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Mathematics presentation "Impossible is possible. Penrose triangle"

Mathematics presentation "Impossible is possible. Penrose triangle"

13.06.2019

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

Penrose triangle (in color)

Story

This figure gained wide popularity 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 Waterfall lithographs.

3D printout of the Penrose triangle

sculptures

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

The same sculpture when changing the viewpoint

Other figures

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

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.

Impossible cube

The impossible cube is an impossible figure invented by Escher for his Belvedere lithograph. This is a two-dimensional figure that looks like a three-dimensional cube perspective, incompatible with a real cube. In the Belvedere lithograph, a boy sitting at the base of a 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, with edges (segments) replaced by seemingly solid bars.

In Escher's lithography, the top four joins of the bars and the top intersection of the bars correspond to one of the two interpretations of the Necker cube, while the bottom four joins 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 more visual ambiguity than the Necker cube, which is less likely to be perceived as an impossible object. The illusion plays on the human eye's interpretation of a two-dimensional drawing as a three-dimensional object. Three-dimensional objects may seem impossible when viewed from a certain angle and either by making cuts in the object in the right place or by using an 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 impossible cube pieces.

A fabricated photograph of the supposedly impossible cube was published in the June 1966 issue of Scientific American, where it was called the "Frimisch cage". The impossible cube was featured on an Austrian postage stamp.

Impossible Trident

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

Ruthersward, Oscar

Oskar Rutersvärd (the spelling of the surname accepted in Russian-language literature; more correctly, Reutersverd), Swede. Oscar Reutersvärd (November 29, 1915, Stockholm, Sweden - February 2, 2002, Lund) - "the father of the impossible figure", a Swedish artist who specialized in depicting 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 was further developed as the "Penrose Triangle" (1934). The work of Ruthersward can be compared with the work of Escher, but if the latter used impossible figures as "backbones" to depict fantastic worlds, then Ruthersward was only interested in figures as such. During his lifetime, Rutersvärd depicted about 2,500 figures in isometric projection. Rutersvärd's books have been published in many languages, including Russian.

Escher, Maurits Cornelis

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


Illusions	Penrose Triangle

Also known by the names impossible triangle And tribar.

Story

This figure gained wide popularity 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 - 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 Waterfall lithographs.

sculptures

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

Deutsches Technikmuseum Berlin February 2008 0004.JPG

The same sculpture when changing the viewpoint

Other figures

Write a review on the article "Penrose Triangle"

An excerpt characterizing the Penrose Triangle

Having said everything that he was ordered, Balashev said that Emperor Alexander wanted peace, but would not start negotiations except on the condition that ... Here Balashev hesitated: he remembered those words that Emperor Alexander did not write in a letter, but which he certainly ordered Saltykov to insert them into the rescript and which he ordered Balashev to hand over to Napoleon. Balashev remembered these words: “until not a single armed enemy remains on Russian soil,” but some kind of complex feeling held him back. He couldn't say those words even though he wanted to. He hesitated and said: on the condition that the French troops retreat beyond the Neman.
Napoleon noticed Balashev's embarrassment when uttering his last words; his face trembled, the left calf of his leg began to tremble measuredly. Without moving from his seat, he began to speak in a voice higher and more hasty than before. During the subsequent speech, Balashev, more than once lowering his eyes, involuntarily observed the trembling of the calf in Napoleon's left leg, which intensified the more he raised his voice.
“I wish peace no less than Emperor Alexander,” he began. “Haven't I been doing everything for eighteen months to get it? I've been waiting eighteen months for an explanation. But in order to start negotiations, what is required of me? he said, frowning and making an energetic questioning gesture with his small white and plump hand.
- The retreat of the troops for the Neman, sovereign, - said Balashev.
- For the Neman? repeated Napoleon. - So now you want to retreat behind the Neman - only for the Neman? repeated Napoleon, looking directly at Balashev.
Balashev bowed his head respectfully.
Instead of demanding four months ago to retreat from Numberania, now they demanded to retreat only beyond the Neman. Napoleon quickly turned and began to pace the room.
- You say that I am required to retreat beyond the Neman to start negotiations; but two months ago they demanded of me to retreat across the Oder and the Vistula in exactly the same way, and in spite of this, you agree to negotiate.
He silently walked from one corner of the room to the other and again stopped in front of Balashev. His face seemed to be petrified in its stern expression, and his left leg trembled even faster than before. Napoleon knew this trembling of his left calf. La vibration de mon mollet gauche est un grand signe chez moi, [The trembling of my left calf is a great sign,] he later said.

Today I am opening a new section called “Cutting”, where I will post drawings, templates, as well as a pattern of optical illusions. Today we will make an impossible triangle from paper. Since we cannot create an impossible triangle, we will create a model that we will consider from a certain angle.

Download and print
Follow the instructions in the picture

How to correctly consider an impossible triangle?

Since the illusion is based on the ambiguous drawing of the cube in isometric view. Then in this orientation, the corners closest to the viewer and the far corner from the viewer will coincide. This means that when going down the nearest edge of the cube, and the two bottom edges, we return to starting point, where the path actually ends at the far corner.

This impossible Penrose triangle

In such a field of pictorial art as painting human skin, the latest trend today is the figures of optical illusions, in particular the Penrose triangle, or tribar, which is also called impossible. For the first time, this form was discovered, or invented, by the Swedish painter Oscar Reutersvärd, who presented it to the world in the form of a set of cubes at the turn of 1935. Later, already in the 80s of our century, the tribar pattern was printed in Sweden on a postage stamp.

However, the image of the impossible Penrose triangle, which belongs to the category of optical illusions, became widely known in 1958, after the publication of the publication of the English mathematician Roger Penrose on impossible figures, published in the British Journal of Psychology. Inspired by this publication, the famous Dutch painter Maurits Escher created in 1961 one of his most popular works, Waterfall.

Optical illusion

Optical illusions in painting are a visual illusion of the perception of a real picture, created by the artist with a certain arrangement of lines on a plane. At the same time, the viewer incorrectly assesses the size of the angles of the figure or the length of its sides, which is the subject of study of such subsections of psychology as, for example, gestalt therapy. In addition to Escher, another great artist, the world famous Salvador Dali, was fond of creating optical illusions. A vivid illustration of his passion is, for example, the painting "Swans reflected in elephants."

The aforementioned triangle also refers to optical illusions, more precisely to that part of them, which is called impossible figures. They are called so because of the feeling that arises when looking at such a form that its existence in the real world is simply impossible.

Application of illusions

Due to their unique shape, illusory objects are the subject of close attention not only for artists and tattoo artists - a triangle made by oneself or with the help of professionals can also act as a company logo. Remarkable examples of this use of illusory forms are: the logo of a psychedelic musical band playing folk music, Conundum in Deed, which is an impossible cube, or the brand of the chip manufacturer Digilent Inc, which is the classic triangular image of Penrose.

You can make your own logo yourself, without turning to professionals. To do this, just follow the instructions, following which you can perform both a simple drawing on paper or on a tablet, and make a three-dimensional figure. It can be placed as a sign or outdoor advertising for your store.

How to do it yourself

Step by step instructions on how to draw a tribar using Adobe Illustrator:

First you need to make 3 squares with the Rectangle tool. To do this, you first need to go to the View menu and enable Smart Guides.
Now you need to select everything and go to the Object menu, then to Transform and open Transform each, where in the Scale window you need to put down the value Vertical Scale = 86.6% and click OK.
Now you need to set each face its own angle of rotation, and for this go to Window open Transform. There, first put down the value for the bevel (Shear), and then for the rotation (Rotate): the upper surface of the cube is Shear +30 °, Rotate -30 °; right surface - Shear +30°, Rotate +30°; left surface — Shear -30°, Rotate -30°.
Now, using the Smart Guides lines, you need to dock all parts of the cube together: to do this, hook the corner of one of the sides with the mouse and pull it to the other, aligning them.
At this stage, you need to rotate the cube by 30°: to do this, go to Object, select Transform and Rotate, set the angle value there to 30° and click OK.
Since you need 6 cubes to get a tri-bar, you should select the cube, press Alt and Shift and drag the selected object to the side with the mouse, stretching it in the horizontal direction. Without removing the selection, press CMD + D 6 times. We got 6 cubes.
Leaving the selection on the last cube, press Enter and in the Move window change the angle value to 240 °, then press Copy. Then again press CMD + D until you get 6 copies.
Now repeat everything: press Enter again, select the last cube, only set the angle to 120 ° and make only 5 copies.
Using the Selection Tool, you need to select the upper surface of the shape (you can recolor it to make it clearer), open the menu Object - Arrange - Send to back. Now select the painted surface of the top cube, go to Object - Arrange - Bring to Front.

The Penrose illusion is ready. It can be posted on your page in social networks or blog, or used for business.

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 it is to draw an impossible triangle. We will draw two drawings of this triangle, one will be ordinary, 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 somewhere else, so read the article to the end and very carefully.

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

How easy it is to draw any 3D drawings.

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

And then in a couple of minutes with the help I translated it into 3D . So you can translate almost any image into 3D. For those who want to learn the same, click here.

And we move on to our drawing.

We draw the usual drawing of a triangle.

STEP #1. We translate from the monitor screen.

In order for you 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 complicated, it is enough to put only the main points in all its corners.

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

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

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

We draw a 3D drawing of a triangle.

STEP #1. We translate.

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

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

Here's 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 your mouse wheel.

We can safely say that our 3D drawing is already ready. It took me about 3 minutes to do it. On this, in principle, we can safely finish, but let's decorate our triangle again.