Penrose triangle: paper hands
Invented several impossible figures - a ladder, a triangle and x-prong. These figures are actually quite real. But when an artist projects volume onto paper, objects appear impossible. The triangle, which is also called the "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 draw in the style of imp-art.
Penrose triangle. What it is?
Nothing is impossible. So you can say about the Penrose triangle. This is a geometrically impossible shape whose elements cannot be joined. Still, the impossible triangle became possible. Swedish painter Oscar Reutersvard in 1934 presented the world with an impossible triangle of cubes. A. Reutersvard is considered the discoverer of this visual illusion. In honor of this event on the postage stamp of Sweden printed this picture later.
And in 1958mathematician Roger Penrose published a publication in an English magazine about impossible figures. It was he who created the scientific model of illusion. Roger Penrose was an incredible scientist. He conducted research in the field of the theory of relativity, as well as fascinating quantum theory. He was awarded the Wolf Prize with S. Hawking.
It is known that the artist Maurits Escher, being under the impression of this article, drew his amazing work - the lithograph “Waterfall”. But is it possible to make a Penrose triangle? How to do if it is possible?
Tribar and reality
Although the figure is considered impossible, making a Penrose triangle with your own hands is easier than simple. It can be made from paper. Origami lovers simply could not ignore the tri-bar and still found a way to create and hold in their hands a thing that seemed previously the transcendental imagination of the 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 reality this is not so.
The triangle tribar, as said,actually not a triangle. Penrose's triangle is an illusion. Only at a certain angle the object looks like an equilateral triangle. However, an object in its natural form is 3 faces of a cube. On such an isometric projection, two angles on the plane coincide: the neighbor from the viewer and the far.
Optical illusion, of course, is quickly revealed, only to take this object in hand. And the shadow opens the illusion, since the shadow of the tribar clearly shows that the angles do not coincide in reality.
Tribar from paper. Scheme
How to make a Penrose triangle with your own hands out of paper? Are there any schemes for this model? At present, 2 schedules have been invented in order to fold such an impossible triangle. The fundamentals of geometry suggest exactly how to fold an object.
To fold the Penrose triangle with your own hands, you need to allocate only 10–20 minutes. It is necessary to prepare glue, scissors for several cuts and paper on which the scheme is printed.
From such a blank, the most popular impossible triangle is obtained. Craft-origami is not too difficult to manufacture. Therefore, it will necessarily come out from the first time, even with a student who has just begun to study geometry.
As you can see, it turns out very nice hack.The second blank looks different and is different, but the Penrose triangle itself looks the same.
Stages of creating a Penrose triangle out of paper.
Choose one of 2 convenient for you blanks, copy the file and print. We give here an example of the second model of the allocation, which is performed a little easier.
The origami blank for the Tribar already contains all the necessary clues. In fact, the instruction to the scheme is not required. It is enough only to download on a thick paper carrier, otherwise it will be inconvenient to work and the figure will not work. If you can not immediately print on cardboard, then you need to attach a sketch to the new material and cut out the outline along the contour. For convenience, you can staple.
What to do next? How to make a Penrose triangle with your own hands in stages? You need to follow this plan of action:
- Direct the reverse side of the scissors to those lines where you need to bend, according to the instructions. Bend all lines
- Where necessary, we make cuts.
- Using PVA, we glue those shreds that are intended for fastening the part into a single whole.
The finished model can be repainted in any color, or in advance to take to color cardboard.But even if the object is made of white paper, all the same, everyone who enters your living room for the first time will certainly be discouraged by this craft.
How to draw a penrose triangle? Not everyone likes to do origami, but many people love to draw.
To begin with, an ordinary square of any size is depicted. Then a triangle is drawn inside, the basis of which is the bottom side of the square. At each corner fits a small rectangle, all sides of which are erased; only those sides remain that adjoin the triangle. It is necessary that the lines are smooth. 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 bottom corner. The same line is drawn starting from the bottom left corner and is not brought to the first line 2 of the measurement. Another line is drawn from the right corner parallel to the bottom side of the main figure.
The final stage - the third one is drawn within the second dimension with the help of three more small lines. Small lines start from the second dimension lines and complete the image of the three-dimensional volume.
Other Penrose figures
By the same analogy, other figures can be drawn - a square or a hexagon. Illusion will be respected. But still these figures are not so stunning imagination. Such polygons just seem to be very twisted. Modern graphics allows you to make more interesting versions of the famous triangle.
In addition to the triangle, Penrose stairs are also world-famous. The idea is to deceive the view, when it seems that a person rises continuously upwards when moving clockwise, and if it moves counterclockwise, then down.
The continuous ladder is known more by association with M. Escher’s painting “Ascent and Descent”. It is interesting that when a person passes all 4 flights of this illusory staircase, he invariably finds himself where he started from.
There are other objects that mislead the human mind, such as an impossible bar. Or a box made by 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.
Impossible triangle in Perth
The figure, named after the mathematician, was honored.She erected a monument. In 1999, in one of the cities of Australia (Perth), a large Penrose aluminum triangle was installed, which is 13 meters high. Near the aluminum giant tourists are taking pictures with pleasure. But if you choose a different angle for a photograph, the deception becomes obvious.