contradict each other, there is a failure of "automatic" data processing with the eye. The eye is not ready to handle so strange material, and we testify to the new visual experience for us.

Figure 13A. Harry Turner, drawing from the "Paradoxical Patterns" series, mixed appliances, 1973-78
Figure 13b. Harry Turner, "Corner", Mixed Technique, 1978

We can split the spatial information contained in the image from the retina (when we look only with one eye) into two classes - natural and cultural. The first class contains information on which the human cultural environment does not have any influence, and which is also detected in the paintings. Such a true "unspoken nature" includes the following:

• Objects of the same size look the less, the further they are. This is the main principle of a linear perspective, which plays a major role in the visual arts since revival;
• The object that partially lights up another object is closer to us;
• Objects or parts of the object, connected to each other, are at the same distance from us;
• Objects that are relatively far from us will be less distinguishable and will be hidden by the blue smoke of the spatial perspective;
• The side of the object on which the light falls is brighter than the opposite side, and the shadow indicate in the direction opposite to the light source.

Figure 14. Zenon Kulpa, "Impossible Figures", Mascara / Paper, 30x21 cm, 1980

In the cultural environment, the following two factors play an important role in our assessment of space. People created their living space so that direct corners are dominated. Our architecture, furniture and many tools are essentially composed of rectangles. We can say that we packed our world to a rectangular coordinate system, in the world of straight lines and corners.

Figure 15. Mitsumasa Anno, "Cube section"
Figure 16. Mitsumasa Anno, "Complex Wooden Puzzle"
Figure 17. Monika Buch, "Blue Cube", Acrylic / Wood, 80x80 cm, 1976

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

• The surface is a plane that continues until other details tell us that it has not ended;
• The corners in which there are three planes, determine the three main directions, and therefore, zigzag lines may indicate an extension or narrowing.

Figure 18. Tamas Farcas, Crystal, Irisated Print, 40x29 cm, 1980
Figure 19. Frans Erens, Watercolor, 1985

In our context, the difference between natural and cultural surroundings is very useful. Our visual feeling developed in a natural environment, and also it has an amazing ability to accurately and unmistakably process spatial information from the cultural category.

Impossible objects (at least most of them) exist due to the presence of spatial statements mutually contradictory. For example, on the picture of Josa de Maya "Double-Guarded Gateway To the Wintery Arcadia" (Fig. 20), a flat surface forming the upper part of the wall decays at the bottom of several planes that are at different distances from the observer. The impression of different distances is also formed by overlapping pieces of the figure in Arthur Stibbe "In Front and Behind" (Fig. 10), which are contrary to the rules of the flat surface. In the Watercolor Franks of Frans Erens (Fig. 19), the shelf shown in the run, decreasing in the amount of the end tells us that it is located horizontally, leaving away from us, and it is also attached to the supports in such a way as to be vertically. In the picture "The Five Bearers" FONS de Vogelaere (Fig. 21) we will be stunned by the number of stereographic paradoxes. Although the picture does not contain paradoxical overlap of objects, there are many paradoxical compounds in it. It is an interest in which the central figure is connected to the ceiling. Five pieces supporting the ceiling connect the parapet and the ceiling as large number of paradoxical connections that the eye is sent to an infinite search for a point with which it is better to consider them.

Figure 20. Jos de Mey, "Double-Guarded Gateway To the Wintery Arcadia", Canvas / Acryl, 60x70 CM, 1983
Figure 21. Fons de Vogelaere, "The Five Bearers", Picture Pencil, 80x98 cm, 1985

You might think that with each possible type of stereographic element, which appears in the picture, relatively easy to make a systematic overview of the impossible figures:

• Those that contain elements of perspectives in mutual conflict;
• Those in which the elements of the perspective in the conflict with the spatial information specified by overlapping elements;
• etc.

However, we will soon find that we will not be able to detect existing examples for many such conflicts, while some of the impossible objects will be difficult to enter into a similar system. However, such a classification will allow us to detect many still unknown types of impossible objects.

Figure 22. Shigeo Fukuda, "Images of Illusion", ScreenPrint, 102x73 cm, 1984

Definitions

In conclusion of this chapter, let's try to define the impossible objects.

In my first publication of pictures with impossible objects MK Escher, which appeared in about 1960, I came to the following wording: A possible object can always be considered as a projection - representation of a three-dimensional object. However, in the case of impossible objects, there is no three-dimensional object, whose representation is this projection, and in this case we can call the impossible object - the illusory representation. This definition is not only incomplete, but also wrong (we will return to Chapter 7), as it applies only to the mathematical side of the impossible objects.

Figure 23. Oscar Reutersvärd, "CUBIC ORGANIZATION OF SPACE", painted drawing of ink, 29x20.6 cm.
In fact, this space is not filled, since the larger cubes are not associated with smaller cubes.

Zenon Kulp suggests the following definition: The image of the impossible object is a two-dimensional figure that creates an impression of an existing three-dimensional object, and this figure cannot exist as we interpret it spatially; Thus, any attempt to create it leads to (spatial) contradictions, which are clearly visible to the viewer.

The last remark of Cultips offers one practical way to clarify whether the object is impossible or not: Just try creating it yourself. You will soon see, perhaps even before the design of the design that you cannot do this.

I would prefer the definition that emphasizes that the eye when analyzing the impossible object comes to two contradictory conclusions. I like it exactly exactly this definition, as it covers the reason for these mutually conflicting conclusions, and, in addition, it clarifies the fact that the impossibility is not the mathematical property of the figure, but the property of the interpretation of the figure of the viewer.

Based on this, I suggest the following definition:

The 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, as 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

Candidate of Technical Sciences D. Rakov (Institute of Machine Studies. A. A. Blagonravov RAS).

There is a large class of images, which can be said: "What we see? Something strange." These are drawings with a distorted perspective, and impossible objects in our three-dimensional world, and unthinkable combinations are completely real objects. Appearing at the beginning of the XI century, such "strange" drawings and photos today became a whole direction of art, called imp art.

William Hogard. "Impossible perspective", where intentionally made at least fourteen errors in the future.

Madonna with baby. 1025 year.

Peter Bruegel. "Forty on the gallows". 1568 year.

Oscar Ruetevard. "Opus 1" (№293AA). 1934 year.

Oscar Ruetevard. "Opus 2B". 1940.

Mauritz Cornelius Escher. "Climbing and descent."

Roger Penrose. "Impossible triangle". 1954 year.

Building a "impossible triangle".

Sculpture "impossible triangle", view from different sides. It is built of curvilinear elements and looks impossible only from one point.

Ill. 1. Morphological table classification of impossible objects.

The person begins to inspect the picture from the lower left corner (1), then the glance first turns to the middle (2), and then to the point 3.

Depending on the direction of the view, we see different objects.

The impossible alphabet is a combination of possible and impossible figures, among which there is even a frame element. Figure author.

Science and life // illustration

"Moscow" (Metro lines scheme) and "Two Lines of Fate". Drawings of the author; Computer processing. 2003. Figures demonstrate new opportunities for constructing schemes and graphs.

Science and life // illustration

Cube in Cuba ("Three Snails"). The rotated image has a greater degree of "impossibility" than the initial one.

"Chertova fork". Based on this figure, many impossible images have been created.

What do we see - a pyramid or opening?

A bit of history

Pictures with a distorted perspective are found already at the beginning of the first millennium. On a miniature of Henry II, created up to 1025 and stored in the Bavarian State Library in Munich, Madonna is drawn with a baby. The picture shows a set consisting of three columns, and the average column according to the laws of the perspective should be located ahead of Madonna, but is behind it, which gives the picture the effect of surrealism. We, unfortunately, never know whether this admission was the conscious act of the artist or his mistake.

The images of the impossible figures, not as a conscious direction in painting, but as techniques that enhance the effect of image perception are found in a number of middle-aged painters. On the Pieter Breughel's Piter (Pieter Breughel), created in 1568, the gallows are visible for the impossible design, which gives the effect of the whole picture as a whole. On the widely known engraving of the English artist of the XVIII century, William Hogarth (William Hogarth) "Fake Perspective" is shown to what absurdity can the artist can bring ignorance of the laws of perspective.

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

The founder of the direction of impossible art - Imp-Art (Imp-Art, Impossible ART) is rightly called the Swedish artist Oscar Ruetesevda (Oscar Reutersvard). The first impossible figure "Opus 1" (N 293AA) is drawn by a master in 1934. The triangle is made up of nine cubes. Experiments with unusual objects The artist continued and in 1940 created a figure of "Opus 2B", representing a reduced impossible triangle consisting of only three cubes. All cubes are real, but their location is impossible in three-dimensional space.

The same artist created a prototype "impossible stairs" (1950). The most famous classic figure "Impossible triangle" English mathematician Roger Penrose (Roger Penrose) created in 1954. He used a linear perspective, and not parallel, like Ruesevard, which gave the picture a depth and expressiveness and, therefore, a greater degree of impossibility.

M. K. Escher became the most famous artist Ipt Art (M. C. Escher). Among its most famous works are the paintings "Waterfall" ("Waterfall") (1961) and "climbing and descent" ("Ascending and Descending"). The artist used the effect of "infinite staircase", Open Ruetyvard and further supplemented by Penrose. On the canvase, two rows of men are depicted: when moving clockwise, the little people are constantly raised, and when moving counterclockwise, descended.

A little geometry

There are many ways to create optical illusions (from the Latin word "iliusio" - a mistake, error - inadequate perception of the subject 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 the plane (two-dimensional images), executed in such a way that the viewer creates an impression of the impossibility of the existence of a similar structure in our real three-dimensional world. Classical, as mentioned, and one of the most simple similar shapes is an impossible triangle. Each part of the figure (triangle angles) separately exists in our world, but their combination is impossible in three-dimensional space. The perception of the whole figure as the composition of incorrect connections between its real parts leads to the deceptive effect of the impossible structure. The glance slides on the edges of the impossible figure and is not able to perceive it as a logical integer. In fact, the look attempts to restore the real three-dimensional structure (see Figure), but encounters the inconsistency.

From a geometric point of view, the impossibility of the triangle is that three beams connected in pairs are alone on the other, but in three different axes of the Cartesian coordinate system, form a closed figure!

The process of perception of impossible objects is divided into two stages: identification of the shape as a three-dimensional object and awareness of the "incorrectness" of the object and the impossibility of its existence in the three-dimensional world.

The existence of impossible figures

Many believe that impossible figures are really impossible and they cannot be created in the real world. But it must be remembered that any drawing on a sheet of paper is a projection of a three-dimensional figure. Consequently, any figure drawn on a sheet of paper should exist in three-dimensional space. Impossible objects in pictures are projections of three-dimensional objects, and therefore objects can be implemented as sculptural compositions (three-dimensional objects). There are many ways to create them. One of them is to use curves of lines as the parties of the impossible trial. The created sculpture looks impossible only from a single point. From this point, the side curves look straight, and the target will be achieved - a real "impossible" object has been created.

About the benefits of imp art

Oscar Ruetyvard tells in the book "Omojliga Figurer" (there is a Russian translation) on the use of imp-art drawings for psychotherapy. He writes that paintings with their paradoxes are surprising, sharpen attention and desire to decipher. In Sweden, they are used in dental practice: considering paintings in the reception, patients are distracted from unpleasant thoughts in front of the dentist's cabinet. Remembering how much time it is necessary to wait for the reception in various kinds of Russian bureaucratic and other establishments, it can be assumed that the impossible pictures on the walls of the receptions may kill waiting time, soothing visitors and thereby reducing social aggression. Another option would be to install in receiving slot machines or, for example, mannequins with appropriate physiognomies as targets for darts, but, unfortunately, this kind of innovation was never encouraged in Russia.

Using phenomenon perception

Is it possible to somehow strengthen the effect of the impossibility? "Is it impossible" Are there any objects than others? And here the features of human perception come to the rescue. Psychologists have been established that the eye begins to inspect the object (pattern) from the left left corner, then the glance slides right to the center and falls into the lower right corner of the picture. Such a trajectory may be due to the fact that our ancestors at a meeting with the enemy first looked at the most dangerous right hand, and then the glance moved to the left, on the face and figure. Thus, artistic perception will significantly depend on how the composition of the painting is built. This feature in the Middle Ages is clearly manifested in the manufacture of tapestries: their drawing was a mirror reflection of the original, and the impression that tapestries and originals produce.

This property can be successfully used when creating creations with impossible objects, increasing or reducing the "degree of impossibility". The prospect of obtaining interesting compositions using computer technologies or from several paintings turned (maybe using a different type of symmetry) is one relative to another creating different impression from the viewers from the object and a deeper understanding of the essence of the plan, or from one, rotating ( Constantly or jerks) with a simple mechanism for some angles.

This direction can be called polygonal (polygonal). There are images that turned one relative to the other. The composition was created as follows: the drawing on paper, made in mascara and a pencil, was scanned, was translated into a digital form and processed in a graphic editor. It can be noted a pattern - the rotated picture has a greater "degree of impossibility" than the initial one. This is easily explained: the artist during the work subconsciously seeks to create a "correct" image.

Combinations, combination

There is a group of impossible objects, the sculptural implementation of which is impossible. The most, perhaps, known from them is the "impossible trident", or the "damn plug" (P3-1). If you carefully look at the object, you can see that three teeth are gradually moving into two on a total basis, leading to a conflict of perception. We compare the number of teeth on top and bottom and come to the conclusion about the impossibility of the object. Based on the "fork", a great set of impossible objects was created, including those where the cylindrical on one end the part becomes square on the other.

In addition to this illusion, there are many other types of optical deceptions of view (illusions of size, movement, colors, etc.). The illusion of the perception of depth is one of the most long-standing and well-known optical illusions. This group belongs to Necker cube (1832), and in 1895, Armand Thierry (Armand Thiery) published an article about the special form of impossible figures. This article first drawn the object, subsequently received the name of Thierry and countless times used by artists of OP-art. The object consists of five identical rhombuses with parties 60 and 120 degrees. In the figure you can see two cubes connected on one surface. If you look at the bottom up, the lower cube with two walls is clearly visible at the top, and if you turn down to top to bottom - the upper cube with the walls below.

The simplest figure of Thierry-like is, apparently, the illusion of the "Pyramid-Overview", which is the right rhombus with the line in the middle. It is impossible to say exactly what we see - a pyramid that rises above the surface, or the opening (depression) on it. This effect is used in the chart "Labyrinth (Pyramid Plan)" 2003. The picture received a diploma at the International Mathematical Conference and Exhibition in Budapest in 2003 "ARS (DIS) Symmetrica" \u200b\u200b03 ". The work used combinations of the illusion of the perception of depth and impossible figures.

In conclusion, we can say that the direction of impressed as an integral part of optical art is actively developing, and in the near future we will undoubtedly expect new discoveries in this area.

LITERATURE

Ruetevard O. Impossible Figures. - M.: Stroyzdat, 1990.

Signatures to illustrations

Ill. 1. The table built by the author of the article does not pretend to complete and strict order, but makes it possible to assess all the variety of impossible figures. The table is more than 300 thousand combinations of various elements. As illustrations, the graphics of the author and materials of the Vlad Alekseev site are used.