The impossible is still possible. And a vivid confirmation of this is the impossible Penrose triangle. Discovered in the last century, it is still often found in the scientific literature. And no matter how surprising it may sound, you can even make it yourself. And it's quite easy to do so. Many lovers of drawing or collecting origami have been able to do this for a long time.
There are several names for this figure. Some call it an impossible triangle, others just a tribar. But most often you can find the definition of exactly the "Penrose triangle".
Under these definitions, one of the main impossible figures is understood. Judging by the name, it is impossible to get such a figure in reality. But in practice, it has been proven that it is still possible to do this. It will only take shape if you look at it from a certain point at the right angle. From all other sides, the figure is quite real. It represents three edges of a cube. And it is easy to make such a design.
The Penrose Triangle was discovered back in 1934 by Swedish artist Oscar Reutersvärd. The figure was presented in the form of cubes assembled together. In the future, the artist began to be called the "father of impossible figures."
Perhaps the Reutersvärd drawing would have remained little known. But in 1954, Swedish mathematician Roger Penrose wrote a paper on impossible figures. This was the second birth of the triangle. True, the scientist presented it in a more familiar form. He did not use cubes, but beams. Three beams were connected to each other at an angle of 90 degrees. The difference was also that Reutersvärd used parallel perspective while painting. And Penrose applied a linear perspective, which made the drawing even more impossible. Such a triangle was published in 1958 in a British psychology journal.
In 1961, the artist Maurits Escher (Holland) created one of his most popular lithographs, Waterfall. It was created under the impression that was caused by the article on impossible figures.
In the 1980s, tribar and other impossible figures were depicted on the state postage stamps of Sweden. This went on for several years.
At the end of the last century (more precisely in 1999), an aluminum sculpture was created in Australia, depicting the impossible Penrose triangle. It reached a height of 13 meters. Similar sculptures, only smaller in size, are also found in other countries.
As you might have guessed, the Penrose triangle is not really a triangle in the usual sense. It is three sides of a cube. But if you look from a certain angle, you get the illusion of a triangle due to the fact that 2 angles completely coincide on the plane. The near from the viewer and the far corners are visually combined.
If you are careful, you can guess that the tribar is nothing more than an illusion. The actual appearance of the figure can give out a shadow from it. It shows that in fact the corners are not connected. And, of course, everything becomes clear if you pick up the figure.
Penrose triangle can be assembled independently. For example, from paper or cardboard. And the diagrams will help in this. They just need to be printed and glued. There are two diagrams on the Internet. One of them is a little easier, the other is more difficult, but more popular. Both are shown in the pictures.
The Penrose triangle will be an interesting product that guests will definitely like. It certainly won't go unnoticed. The first step to create it is to prepare the schema. It is transferred to paper (cardboard) using a printer. And then it's even easier. It just needs to be cut around the perimeter. The diagram already has all the necessary lines. It will be more convenient to work with thicker paper. If the diagram is printed on thin paper, but you want something denser, the blank is simply applied to the selected material and cut out along the contour. To prevent the circuit from moving, it can be attached with paper clips.
Next, you need to determine the lines along which the workpiece will bend. As a rule, it is represented in the diagram. We bend the part. Next, we determine the places that are subject to gluing. They are coated with PVA glue. The part is combined into a single figure.
Detail can be painted. And you can initially use colored cardboard.
The Penrose Triangle can also be drawn. To begin with, a simple square is drawn on the sheet. Its size doesn't matter. With the base on the bottom side of the square, a triangle is drawn. Small rectangles are drawn in its corners inside. Their sides will need to be erased, leaving only those that are in common with the triangle. The result should be a triangle with truncated corners.
A straight line is drawn from the left side of the upper lower corner. The same line, but slightly shorter, is drawn from the lower left corner. A line extending from the right corner is drawn parallel to the base of the triangle. It turns out the second dimension.
According to the principle of the second, the third dimension is drawn. Only in this case, all lines are based on the angles of the figure, not the first, but the second dimension.
Dmitry Rakov
Our eyes cannot see
the nature of the objects.
So don't force them
mental delusions.
Titus Lucretius Kar
The common expression "deception of the eye" is essentially wrong. The eyes cannot deceive us, because they are only an intermediate link between the object and the human brain. Optical deception usually arises not because of what we see, but because we unconsciously reason and involuntarily err: "through the eye, and not with the eye, the mind knows how to look at the world."
One of the most spectacular trends in the artistic flow of optical art (op-art) is imp-art (imp-art, impossible art), based on the image of impossible figures. Impossible objects are drawings on a plane (any plane is two-dimensional), depicting three-dimensional structures, the existence of which is impossible in the real three-dimensional world. The classic and one of the simplest shapes is the impossible triangle.
In an impossible triangle, each corner is itself possible, but a paradox arises when we consider it as a whole. The sides of the triangle are directed both towards the viewer and away from him, so its individual parts cannot form a real three-dimensional object.
As a matter of fact, our brain interprets a drawing on a plane as a three-dimensional model. Consciousness sets the "depth" at which each point of the image is located. Our ideas about the real world are in conflict, with some inconsistency, and we have to make some assumptions:
The human mind first creates a general image of the object, and then examines the individual parts. Each angle is compatible with spatial perspective, but when reunited, they form a spatial paradox. If you close any of the corners of the triangle, then the impossibility disappears.
Errors in spatial construction were encountered by artists a thousand years ago. But the first to build and analyze impossible objects is considered to be the Swedish artist Oscar Reutersvärd, who in 1934 painted the first impossible triangle, which consisted of nine cubes.
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"Moscow", graphics (ink, pencil), 50x70 cm, 2003 |
Independently of Reutersvaerd, the English mathematician and physicist Roger Penrose rediscovers the impossible triangle and publishes its image in the British Psychology Journal in 1958. The illusion uses "false perspective". Sometimes such a perspective is called Chinese, since a similar way of drawing, when the depth of the drawing is "ambiguous", was often found in the works of Chinese artists.
In the "Three Snails" drawing, the small and large cubes are not oriented in the normal isometric view. The smaller cube mates with the larger one on the front and back sides, which means, following three-dimensional logic, it has the same dimensions of some sides as the large one. At first, the drawing seems to be a real representation of a solid body, but as the analysis proceeds, the logical contradictions of this object are revealed.
Drawing "Three snails" continues the traditions of the second famous impossible figure - the impossible cube (box).
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"IQ", graphics (ink, pencil), 50x70 cm, 2001 |
"Up and down", M. Escher |
The combination of different objects can also be found in the not-so-serious "IQ" (intelligence quotient) figure. It is interesting that some people do not perceive impossible objects due to the fact that their consciousness is not able to identify flat pictures with three-dimensional objects.
Donald E. Simanek has opined that understanding visual paradoxes is one of the hallmarks of the kind of creativity possessed by the best mathematicians, scientists, and artists. Many works with paradoxical objects can be attributed to "intellectual mathematical games". Modern science speaks of a 7-dimensional or 26-dimensional model of the world. It is possible to model such a world only with the help of mathematical formulas; a person is simply not able to imagine it. This is where impossible figures come in handy. From a philosophical point of view, they serve as a reminder that any phenomena (in systems analysis, science, politics, economics, etc.) should be considered in all complex and non-obvious relationships.
A variety of impossible (and possible) objects are presented in the painting "The Impossible Alphabet".
The third popular impossible figure is the incredible staircase created by Penrose. You will continuously either ascend (counterclockwise) or descend (clockwise) along it. Penrose's model formed the basis of M. Escher's famous painting "Up and Down" ("Ascending and Descending").
There is another group of objects that cannot be implemented. The classic figure is the impossible trident, or "devil's fork".
Upon careful study of the picture, you can see that three teeth gradually turn into two on a single basis, which leads to a conflict. We compare the number of teeth from above and below and come to the conclusion that the object is impossible.
Is there any greater use for impossible drawings than mind games? In some hospitals, images of impossible objects are specially hung up, since their examination can occupy patients for a long time. It would be logical to hang such drawings at the box office, in the police and other places where waiting for one's turn sometimes takes forever. The drawings could act as a kind of "chronophages", i.e. time wasters.
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 closer 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 most famous impossible figures are the impossible triangle, the endless staircase and the impossible trident.
Impossible Perrose Triangle
The Reutersvard Illusion (Reutersvard, 1934)
Note also that the change in the figure-ground organization made it possible to perceive the centrally located "star".
_________
Escher's impossible cube
In fact, all impossible figures can exist in the real world. So, all objects drawn on paper are projections of three-dimensional objects, therefore, it is possible to create such a three-dimensional object that, when projected onto a plane, will look impossible. When looking at such an object from a certain point, it will also look impossible, but when viewed from any other point, the effect of impossibility will be lost.
The 13-meter aluminum sculpture of the impossible triangle was erected in 1999 in the city of Perth (Australia). Here the impossible triangle was depicted in its most general form - in the form of three beams connected to each other at right angles.
Devil's fork
Among all the impossible figures, the impossible trident ("devil's fork") occupies a special place. 
If you close the right side of the trident with your 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 forward. In addition to changing the foreground and background, this drawing has another effect - the flat edges of the right side of the trident become round in the left.
The effect of impossibility is achieved due to the fact that our brain analyzes the contour of the figure and tries to count the number of teeth. The brain compares the number of teeth of the figure in the left and right parts of the picture, which causes a feeling of the impossibility of the figure. If the figure had a significantly larger number of teeth (for example, 7 or 8), then this paradox would be less pronounced.
Some books claim that the impossible trident belongs to a class of impossible figures that cannot be recreated in the real world. Actually it is not. ALL impossible figures can be seen in the real world, but they will look impossible only from one single point of view.
______________
impossible elephant
How many legs does an elephant have?
Stanford psychologist Roger Shepard used the idea of a trident for his picture of the impossible elephant.
______________
Penrose stairs(endless staircase, impossible staircase)
The Infinite Stair is one of the most famous classical impossibilities.
It is a staircase design in which, in the case of movement along it in one direction (counterclockwise in the figure to the article), a person will rise indefinitely, and when moving in the opposite direction, he will constantly descend.
In other words, we see a staircase leading, it would seem, 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 be at the beginning of the path. If you really had to walk up that ladder, you would go up and down it aimlessly an infinite number of times. You can call it an endless Sisyphean labor!
Since the Penroses published this figure, it has appeared in print more often than any other impossible object. The "Endless Stair" can be found in books about games, puzzles, illusions, textbooks on psychology and other subjects.
"Ascent and Descent"
The "Endless Stairway" was successfully used by the artist Maurits K. Escher, this time in his charming 1960 Ascending and Descent lithograph.
In this drawing, which reflects all the possibilities of the Penrose figure, the quite recognizable Endless Staircase is neatly inscribed in the roof of the monastery. The hooded monks move continuously up the stairs in a clockwise and counter-clockwise direction. They go towards each other on an impossible path. They never manage to go up or down.
Accordingly, The Endless Stair became more often associated with Escher, who redrawn it, than with the Penroses, who conceived it.
How many shelves are there?
Where is the door open?
Out or in?
Impossible figures occasionally appeared on the canvases of the masters of the past, for example, such is the gallows in the painting by Pieter Brueghel (the Elder)
"Magpie on the gallows" (1568)
__________
Impossible arch
Jos de Mey is a Flemish artist who studied at the Royal Academy of Fine Arts in Ghent (Belgium) and then taught interior design and color to students for 39 years. Beginning in 1968, drawing became his focus. He is best known for his meticulous and realistic execution of impossible structures.
The most famous impossible figures in the works of the artist Maurice Escher. When considering such drawings, each individual detail seems quite plausible, however, when trying to trace the line, it turns out that this line is already, for example, not the outer corner of the wall, but the inner one.
"Relativity"
This lithograph by the Dutch artist Escher was first printed in 1953.
The lithograph depicts a paradoxical world in which the laws of reality do not apply. Three realities are united in one world, three forces of gravity are directed perpendicular to one another.
An architectural structure has been created, realities are connected by stairs. For people living in this world, but in different planes of reality, the same ladder will be directed either up or down.
"Waterfall"
This lithograph by the Dutch artist Escher was first printed in October 1961.
This work by Escher depicts a paradox - the falling water of a waterfall controls a wheel that directs water to the top of the waterfall. The waterfall has the structure of the "impossible" Penrose triangle: the lithograph was created based on an article in the British Journal of Psychology.
The design is made up of three crossbars laid on top of each other at right angles. The waterfall on the lithograph works like a perpetual motion machine. It also seems that both towers are the same; actually the one on the right, one floor below the left tower.
Well, more modern work: o)
Endless photography
Amazing construction
Chess board
What do you see: a huge crow with prey or a fisherman in a boat, a fish and an island with trees?
Rasputin and Stalin
Youth and old age
_________________
Noble and Queen
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 the first to the final grade. 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, then the idea of mathematics will change, there will be a desire to study more than is studied in the school mathematics course.
Objective:
to show that the existence of impossible figures will broaden one's horizons, develop spatial imagination, 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 a plane.
3. Unfold the impossible triangle.
4. Consider examples of the use of the impossible triangle in fine art.
Introduction
Historically, mathematics has played an important role in the visual arts, in particular in the depiction of perspective, which involves realistically depicting a three-dimensional scene on a flat canvas or sheet of paper. According to modern views, mathematics and fine arts are very distant disciplines from each other, the first is analytical, the second is emotional. Mathematics does not play an obvious role in most contemporary art work, and, in fact, many artists rarely or never even use perspective. However, there are many artists who focus 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 topics in mathematical art, such as impossible figures, the Möbius strip, distortion or unusual systems of perspective, and fractals.
History of impossible figures
Impossible figures are a certain kind of mathematical paradox, consisting of regular pieces connected in an irregular complex. If you try 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 looking at them is much more pleasant, drawing up definitions.
Errors in spatial construction were encountered by artists a thousand years ago. But the first to build and analyze impossible objects is considered to be the Swedish artist Oscar Reutersvärd, who painted in 1934. the first impossible triangle, consisting of nine cubes.
Reutersvärd triangle
Independent of Reutersvaerd, the English mathematician and physicist Roger Penrose rediscovers the impossible triangle and publishes its image in the British Psychological Journal in 1958. The illusion uses "false perspective". Sometimes such a perspective is called Chinese, since a similar way of drawing, when the depth of the drawing is “ambiguous”, was often found in the works of Chinese artists.
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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 falls back to the starting point. In fact, this is an image of a perpetual motion machine, but any attempt in reality to build this design is doomed to failure.
Another example of impossible figures is presented in the drawing "Moscow", which depicts an unusual scheme of the Moscow metro. At first, we perceive the image as a whole, but tracing the individual lines with our eyes, we are convinced of the impossibility of their existence.
«
Moscow”, graphics (ink, pencil), 50x70 cm, 2003
Drawing "Three snails" continues the traditions of the second famous impossible figure - an impossible cube (box).
"Three snails" Impossible cube
The combination of various objects can also be found in the not-so-serious "IQ" (intelligence quotient) figure. It is interesting that some people do not perceive impossible objects due to the fact that their consciousness is not able to identify flat pictures with three-dimensional objects.
Donald Simanek opined that understanding visual paradoxes is one of the hallmarks of the kind of creativity that the best mathematicians, scientists, and artists possess. Many works with paradoxical objects can be classified as "intellectual mathematical games". Modern science speaks of a 7-dimensional or 26-dimensional model of the world. It is possible to model such a world only with the help of mathematical formulas; a person is simply not able to imagine it. This is where impossible figures come in handy.
The 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 of the famous painting by M. Escher "Up and Down" The Incredible Penrose Stairs
Impossible Trident
"Damn Fork"
There is another group of objects that cannot be implemented. The classic figure is the impossible trident, or "devil's fork". Upon careful study of the picture, you can see that three teeth gradually turn into two on a single basis, which leads to a conflict. We compare the number of teeth from above and below and come to the conclusion that the object is impossible. If you close the upper part of the trident with your 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 forward. In addition to changing the foreground and background, this drawing has another effect - the flat edges of the upper part of the trident become round at the bottom.
Main part.
Triangle- a figure consisting of 3 adjoining parts, which, with the help of unacceptable connections of these parts, creates the illusion of an impossible structure from a mathematical point of view. In another way, this three-bar is also called 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 bars of square section, located in 3 mutually perpendicular directions; each one connects to the next at right angles, all of which fit into three-dimensional space. Here is a simple recipe for how to draw this isometric view of a Penrose square:
Trim the corners of an equilateral triangle along lines parallel to the sides;
Draw parallels to the sides inside the cropped triangle;
Trim the corners again
Once again, draw inside the parallels;
· Imagine one of the two possible cubes in one of the corners;
· Continue it with an L-shaped “thing”;
Run this design in a circle.
If we chose another cube, then the square would be “twisted” in the other direction .
Development of an impossible triangle.
break line
cutting line
What elements make up an impossible triangle? More precisely, from what elements does it seem to us (it seems!) Built? The design is based on a rectangular corner, which is obtained by connecting two identical rectangular bars at a right angle. Three such corners are required, and the bars, therefore, six pieces. These corners must be visually “connected” to each other in a certain way so that they form a closed chain. What happens is the impossible triangle.
Place the first corner in a horizontal plane. We will attach the second corner to it, directing one of its edges up. Finally, we add 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.
And now let's try to soapy look at the figure from different points in space (or make a real model of wire). Imagine how it looks from one point, from another, from a third ... When changing the observation point (or - which is the same - when the structure is rotated in space), it will seem that the two "end" edges of our corners move relative to each other. It is not difficult to find a position in which they will connect (of course, in this case, 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 are viewing 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, then we will not see a closed circuit there.
And from the chosen point of observation, we see with our own eyes a 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 you 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 used the possibilities that arise when mapping three-dimensional space and three-dimensional objects onto a two-dimensional plane (that is, when designing) and drew attention to some design uncertainty - an open design of three corners can be perceived as a closed circuit.
As already mentioned, the simplest model can be easily made from wire, which explains in principle the observed effect. Take a straight piece of wire and divide it into three equal parts. Then bend the extreme parts so that they form a right angle with the middle part, and rotate relative to each other by 900. Now turn this figurine and observe it with one eye. At a certain position, it will seem that it is formed from a closed piece of wire. Turning on the table lamp, you can watch the shadow falling on the table, which also turns into a triangle at a certain position 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 get one turn of a cylindrical spiral. This loop is, of course, open. But when projecting it onto a plane, you can get a closed line.
We have once again seen that the projection onto the plane, according to the drawing, the three-dimensional figure is restored ambiguously. That is, the projection contains some ambiguity, understatement, which give 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
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 “placed” 1800.
Moreover, even if we consider an impossible triangle glued together from corners less than 900, then in this case it can be proved that the impossible triangle does not exist.
Consider another triangle, which consists of several parts. If the parts of which it consists are arranged differently, then exactly the same triangle will be obtained, but with one small flaw. One square will be missing. How is this possible? Or is it just an illusion.
https://pandia.ru/text/80/021/images/image016_2.jpg" alt="(!LANG:Impossible triangle" width="298" height="161">!}
Is there any way to increase the impossibility effect? Are some objects "impossible" than others? And here the features of human perception come to the rescue. Psychologists have established that the eye begins to examine the object (picture) from the lower left corner, then the gaze slides to the right to the center and descends to the lower right corner of the picture. Such a trajectory may be due to the fact that our ancestors, when meeting with the enemy, first looked at the most dangerous right hand, and then their gaze moved to the left, at the face and figure. Thus, artistic perception will significantly depend on how the composition of the picture is built. This feature in the Middle Ages was clearly manifested in the manufacture of tapestries: their design was a mirror image of the original, and the impression made by tapestries and originals differs.
This property can be successfully used when creating creations with impossible objects, increasing or decreasing the "degree of impossibility". It also opens up the prospect of obtaining interesting compositions using computer technology, either from several pictures rotated (perhaps using different types of symmetries) one relative to the other, creating a different impression of the object and a deeper understanding of the essence of the idea, or from one that is rotated ( constantly or jerkily) using a simple mechanism at some angles.
Such a direction can be called polygonal (polygonal). The illustrations show images rotated one relative to the other. The composition was created as follows: a drawing on paper, made in ink and pencil, was scanned, digitized and processed in a graphics editor. We can note a regularity - the rotated picture has a greater "degree of impossibility" than the original one. This is easily explained: in the process of work, the artist subconsciously strives to create the "correct" image.
The use of various mathematical figures and laws is not limited to the above examples. By carefully studying all the given figures, you can also find other geometric bodies that are not mentioned in this article or a visual interpretation of mathematical laws.
Mathematical visual arts are flourishing today, and many artists create paintings in Escher's style and in their own style. These artists work in a variety of mediums, including sculpture, painting on flat and three-dimensional surfaces, lithography and computer graphics. And the most popular topics of mathematical art are polyhedra, impossible figures, Möbius strips, distorted systems of perspective and fractals.
Conclusions:
1. So, the consideration of impossible figures develops our spatial imagination, helps to “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 the plane.
3. Consideration of mathematical sophisms and paradoxes instills interest in mathematics.
When doing 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 projections on a plane, saw the paradox of this figure.
3. Examined the reproductions of artists, which depict these figures
4. My studies interested my classmates.
In the future, I will use the acquired knowledge in mathematics lessons and I was interested, but are there other paradoxes?
LITERATURE
1. Candidate of Technical Sciences D. RAKOV History of impossible figures
2. Impossible figures.- M.: Stroyizdat, 1990.
3. Alekseeva Illusions · 7 Comments
4. J. Timothy Anrach. - Amazing figures.
(LLC "Publishing House AST", LLC "Publishing House Astrel", 2002, 168 p.)
5. . - Graphic arts.
(Art-Spring, 2001)
6. Douglas Hofstadter. - Gödel, Escher, Bach: this endless garland. (Publishing house "Bahrakh-M", 2001)
7. A. Konenko - Secrets of impossible figures
(Omsk: Lefty, 199)
Several impossible figures were invented - a ladder, a triangle and an x-prong. These figures are actually quite real in a three-dimensional image. But when an artist projects volume onto paper, objects seem impossible. The triangle, which is also called "tribar", has become a wonderful example of how the impossible becomes possible when you make an effort.
All these figures are beautiful illusions. The achievements of the human genius are used by artists who paint in the style of imp art.
Nothing is impossible. The same can be said about the Penrose Triangle. This is a geometrically impossible figure, the elements of which cannot be connected. Still, the impossible triangle became possible. The Swedish painter Oscar Reutersvärd presented the world with an impossible triangle of cubes in 1934. O. Reutersvärd is considered the discoverer of this visual illusion. In honor of this event, this drawing was later printed on a postage stamp in Sweden.
And in 1958, the mathematician Roger Penrose published a publication in an English journal about impossible figures. It was he who created the scientific model of the illusion. Roger Penrose was an incredible scientist. He did research in the theory of relativity, as well as the fascinating quantum theory. He was awarded the Wolf Prize together with S. Hawking.
It is known that the artist Maurits Escher, under the influence of this article, painted his amazing work - the lithograph "Waterfall". But is it possible to make a Penrose triangle? How to do it if possible?
Although the figure is considered impossible, making a Penrose triangle with your own hands is easier than ever. It can be made from paper. Origami lovers simply could not ignore the tri-bars and nevertheless found a way to create and hold in their hands a thing that previously seemed like an outrageous fantasy of a scientist.
However, we are deceived by our own eyes when we look at the projection of a three-dimensional object from three perpendicular lines. It seems to the observer that he sees a triangle, although in fact it is not.
Tribar triangle, as said, is not really a triangle. The Penrose Triangle is an illusion. Only at a certain angle does the object look like an equilateral triangle. However, the object in its natural form is 3 faces of a cube. On such an isometric projection, 2 angles coincide on the plane: the nearest from the viewer and the far one.
The optical illusion, of course, is quickly revealed, as soon as you pick up this object. And the shadow also reveals the illusion, since the shadow of the tribar clearly shows that the angles do not match in reality.
How to make a Penrose triangle with your own hands out of paper? Are there any schematics for this model? To date, 2 layouts have been invented in order to fold such an impossible triangle. The basics of geometry tell you exactly how to fold an object.
To fold the Penrose triangle with your own hands, you will need to allocate only 10-20 minutes. You need to prepare glue, scissors for several cuts and paper on which the diagram is printed.
From such a blank, the most popular impossible triangle is obtained. The origami craft is not too difficult to make. Therefore, it will definitely turn out the first time, and even for a schoolboy who has just begun to study geometry.
As you can see, it turns out a very nice craft. The second blank looks different and folds differently, but the Penrose triangle itself ends up looking the same.
Choose one of 2 blanks convenient for you, copy the file and print. We give here an example of the second layout model, which is performed a little easier.
The Tribar origami blank itself already contains all the necessary tips. In fact, instructions for the circuit are not required. It is enough just to download it on a thick paper carrier, otherwise it will be inconvenient to work and the figure will not work. If it is impossible to immediately print on cardboard, then you need to attach a sketch to the new material and cut out the drawing along the contour. For convenience, you can fasten with paper clips.
What to do next? How to fold the Penrose triangle with your own hands in stages? You need to follow this action plan:
The finished model can be repainted in any color, or you can take colored cardboard for work in advance. But even if the object is made of white paper, anyway, everyone who enters your living room for the first time will certainly be discouraged by such a craft.
How to draw a Penrose triangle? Not everyone likes origami, but many people love to draw.
To begin with, a regular square of any size is depicted. Then a triangle is drawn inside, the basis of which is the lower side of the square. A small rectangle fits into each corner, all sides of which are erased; only those sides that are adjacent to the triangle remain. This is necessary to keep the lines straight. It turns out a triangle with truncated corners.
The next stage is the image of the second dimension. A strictly straight line is drawn from the left side of the upper lower corner. The same line is drawn starting from the lower left corner, and is slightly not brought to the first measurement line 2. Another line is drawn from the right corner parallel to the bottom side of the main figure.
The final step is to draw the third dimension inside the second dimension using three more small lines. Small lines start from the lines of the second dimension and complete the image of the three-dimensional volume.
By the same analogy, you can draw other shapes - a square or a hexagon. The illusion will be maintained. But still, these figures are no longer so amazing. Such polygons just seem to be heavily twisted. Modern graphics allow you to make more interesting versions of the famous triangle.
In addition to the triangle, the Penrose staircase is also world famous. The idea is to trick the eye so that it seems that the person is constantly moving upwards when moving clockwise, and if moving counterclockwise, then downwards.
The continuous staircase is more known by association with M. Escher's painting Ascending and Descent. Interestingly, when a person goes through all 4 flights of this illusory staircase, he invariably ends up where he started from.
Other objects are known to mislead the human mind, such as an impossible bar. Or a box made according to the same laws of illusion with intersecting edges. But all these objects have already been invented on the basis of an article by a remarkable scientist - Roger Penrose.
The figure named after the mathematician is honored. She erected a monument. In 1999, in one of the cities of Australia (Perth), a large aluminum Penrose triangle was installed, which is 13 meters high. Tourists are happy to take pictures next to the aluminum giant. But if you choose a different angle of view for photography, then the deception becomes obvious.
