Impossible triangle. Penrose triangle: do it yourself from paper Do-it-yourself Impossible Penrose triangle

Also known as impossible triangle And tribar.

Story

This figure became widely known after the publication of an article on impossible figures in the British Journal of Psychology by the English mathematician Roger Penrose in 1958. In this article, the impossible triangle was depicted in its most general form- V the form of three beams connected to each other at right angles. Influenced by this article in Dutch artist Maurits Escher created one of his famous lithographs "Waterfall".

Sculptures

A 13-meter sculpture of an impossible triangle made of aluminum was erected in 1999 in Perth (Australia)

Deutsches Technikmuseum Berlin February 2008 0004.JPG

The same sculpture when changing the viewpoint

Other figures

Although it is quite possible to construct analogues of the Penrose triangle based on regular polygons, the visual effect from them is not so impressive. As the number of sides increases, the object simply appears bent or twisted.

see also

• Three rabbits (English) Three hares )

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An excerpt characterizing the Penrose Triangle

Greetings, dear readers of the blog site. Rustam Zakirov is in touch and I have another article for you, the topic of which is how to draw a Penrose triangle. Today I want to show you how easy and simple it is to draw an impossible triangle. We will draw two drawings of this triangle, one will be a regular one, and the second will be a real 3D drawing. And all this will be surprisingly simple. You can get a real 3D drawing of this triangle. I doubt that this will be shown to you anywhere else, so read the article to the end and very carefully.

For our drawings, as always, we will need: a piece of paper simple pencils(preferably one “medium”, “the other soft”) and several colored pencils or markers.

How to easily draw any 3D drawings.

I pulled out this impossible triangle from this ordinary picture, which I simply found on the Internet. Here she is.

And then in a couple of minutes I converted it to 3D with help . This way you can convert almost any image into 3D. If you want to learn the same way, click here.

And we move on to our drawing.

Draw a regular triangle pattern.

STEP #1. We translate from the monitor screen.

In order to draw a triangle, you will need to do the following. You take your piece of paper and lean it against the triangle on the monitor screen, and simply translate it.

And since our triangle is not at all complex, it is enough to put only the main points in all its corners.

And then we look at the original and connect these points using a ruler. I got it like this.

Our triangle is all ready. You can leave it like this, but let's decorate it a little more. I did this using colored pencils. After we have completely decorated our triangle, we completely outline it again with a simple soft pencil.

At this point, our usual Penrose triangle is completely ready, and we move on to the same triangle.

Draw a 3D drawing of a triangle.

STEP #1. We translate.

We proceed according to the same scheme as with a regular pattern. I give you a ready-made triangle, already translated into 3D format. Here he is.

And you translate it. We do everything the same as with a regular pattern. You take your sheet of paper, lean it against the monitor screen, the sheet of paper shines through, and you simply transfer the finished 3D drawing onto your sheet of paper.

This is what happened to me.

The size of the triangle can be increased or decreased. To do this, you just need to change the scale of your monitor. Hold down the Ctrl key and roll the mouse wheel.

We can safely say that our 3D drawing is already ready. It took me about 3 minutes. In principle, we can safely finish here, but let’s decorate our triangle some more.

Penrose triangle- one of the main impossible figures, also known as impossible triangle And tribar.

Penrose triangle (in color)

Story

This figure became widely known after the publication of an article on impossible figures in the British Journal of Psychology by the English mathematician Roger Penrose in 1958. Also in this article, the impossible triangle was depicted in its most general form - in the form of three beams connected to each other at right angles. Influenced by this article, the Dutch artist Maurits Escher created one of his famous lithographs, “Waterfall”.

3D print of a Penrose triangle

Sculptures

A 13-meter sculpture of an impossible triangle made of aluminum was erected in 1999 in Perth (Australia)

The same sculpture when changing the viewpoint

Other figures

Although it is quite possible to construct analogues of the Penrose triangle based on regular polygons, the visual effect from them is not so impressive. As the number of sides increases, the object simply appears bent or twisted.

see also

• Three rabbits (English) Three hares)

Illusionism - in a broad sense, is the name for a philosophical position regarding certain phenomena; for the way of considering such phenomena; in a narrow sense, it is the name for several specific philosophical theories.

The Cafe Wall Illusion is an optical illusion created by synergy. different levels neural mechanisms: retinal neurons and visual cortex neurons.

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

The Impossible Cube is an impossible figure invented by Escher for his lithograph Belvedere. This is a two-dimensional figure that superficially resembles the perspective of a three-dimensional cube, which is incompatible with a real cube. In the Belvedere lithograph, a boy sitting at the base of the building holds an impossible cube. A drawing of a similar Necker cube lies at his feet, while the building itself contains the same properties of an impossible cube.

The impossible cube borrows the ambiguity of the Necker cube, in which the edges are drawn as line segments and which can be interpreted in one of two different three-dimensional orientations.

The impossible cube is usually drawn as a Necker cube, in which the edges (segments) are replaced by seemingly solid bars.

In the Escher lithograph, the top four joints of the bars and the top intersection of the bars correspond to one of two interpretations of the Necker cube, while the bottom four connections and the bottom intersection correspond to the other interpretation. Other variations of the impossible cube combine these properties in other ways. For example, one of the cubes in the figure contains all eight connections according to one interpretation of the Necker cube, and both intersections correspond to another interpretation.

The apparent solidity of the bars gives the impossible cube greater visual ambiguity than the Necker cube, which is less likely to be perceived as an impossible object. Illusion plays on interpretation by the human eye two-dimensional drawing as a three-dimensional object. Three-dimensional objects can appear impossible when viewed from a certain angle and either by cutting the object in the right place or by using altered perspective, but human experience with rectangular objects makes impossible perceptions more likely than illusions in reality.

Other artists, including Jos De Mey, also painted works with the impossible cube.

A fabricated photograph of the supposedly impossible cube was published in the June 1966 issue of Scientific American, where it was called the "Frimish Cage." The impossible cube was placed on the Austrian postage stamp.

Blivet, also known as poyut or devil's pitchfork, is an inexplicable figure, an optical illusion, and an impossible figure. It seems that three cylindrical rods turn into two bars.

Oscar Rutersvärd (usual spelling of the surname in Russian-language literature; more correctly Reutersvärd), Swede. Oscar Reutersvärd (November 29, 1915, Stockholm, Sweden - February 2, 2002, Lund) - "father of the impossible figure", a Swedish artist who specialized in the depiction of impossible figures, that is, those that can be depicted (given the inevitable violations of perspective when representing 3-dimensional space on paper), but cannot be created. One of his figures received further development as the "Penrose triangle" (1934). Ruthersvard's work can be compared with Escher's work, however, if the latter used impossible figures as “bones” for the image fantasy worlds, then Rutersvärd was only interested in the figures as such. During his life, Ruthersvard depicted about 2,500 figures in isometric projection. Ruthersvard's books have been published in many languages, including Russian.

Maurits Cornelis Escher [ˈmʌu̯rɪts kɔrˈneːlɪs ˈɛʃər̥]; June 17, 1898, Leeuwarden, the Netherlands - March 27, 1972, Hilversum, the Netherlands) - a Dutch graphic artist. Known primarily for his conceptual lithographs, wood and metal engravings, in which he masterfully explored the plastic aspects of the concepts of infinity and symmetry, as well as the peculiarities of the psychological perception of complex three-dimensional objects, he is the most prominent representative of imp art.

supervisor

mathematic teacher

1.Introduction………………………………………………….……3

2. Historical background……………………………………..…4

3. Main part…………………………………………………………….7

4. Proof of the impossibility of the Penrose triangle......9

5. Conclusions……………………………………………………………..…………11

6. Literature……………………………………………….…… 12

Relevance: Mathematics is a subject studied from first to high school. Many students find it difficult, uninteresting and unnecessary. But if you look beyond the pages of the textbook, read additional literature, mathematical sophisms and paradoxes, your idea of ​​mathematics will change, and you will have a desire to study more than is studied in the school mathematics course.

Goal of the work:

show that the existence of impossible figures expands horizons, develops spatial imagination, and is used not only by mathematicians, but also by artists.

Tasks :

1. Study the literature on this topic.

2. Consider impossible figures, make a model of an impossible triangle, prove that an impossible triangle does not exist on the plane.

3. Make a development of an impossible triangle.

4. Consider examples of the use of the impossible triangle in the visual arts.

Introduction

Historically, mathematics has played an important role in the visual arts, particularly in perspective painting, which involves depicting a three-dimensional scene realistically on a flat canvas or piece of paper. According to modern views, mathematics and art disciplines very distant from each other, the first is analytical, the second is emotional. Mathematics does not play an obvious role in most jobs contemporary art, and, in fact, many artists rarely or never even use perspective. However, there are many artists whose focus is on mathematics. Several significant figures in the visual arts paved the way for these individuals.

In general, there are no rules or restrictions on the use of various themes in mathematical art, such as impossible figures, Möbius strips, distortion or unusual perspective systems, and fractals.

History of impossible figures

Impossible figures are a certain type of mathematical paradox, consisting of regular parts connected in an irregular complex. If we tried to formulate a definition of the term “impossible objects,” it would probably sound something like this - physically possible figures assembled in an impossible form. But it’s much more pleasant to look at them, drawing up definitions.

Errors in spatial construction were encountered by artists even a thousand years ago. But the Swedish artist Oscar Reutersvärd, who painted in 1934, is rightfully considered to be the first to construct and analyze impossible objects. the first impossible triangle, consisting of nine cubes.

Reutersvaerd's triangle

Independent of Reuters, English mathematician and physicist Roger Penrose rediscovers the impossible triangle and publishes its image in a British psychology journal in 1958. The illusion uses “false perspective.” Sometimes this perspective is called Chinese, since a similar method of drawing, when the depth of the drawing is “ambiguous,” was often found in the works of Chinese artists.

Escher Falls

In 1961 Dutchman M. Escher, inspired by the impossible Penrose triangle, creates the famous lithograph “Waterfall”. The water in the picture flows endlessly, after the water wheel it passes further and ends up back at the starting point. In essence, this is an image of a perpetual motion machine, but any attempt to actually build this structure is doomed to failure.

Another example of impossible figures is presented in the drawing “Moscow”, which depicts an unusual diagram of the Moscow metro. At first we perceive the image as a whole, but when we trace the individual lines with our gaze, we become convinced of the impossibility of their existence.

« Moscow", graphics (ink, pencil), 50x70 cm, 2003.

The “Three Snails” drawing continues the tradition of the second famous impossible figure - the impossible cube (box).

"Three Snails" Impossible Cube

Combination various objects can also be found in the not entirely serious picture “IQ” (intelligence quotient). Interestingly, some people do not perceive impossible objects because their minds are unable to identify flat pictures with three-dimensional objects.

Donald Simanek has suggested that understanding visual paradoxes is one of the hallmarks of that kind of creative potential which they have best mathematicians, scientists and artists. Many works with paradoxical objects can be classified as “intellectual” math games». Modern science speaks of a 7-dimensional or 26-dimensional model of the world. Such a world can only be modeled using mathematical formulas; humans simply cannot imagine it. This is where impossible figures come in handy.

A third popular impossible figure is the incredible staircase created by Penrose. You will continuously either ascend (counterclockwise) or descend (clockwise) along it. The Penrose model formed the basis famous painting M. Escher "Up and Down" The Incredible Penrose Staircase

Impossible trident

"Devil's Fork"

There is another group of objects that cannot be implemented. The classic figure is the impossible trident, or "devil's fork". If you carefully study the picture, you will notice that three teeth gradually turn into two on a single base, which leads to a conflict. We compare the number of teeth above and below and come to the conclusion that the object is impossible. If we close the upper part of the trident with our hand, we will see completely real picture- three round teeth. If we close the lower part of the trident, we will also see the real picture - two rectangular teeth. But, if we consider the entire figure as a whole, it turns out that three round teeth gradually turn into two rectangular ones.

Thus, it can be seen that the front and background of this picture conflict. That is, what was originally in the foreground goes back, and the background (middle tooth) comes forward. In addition to the change of foreground and background, there is another effect in this drawing - the flat edges of the upper part of the trident become round at the bottom.

Main part.

Triangle- a figure consisting of 3 adjacent parts, which, through unacceptable connections of these parts, creates the illusion of a mathematically impossible structure. This three-beam structure is also called differently square Penrose

The graphic principle behind this illusion owes its formulation to a psychologist and his son Roger, a physicist. The Penruzov square consists of 3 square bars located in 3 mutually perpendicular directions; each connects to the next at right angles, all of this is placed in three-dimensional space. Here's a simple recipe on how to draw this isometric projection of the Penrose square:

· Trim the corners of an equilateral triangle along lines parallel to the sides;

· Draw parallels to the sides inside the trimmed triangle;

· Trim the corners again;

· Draw parallels inside again;

· Imagine in one of the corners any of the two possible cubes;

· Continue it with an L-shaped “thing”;

· Run this design in a circle.

· If we had chosen a different cube, the square would have been “twisted” in the other direction .

Development of an impossible triangle.

Inflection line

Cut line

What elements are used to construct an impossible triangle? More precisely, from what elements does it seem to us (precisely it seems!) built? The design is based on a rectangular corner, which is obtained by connecting two identical rectangular bars at right angles. Three such corners are required, and therefore six pieces of bars. These corners must be visually “connected” to one another in a certain way so that they form a closed chain. What happens is an impossible triangle.

Place the first corner in the horizontal plane. We will attach a second corner to it, directing one of its edges upward. Finally, we attach a third corner to this second corner so that its edge is parallel to the original horizontal plane. In this case, the two edges of the first and third corners will be parallel and directed in different directions.

Now let’s try to look at the figure from different points in space (or make a real wire model). Imagine what it looks like from one point, from another, from a third... When the observation point changes (or - which is the same thing - when the structure is rotated in space), it will seem that the two “end” edges of our corners are moving relative to each other. It is not difficult to choose a position in which they will connect (of course, the near corner will seem thicker to us than the longer one).

But if the distance between the ribs is much less than the distance from the corners to the point from which we view our structure, then both ribs will have the same thickness for us, and the idea will arise that these two ribs are actually a continuation of one another.

By the way, if we simultaneously look at the display of the structure in the mirror, we will not see a closed circuit there.

And from the chosen observation point we see with our own eyes the miracle that has happened: there is a closed chain of three corners. Just do not change the point of observation so that this illusion (in fact, it is an illusion!) does not collapse. Now you can draw an object that you can see or place a camera lens at the found point and get a photograph of an impossible object.

The Penroses were the first to become interested in this phenomenon. They took advantage of the possibilities that arise when mapping three-dimensional space and three-dimensional objects onto a two-dimensional plane (that is, design) and drew attention to some of the uncertainty of design - an open structure of three corners can be perceived as a closed circuit.

As already mentioned, a simple model can be easily made from wire, which in principle explains the observed effect. Take a straight piece of wire and divide it into three equal parts. Then bend the outer parts so that they form a right angle with the middle part and rotate 900 relative to each other. Now turn this figure and watch it with one eye. At some position it will seem that it is formed from a closed piece of wire. By turning on the table lamp, you can observe the shadow falling on the table, which also turns into a triangle at a certain location of the figure in space.

However, this design feature can be observed in another situation. If you make a ring of wire and then spread it in different directions, you will get one turn of a cylindrical spiral. This loop, of course, is open. But when projecting it onto a plane, you can get a closed line.

We were once again convinced that from a projection onto a plane, from a drawing, a three-dimensional figure is reconstructed ambiguously. That is, the projection contains some ambiguity, understatement, which gives rise to the “impossible triangle.”

And we can say that the “impossible triangle” of the Penroses, like many other optical illusions, is on a par with logical paradoxes and puns.

Proof of the impossibility of the Penrose triangle

By analyzing the features of a two-dimensional image of three-dimensional objects on a plane, we understood how the features of this display lead to an impossible triangle.

It is extremely easy to prove that an impossible triangle does not exist, because each of its angles is right, and their sum is 2700 instead of the “positioned” 1800.

Moreover, even if we consider an impossible triangle glued together from angles less than 900, then in this case we can prove that an impossible triangle does not exist.

Let's consider another triangle, which consists of several parts. If the parts of which it consists are arranged differently, you will get exactly the same triangle, but with one small flaw. One square will be missing. How is this possible? Or is it still an illusion?

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Using the phenomenon of perception

Is there any way to enhance the effect of impossibility? Are some objects more "impossible" than others? And this is where features come to the rescue. human perception. Psychologists have found that the eye begins to examine an object (picture) from the lower left corner, then the gaze slides to the right to the center and drops to the lower right corner of the picture. This trajectory may be due to the fact that our ancestors, when meeting an enemy, first looked at the most dangerous right hand, and then the gaze moved to the left, to the face and figure. Thus, artistic perception will significantly depend on how the composition of the picture is constructed. This feature was clearly manifested in the Middle Ages in the manufacture of tapestries: their design was a mirror image of the original, and the impression produced by the tapestries and the originals differs.

This property can be successfully used when creating creations with impossible objects, increasing or decreasing the “degree of impossibility”. 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.

Conclusion

The use of various mathematical figures and laws is not limited to the above examples. By carefully studying all the given figures, you can find others not mentioned in this article. geometric bodies or visual interpretation of mathematical laws.

Mathematical fine arts are flourishing today, and many artists create paintings in Escher's style and in their own own style. These artists work in various directions, including sculpture, painting on flat and three-dimensional surfaces, lithography and computer graphics. And the most popular themes in mathematical art remain polyhedra, impossible figures, Möbius strips, distorted perspective systems and fractals.

Conclusions:

1. So, consideration of impossible figures develops our spatial imagination, helps us “get out” of the plane into three-dimensional space, which will help in the study of stereometry.

2. Models of impossible figures help to consider projections on a plane.

3. Consideration of mathematical sophisms and paradoxes instills interest in mathematics.

When performing this work

1. I learned how, when, where and by whom impossible figures were first considered, that there are many such figures, artists are constantly trying to depict these figures.

2. Together with my dad, I made a model of an impossible triangle, examined its projection onto a plane, and saw the paradox of this figure.

3. Examined reproductions of artists depicting these figures

4. My classmates were interested in my research.

In the future, I will use the acquired knowledge in mathematics lessons and I was interested in whether there are other paradoxes?

LITERATURE

1. Candidate of Technical Sciences D. RAKOV History of impossible figures

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

3. Website of V. Alekseev Illusions · 7 Comments

4. J. Timothy Unrach. – Amazing figures.
(AST Publishing House LLC, Astrel Publishing House LLC, 2002, 168 p.)

5. . - Graphic arts.
(Art-Rodnik, 2001)

6. Douglas Hofstadter. – Gödel, Escher, Bach: this endless garland. (Publishing house "Bakhrakh-M", 2001)

7. A. Konenko – Secrets of impossible figures
(Omsk: Levsha, 199)