Impossible figures in the real world. Impossible reality What is the name of the infinite figure

29.06.2019

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Pride and Prejudice

We draw pictures ourselves. How to draw pictures. Required materials and tools

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GU Osmeryzhskaya basic secondary school

Impossible figures

Direction: Physics and Mathematics

Work performer : Dippel Sergey pupil of the 6th grade Osmeryzhskaya secondary school Pavlodar region Kachirsky district Osmeryzhsk village

Work supervisor: Dovzhenko Natalia Vladimirovna teacher of mathematics Osmeryzhskaya secondary school

year 2013

Summary / abstract / ……………………………………………………………… 2

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

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

2. Kinds of impossible figures ……………. …………………………………… .9

3. Oscar Ruthersward - the father of an impossible figure ………. ………………… ..16

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

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

References ……………………………………………………………… 22

Summary / abstract /

Project stages:

Stage 1.

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

Conducting conversations about impossible figures;

Statement of a problematic issue, motivation for the implementation of the project;

Carrying out preliminary work on the topic "Impossible figures";

Discussion and preparation of a step-by-step work plan, creation of a bank of ideas and proposals. Selection of sources of information.

Stage 2. Project implementation activities.

Information and educational conversations;

Information retrieval work;

Experimental work;

Literature review

Achievements of goals

Introduction

For some time now, I have been interested in such figures that at first glance seem to be ordinary, and if you look closely, you can see that something is wrong in them. The main interest for me was the so-called impossible figures, looking at which it seems that they cannot exist in the real world. I wanted to know more about them.

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

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

Even such simple volumetric forms as a cube, pyramid, parallelepiped can be represented as a combination of several figures located at different distances from the observer's eye. At the same time, there should always be a line along which the image of individual parts combining into a coherent 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 close examination of which, contradictory connections of the figure's elements become visible. The illusion of the impossibility of the existence of such a figure in three-dimensional space is created.

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

The question arose before me: "Are there impossible figures in the real world?"

Objectives of the project:

1. Find out how unreal figures are created.

2. Find areas of application of impossible figures.

Project objectives:

1. To study the literature on the topic "Impossible figures".

2. Make a classification of impossible figures.

3. Consider ways to construct impossible figures.

4. Create an impossible figure.

The topic of my work is relevant because understanding paradoxes is one of the signs of the kind of creative potential that the best mathematicians, scientists and artists possess. Many works with unreal objects can be attributed to "intellectual mathematical games". Such a world can be modeled only with the help of 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 that which will be simple and understandable for him. He cannot even imagine that some of the objects surrounding him may be "impossible". In fact, the world is one, but it can be viewed from different angles.

Impossible figures

A bit of history

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

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

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

In medieval Japanese and Persian painting, impossible objects are an integral part of the oriental art style, which gives only a general outline of the picture, the details of which "have to" be thought out by the viewer independently, in accordance with their preferences. Here is a school in front of us. Our attention is drawn to the architectural structure in the background, the geometric inconsistency of which is obvious. It can be interpreted both as the inner wall of the room and as the outer wall of a building, but both of these interpretations are wrong, 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 a distorted perspective are 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 the Madonna and Child. The painting depicts a vault consisting of three columns, and the middle column, according to the laws of perspective, should be located in front of the Madonna, but is behind her, which gives the painting the effect of unreality.

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

Meanwhile, everything is simple. Our perception is arranged in such a way that when processing a two-dimensional figure that has signs of perspective (i.e. volumetric space), the brain perceives it as three-dimensional, choosing the simplest way to convert 2D to 3D, guided by life experience, and as shown above, real prototypes "impossible" figures are rather sophisticated constructions with which our subconscious mind is unfamiliar, but even after meeting them, the brain continues to choose the simplest (from its point of view) transformation option and only after long training the subconscious mind finally "enters the situation" and the seeming abnormality of the "impossible figures" disappears.

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

At first glance, the architectural structure seems impossible, but after a second hitch, the consciousness finds a rescue option: the brickwork is in a plane perpendicular to the observer and rests on three columns, the tops of which seem to be located at an equal distance from the masonry, but in fact the empty space simply "disappears "due to the" well "chosen projection. After consciousness "decoded" the picture, it (and all images like it) is perceived as completely normal and geometric contradictions disappear as imperceptibly as they appear.

Impossible painting by Jos de Mei

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

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

Computer model of Escher's perpetual motion machine.

When you try to build an engine according to a drawing (or with a careful analysis of the latter), the "deception" pops up immediately - in three-dimensional space, such designs are geometrically contradictory and can exist only on paper, that is, on a plane, and the illusion of "volume" is created only due to the signs of perspective ( in this case - deliberately distorted) and in the drawing lesson for such a masterpiece we will easily be given two points, indicating the errors in the projection.

Types of impossible figures.

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

The amazing triangle is a tri-bar.

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

Here are some more examples of impossible tribar-based shapes.

Triple deformed tribar Triangle of 12 cubes

Winged Tribar Triple Domino

The acquaintance with impossible figures (especially in the performance of Escher) is, of course, overwhelming, but the fact that any of the impossible figures can be constructed in a real three-dimensional world is perplexing.

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

Let's go back to the Penrose Triangle and try to construct a three-dimensional figure, the projection of which on a two-dimensional plane would look like this. Naturally, such a problem cannot be solved directly, but if you think well and choose the right angle, then ... one of the possible options is shown in the figure.

Possible Impossible Penrose Triangle.

And here's another display from Mathieu Hemakers. There are many possible reverse mapping options. Lots of. Infinitely many!

The same Penrose Triangle from different angles.

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

Perose triangle in Australia

But it is worth changing the angle of view, as a triangle from "impossible" turns into a real and aesthetically unattractive structure that has nothing to do with triangles.

This is what the Penrose Triangle looks like in reality.

Endless staircase

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

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

The "Endless Staircase" was successfully used by the artist Maurits K. Escher, this time in his 1960 lithograph "Ascent and Descent".

Ladder with four or seven steps. To create this figure with many steps, the author may have been inspired by a bunch of ordinary railway 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 ladder took advantage of parallel lines when designing the end parts of blocks that are at the same distance; some blocks seem to be twisted to fit the illusion.

Space plug.

The next group of figures under the general name "Space Fork". With this figure, we enter the very core and essence of the impossible. Perhaps this is the most numerous class of impossible objects.

This notorious impossible object with three (or two?) Prongs 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 "The Brace Consisting of Three Elements".

From a practical point of view, this strange trident or mechanism in the form of a bracket is absolutely inapplicable. Some call it just a "annoying mistake". One of the representatives of the aerospace industry suggested using its properties in the design of an interdimensional space tuning fork.

Impossible boxes

"Crazy Box" is a cube frame 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 a point is either in the foreground or in the background, it jumps from one position to another.

Oscar Ruthersward is the father of an impossible figure.

The "father" of impossible figures is considered to be Swedish artist Oskar Ruthersward. Swedish artist Oskar Ruthersward, a specialist in creating images of impossible figures, claimed that he was poorly versed in mathematics, but, nevertheless, raised his art to the rank of science, creating a whole theory of creating impossible figures according to a certain number of templates.

A pair of impossible figures from Oscar Reutersvard.

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

Another kind is dubious impossible figures. These figures do not represent a single solid body. They are a connection of two or more shapes. They can neither be painted nor light and shadow applied to them.

A true impossible figure consists of a fixed number of possible elements, and a dubious one "loses" a certain number of elements if you follow them with your eyes.

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

In the figure, we see three variants of dubious impossible figures. On the left is a three-seven-bar, built of seven lines, in which three beams turn into seven. A figure in the middle, constructed 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, Ruthersward painted about 2,500 figures. Ruthersward'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 piece of paper is a projection of a three-dimensional figure. Therefore, any shape drawn on a piece of paper must exist in 3D space. Impossible objects in the paintings are projections of three-dimensional objects, which means that objects can be realized in the form of sculptural compositions. There are many ways to create them. One of them is using 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 is created.

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

I made my impossible figure. I took forty-two cubes and glued them together to form a cube in which part of the edge is missing. Note that to create a complete illusion, you need the right angle of view and the right lighting.

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

Applying impossible figures

Impossible figures sometimes find unexpected uses. Oskar Ruthersward talks in his book "Omojliga figurer" about the use of imp-art drawings for psychotherapy. He writes that the pictures, with their paradoxes, cause surprise, sharpen 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: 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 direction in painting, called impossibilism. The Dutch artist Escher is referred to as impossibilists. The famous lithographs "Waterfall", "Ascent and Descent" and "Belvedere" belong to him. The artist used the "endless staircase" effect discovered by Rutesward.

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

The most famous use of impossible figures in popular culture is the Renault logo

Mathematicians argue 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 already in the virtual world, which modern computer technology opens up to us, not this can be done. 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 an answer - what has been done wrong, in which the zest of the paradox is hidden. And sometimes it is not so easy to find the answer - it is hidden in the optical, psychological, logical perception of the drawings.

The development of science, the need to think in a new way, the search for the beautiful - all these requirements of modern life make us 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 by hand. I created the Ames Impossible Cube model. After looking at ways to construct impossible shapes, I was able to draw my impossible shapes. I was able to show that

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

There are many more areas in which impossible shapes 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 the point of view of geometry. The work can be used in mathematics classes to develop students' spatial thinking. For creative people inclined to invention, impossible figures are a kind of lever for creating something new and unusual.

Bibliography

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

Penrose L., Penrose R. Impossible objects, Kvant, no. 5,1971, p. 26

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

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

Internet resources:

http://wikipedia.tomsk.ru

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

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

Ability to create and to operate with spatial images characterizes the level of general intellectual development of a person. IN psychological research has experimentally confirmed that between a person's tendency to relevant professions and by the level of development of spatial representations, there is a statistically reliable connection. The 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 different professions and decide on choice of a future profession.

Keywords: tribar, endless staircase, space plug, impossible boxes, triangle and Penrose ladder, Escher cube, Reuterswerd triangle.

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

Research objectives:

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

Impossible figures concept

There is no objective concept of "impossible figures". From one source impossible figure- a kind of optical illusion, a figure that seems to be a projection of an ordinary three-dimensional object, upon close examination of which, contradictory connections of the figure's elements become visible. And from another source impossible figures are geometrically contradictory images of objects that do not exist in real three-dimensional space. Impossibility arises from the contradiction between the subconsciously perceived geometry of the depicted space and formal mathematical geometry.

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 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 form, orientation and structure, starting with the analysis of individual fragments and hints of depth. Further, these separate parts are combined and coordinated in some order to create a general hypothesis about the spatial structure of the whole object. Usually, despite the fact that a flat image can have an infinite number of spatial interpretations, our interpretation mechanism chooses only one - the most natural for us. It is this interpretation of the image that is further tested for possibility or impossibility, and not the drawing itself. The impossible interpretation turns out to be contradictory in its structure - different partial interpretations do not fit the general consistent whole.

Figures are impossible if their natural interpretations are impossible. However, this does not imply that there is no other interpretation of the same figure that may exist. Thus, finding a method for accurately describing the spatial interpretations of figures is one of the main ways for further work with impossible figures and the mechanisms of their interpretation. If you are able to describe different interpretations, then you can compare them, correlate the figure and its different interpretations (understand the mechanisms of creating interpretations), check their correspondence or determine the types of inconsistencies, etc.

Types of impossible figures

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

In the course of work on the topic, 4 types of impossible figures were studied: a tribar, an endless staircase, impossible boxes and a space fork. They are all unique in their own way.

Tribar (Penrose triangle)

It is a geometrically impossible figure whose elements cannot be connected. After all, the impossible triangle became possible. Swedish painter Oskar Reyteswerd in 1934 for the first time presented to the world an impossible triangle made of cubes. In honor of this event, a postage stamp was issued in Sweden. Tribar can be made from paper. Origami lovers have found a way to create and hold in their hands a thing that seemed previously an out-of-the-box fantasy of a scientist. However, we are deceived by our own eyes when we look at the projection of a three-dimensional object of three perpendicular lines. It seems to the observer that he sees a triangle, although in reality it is not.

Endless staircase.

The design, which has no end or 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 classical impossible figure, and its basic concept was applied in painting, architecture, and psychology. The Penrose stair model has gained the greatest popularity in comparison with other unreal figures in the field of computer games, puzzles, optical illusions. "Up the stairs leading down" - this is how you can describe the Penrose stairs. The idea of this design is that when moving clockwise, the steps lead all the time up, and in the opposite direction - down. Moreover, the "eternal staircase" consists of only four flights. This means that after just four flights of stairs, the traveler finds himself in the same place from where he began to move.

Impossible boxes.

Another impossible object appeared in 1966 in Chicago as a result of the original experiments of the photographer Dr. Charles F. Cochran. Many fans of impossible figures have experimented with the Crazy Box. The author originally called it a "free box" and stated that it was "designed to send large quantities of impossible objects." "Crazy Box" is a cube frame 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 a point is either in the foreground or in the background, it jumps from one position to another.

Space plug.

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, then we will see a very real picture - three round teeth. If we close the lower part of the trident, then we will also see a real picture - two rectangular teeth. But, if we consider the whole figure as a whole, it turns out that three round teeth gradually turn into two rectangular ones.

Thus, you can see that the foreground and background of this drawing are in conflict. That is, what was originally in the foreground goes back, and the background (middle tooth) crawls out forward. In addition to changing the foreground and background, this figure has another effect - the flat edges of the right side of the trident become round in the left. The impossibility effect 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 drawing, which makes the figure feel impossible. If the number of teeth in the figure was significantly greater (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 viewed in space, all cracks and bends become visible in space, 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 exist only in the imagination of a person, but not in space.

Tribar

Paper model:

Impossible bar

Paper model:

Construction of impossible figures inprogramImpossibleConstructor

The Impossible Constructor program is designed to construct images of impossible figures from cubes. The main disadvantages of this program were the complexity of choosing the required cube (it is quite difficult to find one of the 32 cube available in the program), as well as the fact that all the cube options 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 cubes constructor.

Modeling impossible figures.

Printing 3Dmodels of impossible figureson the printer

In the course of work, models of four impossible figures were printed on a 3D printer.

Penrose triangle

Tribar creation process:

Here's what I ended up with:

Escher's cube

The process of creating a cube: Finally, a model is obtained:

Penrose Ladder(after just four flights of stairs, the traveler finds himself in the same place from where he began to move):

Reuterswärd Triangle(the first impossible triangle of nine cubes):

The process of preparing for printing made it possible in practice to learn how to build stereometric figures on a plane, to perform projections of figure elements on a given plane and to think over algorithms for constructing figures. The created models helped to visually see and analyze the properties of impossible figures, compare them with the well-known stereometric figures.

"If you can't change the situation, look at it from a different angle."

This quote is directly related 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 have existed from ancient times to our time. They can be found almost everywhere: in art, architecture, mass culture, painting, icon painting, philatelicism. Impossible figures are of great interest to psychologists, cognitive scientists, and evolutionary biologists, helping to learn more about our vision and spatial thinking. Today, computer technology, virtual reality and projection are expanding possibilities so that conflicting objects can be looked at with renewed interest. There are many professions that are somehow associated with impossible figures. All of them are in demand in the modern world, and therefore the study of impossible figures is relevant and necessary.

Literature:

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

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

Annotation: The ability to create and operate with spatial images characterizes the level of a person's general intellectual development. In psychological studies, it has been experimentally confirmed that there is a statistically reliable relationship between a person's inclination to appropriate professions and the level of development of spatial representations. The widespread use of impossible figures in architecture, painting, psychology, geometry and in many other areas of practical life makes it possible to learn more about various professions and decide on the choice of a future profession.

Picture 1.

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

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

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

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

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

The second paradoxical feature of the impossible tribar arises from reasoning about its construction. If bar A is directed towards us, and bar B is directed away from us, and yet they join, then the angle that they form should lie in two places at the same time, one closer to the observer, and the other further away. (The same applies to the other two corners, as the object remains the same shape when rotated upward at the other angle.)

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

Reality of impossible objects

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

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

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

The photograph in Figure 2 is real, although a tribar made up of cigar boxes and photographed from a specific angle is not real. This is a visual joke invented by Roger Penrose, co-author of the first article and impossible tribar.

Figure 5.

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

Figure 6.
Figure 7.

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

Figure 8.

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

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

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

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

Figure 9.

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

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

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

Impossible object as a paradox

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

At the beginning of this chapter, we saw an impossible object as a three-dimensional paradox, that is, an image whose stereographic elements contradict each other. Before investigating this paradox in more depth, it is necessary to understand whether such a phenomenon exists as a pictoraphic paradox. It actually exists - think of mermaids, sphinxes, and other fabulous creatures common in the visual arts of the Middle Ages and early Renaissance. But in this case, it is not the work of the EYE that is violated by such a pictographic equation as woman + fish = mermaid, but our knowledge (in particular, knowledge of biology), according to which such a combination is unacceptable. Only where the spatial data on the retinal image mutually contradict each other does the "automatic" processing of the data by the EYE fail. The EYE is not ready to handle such strange material, and we witness a new visual experience for us.

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

We can divide the spatial information contained in the retinal image (when looking with only one eye) into two classes - natural and cultural. The first class contains information that the cultural environment of a person does not have any influence, and which is also found in paintings. Such true "unspoiled nature" includes the following:

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

Picture 14. Zenon Kulpa, "Impossible Figures", ink / paper, 30x21 cm, 1980

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

Figure 15. Mitsumasa Anno, "Cross-section of a cube"
Figure 16. Mitsumasa Anno, "Hard Wooden Puzzle"
Figure 17. Monika Buch, "Blue Cube", acrylic / wood, 80x80 cm, 1976

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

A surface is a plane that continues until other details tell us that it's not finished;
The angles at which the three planes meet define three main directions, and therefore, zigzag lines can indicate expansion or contraction.

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

In our context, the distinction between natural and cultural surroundings is very useful. Our sense of sight evolved in a natural environment, and it also has an amazing ability to accurately and accurately process spatial information from a cultural category.

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

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

You might think that with every possible type of stereographic element that appears in a painting, it is relatively easy to compile a systematic overview of impossible figures:

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

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

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

Definitions

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

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

Figure 23. Oscar Reutersvärd, "Cubic organization of space", colored ink drawing, 29x20.6 cm.
This space is not actually full because the larger cubes are not associated with the smaller cubes.

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

Kulpa's last remark offers one practical way to find out if an object is impossible or not: just try to create it yourself. You will soon see, perhaps even before you start designing, that you cannot do this.

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

Based on this, I propose the following definition:

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

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

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

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

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

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

2.3. Oscar Ruthersward - the father of an impossible figure ……………………… ..11

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

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

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

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

Introduction

"The World of Impossible Figures" is one of the most interesting themes, which received its rapid development only at the beginning of the twentieth century. However, much earlier, many scientists and philosophers have dealt with this issue. Even such simple volumetric forms as a cube, pyramid, parallelepiped can be represented as a combination of several figures located at different distances from the observer's eye. At the same time, there should always be a line along which the image of individual parts combining into a coherent picture.

The question arose before me: "Are there impossible figures in the real world?"

Objectives of the project:

1.Find out tohow createdunreal figures appear.

2. Find applicationsimpossible figures.

Project objectives:

1. To study literature on the topic "Impossible figures".

2 .Create a classificationimpossible figures.

3.PConsider ways to construct impossible figures.

4.It's impossible to createfigure.

Impossiblefigures

A bit of history

In medieval Japanese and Persian painting, impossible objects are an integral part of the oriental art style, which gives only a general outline of the picture, the details of which the viewer “has to think out” on their own, in accordance with their preferences. Here is a school in front of us. Our attention is drawn to the architectural structure in the background, the geometric inconsistency of which is obvious. It can be interpreted both as the inner wall of the room and as the outer wall of a building, but both of these interpretations are wrong, 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.

Viewsimpossible figures.

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

The amazing triangle is a tri-bar.

Here are some more examples of impossible tribar-based shapes.

Triple deformed tribar

Triangle of 12 cubes

Winged tribar

Triple domino

Endless staircase

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

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

The creators of this ladder took advantage of parallel lines when designing the end parts of blocks that are at the same distance; some blocks seem to be twisted to fit the illusion.

Space plug.

The next group of figures under the general name "Space Fork". With this figure, we enter the very core and essence of the impossible. Perhaps this is the most numerous class of impossible objects.

From a practical point of view, this strange trident or mechanism in the form of a bracket is absolutely inapplicable. Some call it just a “annoying mistake”. One of the representatives of the aerospace industry suggested using its properties in the design of an interdimensional space tuning fork.

Impossible boxes

"Crazy Box" is a cube frame 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 a point is either in the foreground or in the background, it jumps from one position to another.

Oscar Rutersward - the father of an impossible figure.

A true impossible figure consists of a fixed number of possible elements, and a dubious one "loses" a certain number of elements if you follow them with your eyes.

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

figures, when talking on the phone, this has already 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 be finished as two beams on one side and three on the other.

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

Impossible figures are possible!

Russian artist Anatoly Konenko, our contemporary, divided impossible figures into 2 classes: some can be modeled 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, it turned out a cube, in which part of the edge is missing. Note that to create a complete illusion, you need the right angle of view and the right lighting.

I studied impossible figures using Euler's theorem and came to the following conclusion: Euler's theorem, which is true for any convex polyhedron, is not true for impossible figures, but is true for their Ames models.

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

Applying impossible figures

Impossible figures sometimes find unexpected uses. Oscar Ruthersward talks in the book “Omojliga figurer” about the use of imp-art drawings for psychotherapy. He writes that the pictures, with their paradoxes, cause surprise, sharpen 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: 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 direction in painting, called impossibilism. The Dutch artist Escher is referred to as impossibilists. The famous lithographs "Waterfall", "Ascent and Descent" and "Belvedere" belong to him. The artist used the “endless staircase” effect discovered by Rutesward.

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

The most famous use of impossible figures in popular culture is Renault carmaker logo

Conclusion.

After studying the literature on the topic, I was able to answer the question "Are there impossible figures in the real world?" I realized that the impossible is possible and unreal figures can be made by hand. I created the Ames Impossible Cube model and tested Euler's theorem on it. After looking at ways to construct impossible shapes, I was able to draw my impossible shapes. I was able to show that

Conclusion 1: All impossible figures can exist in the real world.

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

Conclusion3: There will be many more areas in which impossible shapes will be used.