Ladies and gentlemen, young and old! In the next zhang big less.
Many of us know that Escher's art is intimately related to mathematics: there are many creatures in his prints that lock together with perfect regularity, or deform in landscapes that connect day and night, heaven and earth; buildings and scenes are depicted on paper in a way that architects cannot build. Fewer people know that he had been associated with more than one, but a trineur, and each of these encounters profoundly influenced his art. We can say that it was mathematics that introduced Escher and his art to many more people.
Plane rule segmentation
From an early age, Escher experimented with and invented the design of seamless, densely paved floors, which became recurring themes in his artistic career. An early example, Eight Faces (1922), was artistically creative, but did not show the mathematical artistry with which we are now associated. It shows eight figures, four male, four female, four upright, four upside down, four dark, four light colored, all alternating in a checkerboard-like pattern, but interlaced with each other, for example, the shoulders of one person forming the feathers of another person's hat. The eight men were arranged in a square, which itself was stamped four times to become a larger square. Escher lets us understand that this pattern can be repeated endlessly to cover an infinite plane. However, the overall effect is not strictly dense.
Eight Faces
In 1922, Escher visited the Alhambra, a Spanish castle complex rebuilt by the Moors, famous for its mosaic tiles. Like many artists, Escher painted a variety of paved patterns, but he didn't use them much, apparently because he was more interested in figures, animals, flowers, birds, fish, and insects than in pure geometric shapes. He commented, "The Moors were masters of laying surfaces with full-scale graphics and did not leave any gaps. But "I find this [geometric] restriction even more unacceptable, because the recognizability of the components of my own pattern is the reason I never stopped being interested in this field." Thus, Escher describes the delicate relationship between the strict geometric order of the motifs and the interesting shapes of the pieces.
Nevertheless, he continued to experiment with flat paving, creating his own patterns, and along the way, the design of the Alhambra inspired him to twist diamonds and hexagons, squares and triangles into recognizable shapes and keep the same basic relationship unchanged. In 1936, he revisited the Alhambra, this time paying more attention to the patterns he saw. But he still has a year to come across an opportunity that will dramatically change his artistic career.
Before continuing, you should try some small exercises. Use a single regular polygon as a paving element to draw all possible paving of a plane. Now, how do you know you've used all of it and haven't left it out? With a 2x1 rectangle as a tile, this can also be done. Of those paintings you paint, which are essentially the same and which are truly different – how would you explain why to others? Use multiple tile shapes in your design. These exercises brought up a larger task of describing all the basic paving patterns, including those that used multiple shapes as tiles, which differed structurally from each other.
Escher's half-brother Bill, a geology professor, realized that Escher was actually experimenting with the analogy of the two-dimensional plane of three-dimensional crystals, so in 1937 Bill sent Escher a batch of crystallography papers that he had written 20 years earlier. In his reply, Escher wrote that he found them mostly too boring, too theoretical, and too difficult for a layman, except for a Krystall Symmetrie der Ebene (analogy for crystal symmetry on a plane) by Georg Pólya (1924). This is a purely mathematical group theory paper designed to solve the daunting task of describing all the different densely paved relationships. The clever twist is that Pollia promised in that paper, "I will discuss elsewhere the fact that the mathematical study of these ornaments is also interesting from an artistic point of view." ”
Polya was a Hungarian mathematician who was working at the Swiss E.T.H. in Zurich at the time, but fled to the United States in 1939 due to interference from the rise of socialism. While working in mathematics, Polya also had a keen interest in education. His books How to Prove and Mathematics and Reasonable Reasoning should be read by every math teacher.
So how did a technical paper in pure mathematics resonate with the artist Escher and change his? Quite simply, Polya has completely solved the problem that Escher has been exploring for years, but he has no formal means of solving it. Instead, Escher relied on visual intuition and the vision he wanted to achieve. There is no doubt that Escher was engaged in mathematical research; it was just that he was working in isolation, using his own language rather than the formal language of mathematics, and perhaps knowing little or nothing about the role of mathematics.
Georg Pólya exemplifies 17 different possible symmetry groups
The most important thing for Escher, and what he immediately understood, was the chart that contained Polyia. Polya was not interested in the specific shapes used to build the pavement. Instead, his focus is on the symmetry inherent in patterns. For example, there are several symmetries in a checkerboard pattern that extends infinitely in all directions to pave the plane. If the pattern is reflected diagonally or by bisecting the horizontal and vertical axes of the sides of the square, mirror symmetry leaves the pattern unchanged. If the pattern rotates half a circle around the center and corners of the square, rotational symmetry keeps the pattern intact. If you pan the pattern up or down, left or right, two squares, the translation symmetry leaves the pattern unchanged. Finally, there is a special symmetry called slip reflection, which is obtained by translating along a line and then reflecting through that line.
Now, in the language of symmetry, we can say that if two dense pavers have exactly the same group of symmetries, then they are structurally identical, even if the particular shape is different. Surprisingly, it turns out that there are exactly 17 different sets of symmetries, no more and no less, and Polya's chart gives examples of each set of symmetries, mathematically illustrating why only these combinations are unique. Some examples of Polya mimic the Alhambra tiles that Escher was familiar with, while others are of Polya's own design.
After this encounter, Escher began to devote himself to the study of the dense paving pattern, writing page after page of ideas he had gathered from Polya. Some of Escher's motifs are even based on the shapes used by Polya, so D1·gg was turned into an eagle (1938). Remember that Escher's driving force is not geometric shapes, but figurative shapes, and now he has more tools to help him construct these shapes so that they are locked together by symmetry.
On this basis, he could begin to modify the block diagram by transforming the shape from one form to another, as well as shrinking and enlarging the same shape in the design, but still retaining the nature of the dense paving. An example of the former is The Cycle (1938, just one year after seeing the Polya article), but is far removed from Eight Faces in development.
"Cycle"
In The Loop, we see a character with his hands raised and running down the stairs outside. As he descended, the figure became a silhouette on the basis of a hexagon, which continued through the bottom of the print, then upward on the other side, and transformed again, but into the stones that formed the building, which in turn were compressed into the tiles of the patio, causing the figure to return. Escher is playing the game of dense paving, and he is also playing with the ambiguity of three-dimensional objects on a two-dimensional surface.
Penrose and Impossible patterns
Escher plays not only with two-dimensional patterns, but also with three-dimensional space. In many of his prints, he contains the same scene from different angles, connecting different parts by using a single element to achieve multiple purposes, depending on the different perspectives. For example, in Heterogeneous Space (1947), we see the same scene depicted from three different vantage points, but locked together in a way that, depending on the viewpoint, the individual surfaces act as walls, floors, and ceilings at the same time. This combines everything into a scene that cannot be considered as a whole, but in this scenario, the individual small parts are very meaningful in individual cases.
Heterogeneous Space
The Theory of Relativity (1953) was even more ambitious. Not only does the plane act as a wall, slab, and ceiling at the same time, but several staircases are depicted, each of which achieves dual use. When one figure falls on the pedal, the other figure is perpendicular to the first figure, and from the perspective of the first figure, it rises along which riser. These prints adopt the basic concept of Eight Faces, where individual picture elements can serve different purposes in the picture and expand them to another dimension.
The Theory of Relativity
This part of Escher's story is probably better known than The Polya episode. Amsterdam was the host city of the 1954 International Congress of Mathematicians. ICM is a large event; it is the largest gathering of mathematicians and is held every 4 years. One of the highlights of each congress is the announcement of fields medal winners.
Dutch mathematician N.G. de Bruijn, who arranged the Exhibition of Escher for the Amsterdam congress, believes that mathematicians will see in Escher's art the playfulness of ideas that they value in their research. This is an important concept that may be overlooked in classrooms and textbooks, namely that mathematics is interesting, that it is about exploring patterns and discovering relationships, and that it is not about unconsciously memorizing rules.
A young scientist, Roger Penrose, saw The Theory of Relativity and was struck by three of its staircases, which, if taken as a whole, would be physically impossible but perfectly consistent if viewed in isolation. As an inquisitive mathematician, he began to look for his own impossible figures. The result is the famous Penrose Triangle, which graphically looks like three straight rectangular bars connected into a triangle, each corner being a right angle.
Penrose Triangle.
Penrose Triangle. It's impossible to build in three-dimensional space, but it can be drawn in two-dimensional space, which retains local consistency and seems possible until you trace the bars along three right angles. Source: Wiki
When Escher discovered Polyia and began to correspond, Penrose also discovered Escher and began to correspond, which directly influenced Escher's art, especially The Rise and Fall (1960) and The Waterfall (1961). Penrose's father, Lionel, was a psychiatrist, geneticist, and mathematician who, after seeing the impossible triangle, created his own impossible shape, a set of staircases surrounding the four sides of a square that looked like you could rise forever, or fall forever, depending on the shape you chose. Roger gave these ideas to Escher, and we now see in these two prints the direct result of the monks and the infinite cycle of water rising and falling.
"Rise and Fall"
Waterfall
Draw your own impossible graphics
You may want to try designing your own impossible graphics. Are there squares, cubes, or tetrahedrons similar to penrose cubes? There is an impossible cube based on the ambiguous Necker cube, but this illustrates a different impossibility, which Escher used in The Belvedere (1958) and is also accessible to creative people.
Necker cubes and impossible cubes. Source: Wiki
《Observation Tower》
The third impossibility is to draw two coherent ends of parallel lines, using the same number of parallel lines, and then connect them together to form an impossible shape.
There are many variations in the shape below, for example, the cover of MAD magazine, published in March 1965, is called "the MAD poiuyt".
bibliography
1 Ernst, B. (1976). The Magic Mirror of M.C. Escher.Toronto: Random House.
2 Escher, M. C. (1989). Escher on Escher. New York: Abrams, 1989.
3 Polya’s 17 symmetry groups https://en.wikipedia.org/wiki/Wallpaper_group
4 Pólya, G. (1924) on the analogy of the krystall symmetry of the plane. Z. Kristall 60, 278-282.
5 Schattschneider, D. (2004).M.C. Escher: Visions of Symmetry, New ed. New York: Harry N. Abrams.
6 Schattschneider, D. (2010). The Mathematical Side of
7 M. C. Escher.Notices of the American Mathematical Society 57, 706 – 718.
8 Toen Castle, ESCHER AND THE MATHEMATICIANS
The green mountains do not change, and the green water flows for a long time, and retreats under the sun.