Impossible figures and their modeling. Impossible figures in the real world Impossible world in the real world

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:

Introduction:
Throughout history, people have encountered optical illusions of one kind or another. Suffice it to recall the mirage in the desert, illusions created by light and shadow, as well as relative movement. The following example is widely known: the moon rising from the horizon appears much larger than it is high in the sky. All these are just a few interesting phenomena that occur in nature. When these phenomena, which deceive the eyes and the mind, were first noticed, they began to excite the imagination of people.

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 the 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. From a practical point of view, 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 sort out 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.

To create this figure with big amount The author's 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. Step wall.

Student 3:

The next group of figures under common name"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.

Many people believe that impossible figures are truly impossible and they cannot be created in real world. However, from school course In geometry, we know 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 infinite number. The same applies to impossible figures.

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

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

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

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

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

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

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

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

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

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

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

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

While an impossible triangle is relatively easy to construct in the real world, creating an impossible trident in three-dimensional space 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.

Impossible figures - a special type of objects in the fine arts. Typically they are called that because they cannot exist in the real world.

More precisely, impossible figures are geometric objects drawn on paper that give the impression of an ordinary projection of a three-dimensional object, however, upon careful examination, contradictions in the connections of the elements of the figure become visible.

Impossible figures are classified as a separate class of optical illusions.

Impossible constructions have been known since ancient times. They have been found in icons since the Middle Ages. A Swedish artist is considered the “father” of impossible figures Oscar Reutersvard, who drew an impossible triangle made from cubes in 1934.

Impossible figures became known to the general public in the 50s of the last century, after the publication of an article by Roger Penrose and Lionel Penrose, in which two basic figures were described - the impossible triangle (which is also called the trianglePenrose) and an endless staircase. This article came into the hands of a famous Dutch artistM.K. Escher, who, inspired by the idea of ​​impossible figures, created his famous lithographs "Waterfall", "Ascent and Descent" and "Belvedere". Following him, a huge number of artists around the world began to use impossible figures in their work. The most famous among them are Jos de Mey, Sandro del Pre, Ostvan Oros. The works of these, as well as other artists, are distinguished into a separate direction visual arts - " imp-art" .

It may seem that impossible figures really cannot exist in three-dimensional space. Eat certain ways, which allow you to reproduce impossible figures in the real world, although they will only look impossible from one vantage point.

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

Article from the journal Science and Life "Impossible Reality" download

Oscar Ruthersward(the spelling of the surname customary in Russian-language literature; more correctly Reuterswerd), ( 1 915 - 2002) is a Swedish artist who specialized in depicting impossible figures, that is, those that can be depicted, but cannot be created. One of his figures received further development like the Penrose triangle.

Since 1964, professor of history and art theory at Lund University.

Rutersvard was greatly influenced by the lessons of the Russian immigrant, professor at the Academy of Arts in St. Petersburg, Mikhail Katz. He created the first impossible figure - an impossible triangle made from a set of cubes - by accident in 1934. Later, over the years of creativity, he drew more than 2,500 different impossible figures. All of them are made in a parallel “Japanese” perspective.

In 1980, the Swedish government released a series of three postage stamps with paintings by the artist.

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

There is a large class of images about which one can say: “What do we see? Something strange.” These are drawings with a distorted perspective, and impossible in our three dimensional world objects, 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. The well-known engraving by the 18th century English artist William Hogarth, “False Perspective,” shows the absurdity to which an artist’s ignorance of the laws of perspective can lead.

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

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

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

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

A bit of geometry

There are many ways to create optical illusions (from the Latin word “iliusio” - error, delusion - inadequate perception of an object and its properties). One of the most spectacular is the direction of imp art, based on images of impossible figures. Impossible objects are drawings on a plane (two-dimensional images), executed in such a way that the viewer gets the impression that such a structure cannot exist in our real three-dimensional world. Classic, as already mentioned, and one of the simplest such figures is the impossible triangle. Each part of the figure (the corners of the triangle) exists separately in our world, but their combination in three-dimensional space is impossible. Perceiving the entire figure as a composition of irregular connections between its real parts leads to the deceptive effect of an impossible structure. The gaze glides along the edges of 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 is that three beams connected in pairs to one another, but along three different axes Cartesian system coordinates, 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 claim to be complete and strict order, but makes it possible to appreciate 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.

Introduction………………………………………………………………………………..2

Main part. Impossible figures……………….…………………………4

2.1. A little history……………………………………………………….4

2.2. Types of impossible figures…………………………………………….6

2.3. Oscar Ruthersward – father of the impossible figure………………………..11

2.4. Impossible figures are possible!……………………………………..13

2.5. Application of impossible figures……………………………………14

Conclusion…………………………………………………………………………………..15

Bibliography………………………………………………………………16

Introduction

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

“The World of Impossible Figures” is one of the most interesting topics, which received its rapid development only at the beginning of the twentieth century. However, much earlier, many scientists and philosophers dealt with this issue. Even such simple volumetric shapes as a cube, pyramid, parallelepiped can be represented as a combination of several figures located at different distances from the observer’s eye. There should always be a line along which the images of individual parts are combined into a complete picture.

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

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

Project goals:

1. Find out what toak createdUnreal figures appear.

2. Find applicationsimpossible figures.

Project objectives:

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

2 .Make a classificationimpossible figures.

3.PConsider ways to construct impossible figures.

4.It is impossible to createnew figure.

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

Impossiblenew figures

A little history

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

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

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

Kindsimpossible figures.

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

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”. 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.

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.

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

Space fork.

The next group of figures is collectively called “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.”

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

Impossible boxes

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

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

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

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

Oscar Ruthersvard - father of the impossible figure.

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

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

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

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

One version of these impossible figures is very easy to perform, and many of those who automatically draw geometric

figures when talking on the phone, this has been done more than once. You need to draw five, six or seven parallel lines, finish these lines at different ends in different ways - and the impossible figure is ready. If, for example, you draw five parallel lines, then they can end up as two beams on one side and three on the other.

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

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

Impossible figures are possible!

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

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

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

I studied impossible figures using Euler's theorem and came to to the following conclusion: Euler's theorem, true for any convex polyhedron, is false for impossible figures, but true for their Ames models.

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

Application of impossible figures

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

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

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

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

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

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

Conclusion.

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

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

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

Conclusion1: All impossible figures can exist in the real world.

Conclusion2: Euler's theorem, true for any convex polyhedron, is false for impossible figures, but true for their Ames models.

Conclusion 3: There will be many more areas in which impossible figures will be used.

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

Bibliography

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

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

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

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