Ability to create and operating with spatial images characterizes the level of general intellectual development person. IN psychological research has experimentally confirmed that between a person’s tendency to relevant professions and There is a statistically significant connection between the level of development of spatial concepts. Widespread use of impossible figures in architecture, painting, psychology, geometry and in many other areas of practical life provide an opportunity to learn more about various professions and decide on choice of future profession.

Keywords: tribar, endless staircase, space fork, impossible boxes, triangle and Penrose staircase, Escher cube, Reutersvaerd triangle.

Purpose of the study: studying the properties of impossible figures using 3-D models.

Research objectives:

1. Study the types and make a classification of impossible figures.
2. Consider ways to construct impossible figures.
3. Create impossible shapes using computer program and 3D modeling.

Concept of impossible figures

There is no objective concept of “impossible figures”. From one source impossible figure- view optical illusions, a figure that 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. And from another source impossible figures- these 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.

Analyzing different definitions, we come to the conclusion:

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.

When we look at an image that gives the impression of a spatial object, our spatial perception system tries to find spatial shape, determine orientation and structure, starting with the analysis of individual fragments and hints of depth. Next, these individual parts are combined and coordinated in some order to create a general hypothesis about the spatial structure of the entire object. Usually, although a flat image may have an infinite number of spatial interpretations, our interpretation mechanism selects only one - the most natural one for us. It is this interpretation of the image that is further tested for possibility or impossibility, and not the drawing itself. An impossible interpretation turns out to be contradictory in its structure - various partial interpretations do not fit into a common consistent whole.

Figures are impossible if their natural interpretations are impossible. However, this does not imply that there is not some other interpretation of the same figure that may exist. Thus, finding a method accurate description spatial interpretation of figures is one of the main ways for further work with impossible figures and mechanisms for their interpretation. If you are able to describe different interpretations, then you will be able to compare them, correlate the figure and its various interpretations (understand the mechanisms for creating interpretations), check their consistency or determine types of inconsistency, etc.

Types of impossible figures

Impossible figures are divided into two large classes: some have real three-dimensional models, while for others it is impossible to create such ones.

While working on the topic, 4 types of impossible figures were studied: tri-bar, endless staircase, impossible boxes and space fork. They are all unique in their own way.

Tribar (Penrose triangle)

This is a geometrically impossible figure, the elements of which cannot be connected. Still impossible triangle became possible. Swedish painter Oskar Reitesvärd first introduced the impossible triangle made of cubes to the world in 1934. In honor of this event, a Postage Stamp. Tribar can be made from paper. Origami lovers have found a way to create and hold in their hands a thing that previously seemed beyond the imagination of a scientist. However, we are deceived by our own eyes when we look at the projection of a three-dimensional object from three perpendicular lines. The observer thinks he sees a triangle, although in fact he does not.

Endless staircase.

The design, which has neither end nor edge, was invented by biologist Leionel Penrose and his mathematician son Roger Penrose. The model was first published in 1958, after which it gained great popularity, became a classic impossible figure, and its basic concept was used in painting, architecture, and psychology. The Penrose step model has gained the most popularity compared to the others unreal figures in the field computer games, puzzles, optical illusions. “Up the steps leading down” - this is how the Penrose staircase can be described. The idea of ​​this design is that when moving clockwise, the steps lead all the time upward, and in the opposite direction - downward. Moreover, the “eternal staircase” consists of only four flights. This means that after just four flights of stairs, the traveler ends up in the same place from where he started.

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 initially called it a "loose box" and stated that it was "designed for sending impossible objects in large numbers." 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.

Space fork.

Among all the impossible figures, the impossible trident (“space fork”) occupies a special place. If we close the right side of the trident with our hand, we will see completely real picture- three round teeth. If we close the lower part of the trident, we will also see the real picture - two rectangular teeth. But, if we consider the entire figure as a whole, it turns out that three round teeth gradually turn into two rectangular ones.

Thus, it can be seen that the front and background of this picture conflict. That is, what was originally on foreground goes back, and the back (middle tooth) comes forward. In addition to the change in foreground and background, there is another effect in this drawing - the flat edges of the right side of the trident become round on the left. The effect of impossibility is achieved due to the fact that our brain analyzes the contour of the figure and tries to count the number of teeth. The brain compares the number of teeth in the figure on the left and right sides of the picture, which gives rise to the feeling that the figure is impossible. If the number of teeth in the figure were significantly larger (for example, 7 or 8), then this paradox would be less pronounced.

Making models of impossible figures according to drawings

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.

Tribar

Paper model:

Impossible block

Paper model:

Construction of impossible figures inprogramImpossibleConstructor

The Impossible Constructor program is designed for constructing images of impossible figures from cubes. The main disadvantages of this program were the difficulty of choosing the right cube (it is quite difficult to find one desired cube out of 32 available in the program), as well as the fact that all variants of cubes were not provided. The proposed program provides a complete set of cubes to choose from (64 cubes), and also provides a more convenient way to find the required cube using the cube constructor.

Modeling impossible figures.

Seal 3Dmodels of impossible figureson the printer

During the work, models of four impossible figures were 3D printed.

Penrose triangle

Tribar creation process:

This is what I ended up with:

Escher cube

The process of creating a cube: Finally, the model was obtained:

Penrose staircase(after just four flights of stairs, the traveler ends up in the same place from where he started):

Reutersvaerd's triangle(the first impossible triangle, consisting of nine cubes):

The process of preparing for printing provided an opportunity to learn in practice how to construct stereometric figures on a plane, perform projections of the elements of figures onto a given plane, and think through algorithms for constructing figures. The created models helped to clearly see and analyze the properties of impossible figures, and compare them with known stereometric figures.

“If you can’t change the situation, look at it from a different angle.”

This quote directly relates to this work. Indeed, impossible figures exist if you look at them from a certain angle. The world of impossible figures is extremely interesting and diverse. They exist from ancient times to our time. They can be found almost everywhere: in art, architecture, popular culture, in painting, in icon painting, in philatelics. Impossible figures are of great interest to psychologists, cognitive scientists and evolutionary biologists, helping to understand more about our vision and spatial thinking. Today, computer technology, virtual reality and projections are expanding possibilities, so that controversial objects can be looked at with new interest. There are many professions that are somehow connected with impossible figures. All of them are in demand in modern world, and therefore the study of impossible figures is relevant and necessary.

Literature:

1. Reutersvard O. Impossible figures. - M.: Stroyizdat, 1990, 206 p.
2. Penrose L., Penrose R. Impossible objects, Quantum, No. 5, 1971, p. 26
3. Tkacheva M.V. Rotating cubes. - M.: Bustard, 2002. - 168 p.
4. http://www.im-possible.info/russian/articles/reut_imp/
5. http://www.impworld.narod.ru/.
6. Levitin Karl Geometrical Rhapsody. - M.: Knowledge, 1984, -176 p.
7. http://www.geocities.jp/ikemath/3Drireki.htm
8. http://im-possible.info/russian/programs/
9. https://www.liveinternet.ru/users/irzeis/post181085615
10. https://newtonew.com/science/impossible-objects
11. http://www.psy.msu.ru/illusion/impossible.html
12. http://referatwork.ru/category/iskusstvo/view/73068_nevozmozhnye_figury
13. http://geometry-and-art.ru/unn.html

Keywords: tribar, infinite staircase, space fork, impossible boxes, triangle and Penrose ladder, Escher cube, Reutersvaerd triangle.

Annotation: The ability to create and operate with spatial images characterizes the level of general intellectual development of a person. Psychological studies have experimentally confirmed that there is a statistically significant connection between a person’s inclination towards relevant professions and the level of development of spatial concepts. The widespread use of impossible figures in architecture, painting, psychology, geometry and many other areas of practical life make it possible to learn more about various professions and decide on the choice of a future profession.

What are impossible figures?
By entering such a question into a search engine, we will receive the answer: “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. An illusion is created of the impossibility of the existence of such a figure in three-dimensional space. (Wikipedia)"
I think that such an answer will not be enough for us to imagine and understand this concept, so let’s try to study this question better. Let's start with history.

Story
IN antique painting You can come across such a common phenomenon as distorted perspective. It was she who created the illusion of the impossibility of the object’s existence. In Pieter Bruegel the Elder’s painting “The Magpie on the Gallows,” such a figure is the gallows itself. But at that time, the creation of such “fables” was not a flight of fancy, but rather an inability to build a correct perspective.

Great interest in impossible figures arose in the twentieth century.

Swedish artist Oskar Rootesvard, passionate about creating something paradoxical and contrary to the laws of Euclidean geometry, created the following works: a triangle made of cubes “Opus 1”, and later “Opus 2B”.

In the 50s of the twentieth century, an article by the British mathematician Roger Penrose was published, devoted to the peculiarities of the perception of spatial forms depicted on a plane. Interested in the article big circle persons: psychologists began to study how our mind perceives such phenomena, scientists looked at these impossible figures as objects with special topological characteristics. Impossible art or impossibilism appeared - an art direction based on the creation of optical illusions and impossible figures.

Penrose's article inspired Maurits Escher to create several lithographs that brought him fame as an illusionist. One of his most famous works"Relativity". Escher depicted a model of the Penroses' "endless staircase".

Roger Penrose and his father Lionel Penrose invented a staircase that turns 90 degrees and locks itself. Therefore, if a person decided to climb it, he would not be able to rise higher. In the picture below you can see that the dog and the man are standing on the same level, which also adds to the impossibility of the picture. If the characters go clockwise, they will constantly go down, and if they go counterclockwise, they will go up.

It is impossible not to note the impossible Escher cube, which seems impossible because to the human eye It is common to perceive two-dimensional images as three-dimensional objects (you can read more about Escher).

And also a classic example of an impossible figure - the Trident. It is a figure with three round teeth at one end and rectangular ones at the other. This effect is achieved due to the fact that it is difficult to clearly say where the foreground is and where the background is.

Currently, the process of creating impossible figures continues. Below are some of them (the name of the creator is under the figure).

And it’s also impossible not to note the beautiful impossible figures created by our fellow countryman, Omsk resident Anatoly Konenko. For example:

Is it possible to see “impossible figures” in real life?

Many will say that impossible figures are truly unreal and cannot be recreated. Others will argue that the 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. So who is right?

The second ones will be closer to the correct answer. Indeed, it is possible to see “such” figures in reality, you just need to look at them with certain point. Using the pictures below, you can verify this.

Jerry Andrus and his impossible cube:

The impossible clutch of gears, also brought to reality by Jerry Andrus.

Sculpture of the Penrose Triangle (Perth, Australia), all sides of which are perpendicular to each other.

And this is how the sculpture looks from the other side.

If you like impossible figures, you can admire them

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 wooden bars, 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, author

large quantity 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.

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 in 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.

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 with a height of multi-storey building, place it in a wide 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 semi-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.
The impossible is what

that cannot exist... or happen...

The purpose of the lesson: development of three-dimensional vision of students; the ability to explain the impossibility of the existence of a particular figure from the point of view of geometry; development of interest in the subject. Equipment: newspaper based on materials from the site " Impossible world"(Internet), tools for constructing figures,

geometric figures

, illustrations of impossible figures.
During the classes:

Since ancient times, optical illusions have been used to enhance the impact of works of art or improve appearance architectural creations. The ancient Greeks used optical illusions to perfect the appearance of their great temples. During the Middle Ages, shifted perspective was sometimes used in painting. Later, many other illusions were used in graphics. Among them, the only one of its kind and a relatively new type of optical illusion is known as “impossible objects”.

One of the important skills for people working in technical fields is the ability to perceive three-dimensional objects in a two-dimensional plane. "Impossible Objects" is built on the use of tricks with perspective and depth within two-dimensional space. Impossible in real three-dimensional space, they affect our vision through displaced perspective, manipulation of depth and plane, deceptive optical cues, inconsistencies in plans, play of light and shadow, unclear connections, due to incorrect and contradictory directions and connections, altered code points and others. "tricks" that the graphic artist resorts to.

The deliberate use of impossible objects in design dates back to ancient times before the advent of classical perspective. Artists tried to find new solutions. An example is the 15th-century depiction of the Annunciation on the fresco of St. Mary's Cathedral in the Dutch city of Breda. The painting depicts the Archangel Gabriel bringing Mary news of her future Son. The fresco is framed by two arches, supported in turn by three columns. However, you should pay attention to the middle column. Unlike the others, she disappears into the background behind the stove. WITH practical point vision, the artist used this “impossibility” as a special technique to avoid dividing the scene into two halves.

An example of such an arch is shown in Fig. 1

"Impossible figures" are divided into 4 groups. Let's now try to analyze the main figures from each group. So, the first one:

Student 1:

An amazing triangle - 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".

Determine what is geometrically impossible.

(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. Try to explain their impossibility.

Triple warped tribar

Triangle of 12 cubes

Winged Tribar

Triple domino

Student 2:

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."

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

Try to explain what properties the creators of this staircase used.

(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).

It is suggested to look at one more figure. Stepped wall.

Student 3:

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.

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.” Perceiving and resolving (if possible) the inconsistency in this new type of ambiguous figure requires a real shift in visual fixation. 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.

Tower with four twin columns.

Student 4:

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."

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.

The Necker cube was first described in 1832 by Swiss crystallographer Lewis A. Necker, who noticed that crystals sometimes visually change shape when you look at them. 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.

A few more impossible figures.

Teacher:

Now try to create some impossible figure yourself.

The lesson ends with students trying to draw an impossible figure on their own.

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

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.

William Hogard. "Impossible Perspective", where at least fourteen errors in perspective are deliberately made.

Madonna and Child. 1025

Pieter Bruegel. "Magpie on the Gallows" 1568

Oscar Rootesward. "Opus 1" (No. 293aa). 1934

Oscar Rootesward. "Opus 2B". 1940

Maurits Cornelius Escher. "Ascent and descent."

Roger Penrose. "Impossible Triangle" 1954

Construction of the "impossible triangle".

Sculpture "Impossible Triangle", view from different sides. It is built from curved elements and looks impossible from just one point.

Ill. 1. Morphological table for the classification of impossible objects.

A person begins examining the picture from the lower left corner (1), then moves his gaze first to the middle (2), and then to point 3.

Depending on the direction we look, we see different objects.

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

Science and life // Illustrations

"Moscow" (metro line diagram) and "Two Lines of Fate". Drawings by the author; computer processing. 2003 The figures demonstrate new possibilities for creating diagrams and graphs.

Science and life // Illustrations

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

"Damn fork." Many impossible images have been created based on this figure.

What do we see - a pyramid or an opening?

A little history

Paintings with distorted perspective can be found already at the beginning of the first millennium. In a miniature from the book of Henry II, created before 1025 and kept in the Bavarian state library in Munich, Madonna and Child is painted. 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 impossible design that adds to the effect of the entire painting. In a well-known engraving of English artist XVIII century William Hogarth's "False Perspective" shows to what absurdity 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 (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 rather than parallel perspective like Rootesward, which gave the painting 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 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 the 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 objects can be realized in the form 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 here the peculiarities of human perception come to the rescue. Psychologists have found that the eye begins to examine an object (picture) from the lower left corner, then the gaze slides to the right to the center and 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 mirror image original, and the impression produced by tapestries and originals differs.

This property can be successfully used when creating creations with impossible objects, increasing or decreasing the “degree of impossibility”. The prospect of receiving interesting compositions using computer technology or from several pictures rotated (maybe using various types symmetries) one relative to the other, creating in viewers a different impression of the object and a deeper understanding of the essence of the design, or from one that rotates (constantly or jerkily) using a simple mechanism at certain angles.

This direction can be called polygonal (polygonal). The illustrations show images rotated relative to each other. The composition was created as follows: a drawing on paper, made in ink and pencil, was scanned, converted into digital form and processed in graphic editor. A regularity can be noted - the rotated picture has a greater “degree of impossibility” than the original one. This is easily explained: the artist, in the process of work, subconsciously strives to create the “correct” image.

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.

Besides this illusion, there are many other types optical illusions vision (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 special form 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.

LITERATURE

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

Captions for illustrations

Ill. 1. The table constructed by the author of the article does not pretend to be complete and strict in order, but makes it possible to evaluate the whole variety of impossible figures. The table contains more than 300 thousand combinations of various elements. Graphics from the author of the article and materials from Vlad Alekseev’s website were used as illustrations.