Divide the known line segment into two parts, so that one part is the proportional median of the whole line segment, and the other part is the golden section problem mentioned in the middle school geometry textbook. If c is the bifurcation point of the line segment AB that meets the conditions, the AC can be about 0.6 18AB. This division is called golden section in textbooks, and we can sometimes say that line segments are divided into middle-end ratio, Chinese-foreign ratio or external-internal ratio. If represented by g, g is called golden ratio or golden ratio. Both the golden section and the golden section number are called "gold", which shows its importance and universality in application and adds a bit of mystery to it. The famous astronomer Kepler called the golden section "a great treasure in geometry". Let's unveil its mystery and explore this treasure together!
Describe the origin of word names
The earliest understanding of China-Belgium can be traced back to Pythagoras School. This school is familiar with regular pentagons and regular decagons, and takes the "five-pointed star" as the symbol of its members, and the practice of these figures is closely related to the mid-range ratio. If Pythagoras is considered to be familiar with the drawing of regular pentagons and pentagrams, it can be inferred that he has mastered the proportion of the middle end. Odoksos, a famous mathematician and astronomer in ancient Greece, first made a systematic study of the mid-late ratio. When he deeply explored the nature of the five-pointed star, he exclaimed: "The mid-late ratio appears here!" The strict exposition of the ratio between the middle and the end was first seen in Euclid's Elements of Geometry. After the Middle Ages, the middle and late period was covered with a more mysterious coat and gradually covered with a mysterious color.
During the Renaissance, the proportion between China and the West aroused widespread concern. 1509, Pachouri, one of the important figures in the Italian Renaissance, published the book Sacred Proportion. This book systematically introduces the comparison between China and foreign countries in ancient Greece and calls it sacred comparison. He believes that everything in the world must abide by this sacred law of proportion. Kepler called the mid-range ratio "proportional division". He wrote: "The Pythagorean theorem and the midrange ratio are Shuang Bao in geometry. The former is like gold, and the latter is called jade. " He gave the Pythagorean Theorem the metaphor of gold, and described China-Belgium with Zhuyu. The name of the golden section was officially used for the first time in the book as ohm (the brother of G.S. ohm, who is famous for ohm's law). This name was first used in the second edition of Pure Elementary Mathematics published by him 1835. /kloc-After the 0/9th century, this name gradually became popular and became a household name.
Talk about the wonders of nature.
The golden section number G has many interesting properties. The most striking thing is its relationship with Fibonacci sequence.
Fibonacci was a famous scholar in the Middle Ages. He put forward an interesting "rabbit breeding problem" in the book Abacus Book, which led to a wonderful sequence:
1,2,3,5,8, 13,2 1,34,55,89, 144,……
The rule is: from the third item, each item is the sum of the first two items. Later generations called it Fibonacci sequence. What does it have to do with the golden section?
Let's calculate the ratio between the first term and the last term in Fibonacci sequence, and we will find that this ratio is getting closer and closer to the golden section number G, which can be completely used as the n-order approximation of the first, second and ... g, how wonderful! In fact, it can be proved that these ratios are limited to g.
This internal relationship between Chinese-foreign ratio and Fibonacci sequence greatly adds luster to it, and also endows it with a special mystery and charming charm, which attracted many mathematicians later.
Influence and application of rough theory of throwing bricks to attract jade
The golden section has an extremely wide and simple application both in theory and in real life, and it has also had a great influence in history. In ancient times, the ratio between the middle and the end was mainly used as a painting method. By the Renaissance, it had aroused people's great interest and attention at that time, and had a wide influence and been applied in many aspects. For example, in painting and sculpture, painters and sculptors hope to solve the relationship between the most perfect form and its parts from the mathematical scale, so as to guide artistic creation as a scientific artistic theory and reflect the perfect structure of ideal things. Leonardo da vinci, a famous painter, thinks in his book On Painting: "Aesthetic feeling is completely based on the sacred proportional relationship between the parts, and all the features must work at the same time to produce a harmonious proportion that fascinates the audience." Artists in this period were consciously tempted by the charm of the golden section, which closely combined mathematical research with artistic creation and had a great influence on later formal aesthetics and experimental aesthetics.
/kloc-in the 9th century, Cai Xin, a German aesthetician, put forward the principle of golden section, and expounded the problem of golden section in theory, holding that golden section is the key to unlock the mysteries of natural beauty and artistic beauty. He studied aesthetics by mathematical proportion method, which inspired future generations. Fechner, a German philosopher, aesthetician and psychologist, tried experimental aesthetics, established the golden section principle on the basis of extensive psychological tests, and combined aesthetic research with natural science research, which attracted wide attention. Until 1950s, the research of experimental aesthetics was still very active. Until recently, the golden section principle was still a scientific aesthetic problem full of magical mysteries. For example, in the field of quasicrystal structure research in crystallography, the problem of golden section has once again aroused the interest of physicists and mathematicians.
It has many practical applications. The most humanized example is the golden section method in optimality, which was first proposed by Kiefer in the United States in 1953. Since 1970, it has been popularized in China and achieved great results. Another optimization method-fractional method, which adopts the fractional approximation of G, is also widely used in practice.
True, false, mysterious legend
Because of its unique nature, with its growing influence, there are more and more legends about it. These legends are illusory and confusing, but they have always been talked about and widely circulated.
It has been studied that the golden section is a structural principle of human and animal and plant forms. So there is the following statement:
The beauty of the human body itself, that is, the most beautiful figure of the human body, follows the golden ratio of G. It is said that all the beautiful things made before people knew about the golden ratio coincided with Huang Jinlv's. For example, the ratio of the lower body to the whole body of the famous statues of Venus and Athena is close to G.
It is said that the charm of ballet art can not be separated from G, and ballet dancers stand on tiptoe when dancing in order to show the most beautiful artistic image in line with G's figure proportion.
In nature, G is also an important law of beauty. It is said that especially refreshing flowers rely on the beautiful password of G.
In addition, we know that almost all the "stars" on national flags are drawn as five-pointed stars. It is said that the beautiful code G is hidden in many places in the five-pointed star, which makes this figure pleasing to the eye.
It is also said that the announcer will leave a good impression on the audience during the golden section. Some people even say that when playing stringed instruments, the timbre obtained by putting the "thousand pounds" in the golden section of the strings is more beautiful and harmonious.
There is also a popular saying that a golden rectangle (that is, a rectangle with the ratio of two sides equal to G) is more beautiful than a rectangle with any other ratio as its sides. In 1876, Fechner made a large-scale experiment. The results show that the number of people who like the golden rectangle accounts for one-third of the total, and they get the most votes among all kinds of rectangles.
I'm afraid there are many such legends. Bottom line: where there is G, there is beauty. The golden section number g has become the beauty of the universe!