Ladies and gentlemen, young and old! In the next zhang big less.
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Exploring Escher's Little-Known Manuscripts (Prequel): 17 Groups of Plane Symmetry
"The flat regular division is the richest fountain of inspiration I have excavated, and it has not dried up to this day."
—Maurits Cornelis Escher
The concept of the so-called "dense paving" is mathematically completely covering a two-dimensional plane with different geometric shapes, and there is neither overlap nor gap between the figures. In short, it's sticking bricks. Bricklaying technology must have originated very early in the history of civilization, when humans began to build houses, they would use stones to pave the ground and walls, and when humans began to choose stones of various shapes and colors to make beautiful designs, it could be considered that bricklaying was an art. Today, however, bricklaying seems to be just a boring task for renovation workers. On the ground and on the walls, we can see all kinds of stickers everywhere, and they are bland, if not uniform. For example, in Figure 3.1.1 below.
Figure 3.1.1 The dense shop in life
However, if you look at these works in Figure 3.1.2 below, you will think that the tiles are no longer bland, but artistically alive. They all come from the stickers in the palaces of the Islamic world. Islamic craftsmen were old drivers of dense shops, and their artistic impulses were often expressed in the use of tiles of some shape and color to construct intricate geometric designs and to decorate walls and floors as a means. The works of Islamic artists reveal the visual beauty inherent in geometry more than any other cultural heritage, and they draw inspiration from geometry to produce a wealth of aesthetic innovations. This is mainly because Islam does not have the figurative worship of other religions, but rather uses the rational work "geometry" as an important tool for religious expression. Geometric patterns in Islamic art are based on basic figures such as triangles, quadrilaterals, and hexagons, constructing boundless, complex and trivial geometric figures that are orderly and beautiful. These geometric figures change in a cycle, forming a variety of all kinds of patterns. Gazing at them, people will involuntarily feel the cycle of the world, think about the whirling and change of life, understand the omniscient and all-powerful Allah and the magic of beginninglessness and endlessness, and get the pleasure of beauty and the cultivation of thought.
Fig. 3.1.2 Islamic decorative motifs
I think you must be curious, how did the craftsmen of the Islamic world create such exquisite and intricate patterns? How can we appreciate it?
Clearly, the history of Islamic geometric design has undergone a continuous evolution from simple to complex for more than a decade, and the creative methods born during this period are by no means unique. Unfortunately, the craftsmen of ancient times did not write the secret books of development like today's technicians, nor did they leave fanciful manuscripts like painters. The designs of ancient craftsmen are unknown, the place of birth is unknown, the creator is unknown, the development process is unknown - in short, except for the works left on the cultural relics and monuments today, everything is unknown.
The lack of methodological knowledge has left connoisseurs and creators interested in incorporating this design into their work like a cloud of clouds, and these exquisite and intricate patterns seem to be an endless road of suffering, with no bright colors in sight and no end in sight.
Fortunately, contemporary researchers have come up with a variety of claims. After analyzing the comparison, I only liked one of them. This method may not be perfect, but it can best reflect the geometric characteristics of Islamic patterns, almost all patterns can be easily completed with this design method, and it is very easy to master, which can be called the "dragon slaying sword" of Islamic pattern design. The most incredible thing is that it was a bacteriologist who proposed this method!
The unscrupulous bacteriologist was Ernest Hanbury Hankin, an Englishman who traveled thousands of miles to India in 1896. At that time, cholera was raging in India, and the local guide told him, "Only the sacred water of the Ganges can keep you from getting cholera!" "I don't know if he dried the bowl of Ganges, but I do know that he found phages from the Water of the Ganges and used it to treat cholera. In addition, he was passionate about Studying Islamic decoration, which abounded in India. One day, he walked into a local bathroom, soaked in the bath, stared up at the stucco ceiling, and suddenly had an epiphany, and found that Islamic geometric design was constructed from the polygonal dense paving on the ground floor, so he defined it as "a pattern formed on the basis of the polygonal densely paved construction line." He observed that as long as the remaining space between polygons is symmetrical, many different polygon combinations can be used. Since then, Hankin has published a series of related papers, but he did not name this method, preferring to call it the "polygon densely paved line method".
Without further ado, let's use a simple example to illustrate this approach.
The "Polygon Dense Paving Method" is divided into three steps.
The first step is to construct a polygonal dense paving. We know that a regular hexagon can be densely paved, so we draw a paving pattern of a regular hexagon, as shown in Figure 3.1.3.
Fig. 3.1.3 Regular hexagonal dense paving
The second step is to connect the dots. There is a trick to take points, usually taking the midpoints on each side of a polygon, and then drawing lines from these points outward, these lines I call "pattern lines". According to the angle formed by the pattern line, it can be divided into three kinds: sharp angle, right angle and obtuse angle, Fig. 3.1.4a-c are two sharp angle patterns and one obtuse angle pattern. It can be seen that the hexagonal star pattern appears in the center of the hexagon of the three figures, and as the angle increases, the shape of the hexagonal star changes from slender to plump. The point taking method is not only to take the midpoint, but also to take two points, such as the third equal point of each side, and then still lead out of these points straight lines. Similarly, straight lines drawn from two points on each side form various angles and may also be parallel, as shown in Figure 3.1.4d-e.
Figure 3.1.4a Sharp Angle Connection 1
Figure 3.1.4b Sharp angle connection 2
Figure 3.1.4c Obtuse angle wiring
Figure 3.1.4d Two-point line 1
Figure 3.1.4e Two-point connection 2
The third step is to decorate. Erase the underlying mosaic created in the first step, and then decorate the pattern lines. Depending on the modification method, the same design can produce different aesthetics. Generally speaking, the methods of retouching include block coloring, thickening lines and structural weaving. We processed Figure 3.1.4b with these three modification methods and obtained three patterns with very different visual effects, as shown in Figure 3.1.5a-c. This step belongs mainly to the category of fine arts, and has little to do with the geometric construction of the pattern, so it is not the focus of the discussion in this book, and in the following text, the modification steps are only selected for coloring methods to be displayed, and are not listed separately.
Figure 3.1.5a Coloring
Figure 3.1.5b Wide edges
Fig. 3.1.5c Braiding
As Hankin argues, this approach is typically characterized by the use of polygonal paving as the underlying structure for deriving geometric patterns. The pattern line is then placed at the specified point on each side of each polygon, and the angle formed by the intersecting pattern line determines the overall characteristics of the design. After the pattern is created, the polygon-densely paved substrate is erased, leaving only the derived pattern, with no easily recognizable pattern construction indications. Perhaps, this also confirms a common understanding of the world's different religions: behind the tangible world is an imperceptible and far-reaching subtle order. Similarly, the underlying mosaics in Fig. 3.1.5a-c are hidden in the final pattern, and we can only perceive their presence through the outside of the pattern and deep inside. When extracting a geometric pattern from a base polygon dense, any of the four series of sharp angles, right angles, obtuse angles, and two points can be used to construct a kaleidoscopic pattern.
You might say that the above diagrams can be drawn in other ways, and some of them seem to be more concise. Yes, these patterns are not unique, and this "polygonal densely paved line method" is more demanding and slightly cumbersome, but the advantage of this method lies in its inherent flexibility, providing a high level of design diversity and complexity. As far as Islamic geometric patterns are concerned, no other design approach offers so much flexibility and diversity. That's why I call this method the "Dragon Slayer Sword."
It has been more than a decade since the birth of Islam, but is still unknown to the public as the most representative masterpiece of the Islamic world. So let's grab the dragon slaying sword and walk with the geometric geniuses of the ancient Islamic world to experience their creative process first-hand, believe me, it is bound to be a wonderful journey. Please do a good job, polish the eyes of insight, and our journey begins.
The green mountains do not change, and the green water flows for a long time, and retreats under the sun.
Forward at will, please contact Zhang Dashao Buddha-figure.