What is the concept of golden section?

The golden section was first seen in ancient Greece and Egypt. The golden section is also called golden section ratio and Chinese-foreign ratio, that is, a line segment is divided into two sections A and B with different lengths, so that the ratio of long section (a+b) is equal to that of short section B and long section A. The formula is A: (A+B) = B: A, and the ratio is 0.6 180339. . This ratio is pleasing to the eye in shape, so 0.6 18 is also called the golden ratio. The golden section rectangle itself consists of a square and a golden section rectangle. You can divide these two basic shapes infinitely. Because its own proportion can stimulate people's vision moderately, and its length proportion just conforms to people's visual habits, it makes people feel pleasing to the eye. The golden section is widely used in architecture, design and painting. In the development of photography technology, the essence of other art categories has been borrowed and integrated to varying degrees, and the golden section has therefore become the most sacred concept in photographic composition. The simplest method used in photography is to arrange 2, 3, 5, 8, 13, 2 1 according to the golden ratio of 0.6 18, so that we can get 2: 3, 3: 5, 5: 8, 8: 13. These ratios are mainly applicable to the determination of the aspect ratio of the picture (for example, the film width of 135 camera is 24mmX36mm, which is derived from the golden ratio), the selection of horizon position, the distribution of light and shadow tones, the division of picture space and the establishment of the visual center of the picture. The trisection (also known as the well-shaped segmentation method) commonly used in photographic composition is the evolution of the golden section method. The length and width of the upper picture are divided into three equal parts, and the whole picture is divided by a pound sign. The intersection of cross-shaped segmentation is the best position of the main body (visual center) of the picture, and it is the most likely visual beauty to arouse people's visual interest. Many basic laws of photographic composition have evolved on the basis of the golden section. However, it is worth reminding that every photo is completely composed according to the golden section, which is unnecessary and impossible. The sameness will make people feel monotonous. Regarding the golden section, it is important to master its laws and use it flexibly. The golden section divides a line segment into two parts so that the ratio of one part to the total length is equal to the ratio of the other part to this part. Its ratio is an irrational number, and the approximate value of the first three digits is 0.6 18. Because the shape designed according to this ratio is very beautiful, it is called golden section, also called Chinese-foreign ratio. This is a very interesting number. We approximate it with 0.6 18, and we can find it by simple calculation:1/0.618 =1.618 (1-0.618). Let's talk about a series. The first few digits are: 1, 1, 2, 3, 5, 8, 13, 2 1, 34, 55, 89, 144 ... The characteristic is that every number is the sum of the first two numbers except the first two numbers (the numerical value is 1). What is the relationship between Fibonacci sequence and golden section? It is found that the ratio of two adjacent Fibonacci numbers gradually tends to the golden section ratio with the increase of the series. That is f (n)/f (n-1)-→ 0.618. Because Fibonacci numbers are all integers, and the quotient of the division of two integers is rational, it is just approaching the irrational number of the golden ratio. But when we continue to calculate the larger Fibonacci number, we will find that the ratio of two adjacent numbers is really very close to the golden ratio. A telling example is the five-pointed star/regular pentagon. The pentagram is very beautiful. There are five stars on our national flag, and many countries also use five-pointed stars on their national flags. Why? Because the length relationship of all the line segments that can be found in the five-pointed star conforms to the golden section ratio. All triangles that appear after the diagonal of a regular pentagon is full are golden section triangles. Because the vertex angle of the five-pointed star is 36 degrees, it can also be concluded that the golden section value is 2Sin 18. The golden section is equal to about 0.6 18: 1, which means that a line segment is divided into two parts, so that the ratio of the long part to the long part of the original line segment is the golden section. There are two such points on the line segment. Using two golden points on the line segment, a regular pentagram and a regular pentagon can be made. More than 2000 years ago, Odox Sass, the third largest mathematician of Athens School in ancient Greece, first proposed the golden section. The so-called golden section refers to dividing a line segment with length L into two parts, so that the ratio of one part to the whole is equal to the ratio of the other part. The simplest way to calculate the golden section is to calculate the ratio of the last two numbers of Fibonacci sequence 1, 1, 2, 3, 5, 8, 13, 2 1, ... 2/3, 3/5, 4/8, 8/655. They called it the "golden method", and a mathematician in Europe17th century even called it "the most valuable algorithm among all kinds of algorithms". This algorithm is called "three-rate method" or "three-number rule" in India, which is what we often say now. In fact, the "golden section" is also recorded in China. Although it was not as early as ancient Greece, it was independently created by China ancient algebras and later introduced to India. After textual research. European proportional algorithm originated in China, and was introduced to Europe from Arabia via India, not directly from ancient Greece. Because it has aesthetic value in plastic arts, it can arouse people's aesthetic feeling in the design of length and width of arts and crafts and daily necessities, and it is also widely used in real life. The proportion of some line segments in the building adopts the golden section scientifically. The announcer on the stage is not standing in the center of the stage, but standing on the side of the stage. The position at the golden section of the stage length is the most beautiful and the sound transmission is the best. Even in the plant kingdom, the golden section is used. If you look down from the top of a twig, you will see that the leaves are arranged according to the golden section law. In many scientific experiments, a method of 0.6 18 is often used to select the scheme, that is, the optimization method, which enables us to arrange fewer experiments reasonably and find reasonable western and suitable technological conditions. It is precisely because of its extensive and important application in architecture, literature and art, industrial and agricultural production and scientific experiments that people call it the golden section. The golden section is a mathematical proportional relationship. The golden section is strict in proportion, harmonious in art and rich in aesthetic value. Generally, it is 0.6 18 in application, just as pi is 3. 14 in application. The aspect ratio of a golden rectangle is the golden ratio. In other words, the long side of a rectangle is 1.6 18 times of the short side. Golden ratio and golden rectangle can bring beauty to the picture, which can be found in many works of art and nature. The Pasa Shennong Temple in Athens, Greece, is a good example, and it conforms to the golden rectangle. The face also conforms to the golden rectangle, which is also suitable for proportional layout. It is found that the Pythagorean school in ancient Greece studied the drawing methods of regular pentagons and regular decagons in the 6th century BC, so modern mathematicians came to the conclusion that the Pythagorean school had contacted and even mastered the golden section at that time. In the 4th century BC, eudoxus, an ancient Greek mathematician, first studied this problem systematically and established the theory of proportion. When Euclid wrote The Elements of Geometry around 300 BC, he absorbed eudoxus's research results and further systematically discussed the golden section, which became the earliest treatise on the golden section. After the Middle Ages, the golden section was cloaked in mystery. Several Italians, pacioli, called the ratio between China and the destination sacred and wrote books on it. German astronomer Kepler called the golden section sacred. It was not until the19th century that the name golden section gradually became popular. The golden section number has many interesting properties and is widely used by human beings. The most famous example is the golden section method or 0.6 18 method in optimization, which was first proposed by American mathematician Kiefer in 1953 and popularized in China in 1970s.