Maurits Escher:
art by formula?

All people are used to admiring works of art. Many of them can be seen in the exhibition halls, on the shelves of specialized stores, hear, read, see. Delivery from China paintings and other works of art makes them more accessible. But, according to many people, art is contained in everything that surrounds us. Every day we can see sunrises and sunsets, starry skies, overflows of sunlight, an abundance of clouds rolling from one end of the celestial sphere to the other. In the winter, we can consider the virtuoso drawings on the windows, admire the shiny snowdrifts, tree branches, which are shrouded in fluffy hoarfrost. All this fascinates us. And all this is an art of nature. Not just a person makes the world more beautiful and better, it makes nature itself.

But are those things subject to this law which, it would seem, from themselves do not represent anything beautiful? This refers to how space and planes are subject to nature?

The Dutch artist Maurits Escher once decided to prove that even mathematics can be subject to art and to be art itself. Without a special mathematical education, Escher himself created such works that forced even the great mathematicians to puzzle. In his paintings they could find and reproduce complex mathematical laws, and simply see the familiar in a different light. So, Escher could arrange the plane so that it took a long time to think, to combine one part of the picture with another, to consider them more and more. Finally, the viewer began to ripple in his eyes, he threw this lesson and retreated to the next picture. And exactly the same thing was repeated there.

In his paintings, Escher often used such phenomena in mathematics as the partition of a plane, like a mosaic, about the location of three-dimensional objects, the so-called "impossible figures". He drew all this knowledge from mathematical articles. For health reasons, Escher’s dream of becoming an architect was impossible, so he began to paint. After several exhibitions and reports in magazines, his name gained worldwide fame.

As you know, a plane can be divided into regular and irregular mosaics. The mosaics belonging to the first group using simple polygons are divided into a set of closed shapes. But Escher was not interested in it. He was more interested in irregular mosaics, in which with the help of different figures you can create completely different interpretations of the plane. But he did not stop there. In this branch of mathematics, he introduced his own invention - a mosaic, which consisted of figures that can change and change the space around him. It is amazing that a person who does not have a mathematical education could perform such work. Most likely - he was a genius in this, and talent contributed to him.

Escher did not use the well-known correct figures in his works. Using such properties of geometric figures as symmetry, displacement, and others, he distorted them, turning them into birds, animals, and so on. He twisted them as he wanted to achieve the desired result.

Polyhedra and voluminous bodies were of particular interest to Escher. If you look at his works, it seems that he does not draw them, but creates on paper from the air. The image is such that you want to take and touch one or another polyhedron. But this is just a drawing. In almost all of Escher’s works, one can find one or another mathematical figure that has special beauty and charm. The artist clearly did not spare time and energy for his work. Studying along the way, supplementing his knowledge with newer and newer examples, Escher in his paintings created his own mathematical universe, distorting space, time, things, and even geometric shapes.

He created works that fascinated, which made you think. You look at them and think - how is this so? But he created all this with an ordinary pencil, using the ability to draw and amateur mathematical knowledge. Asher’s famous works were seen by everyone: it’s “Drawing Hands”, which draw themselves - we get a cycle, and an image of the Mobius strip, and many, many other works that can be immediately recognized from the crowd - by how much geometry they contain.