Practical application of golden section

Some ancient Greeks tried to solve the problem of golden section by image and achieved beautiful results. Euclid (about 330-257 BC) summarized the experience and research results of predecessors and compiled thirteen volumes of Principles of Geometry. This is the world's earliest mathematical work described by axiomatic method. The problem of golden section geometry has aroused people's extensive interest and been widely used in science, art, architecture and technology. Philosophers and aestheticians have also discussed it repeatedly, and articles have been published continuously.

The golden section is good for health and fitness. "0.6 18", this ratio was applied to plastic arts by ancient Greek aestheticians because of its aesthetic value, because any body that conforms to the golden section law is always the most beautiful. Now it is found that this ratio is inextricably linked with medical care, health and longevity, which can also be called the golden section law of health. In the human body structure, 0.6 18 is everywhere. The ratio of umbilical cord to sole, top of head to umbilical cord; The ratio of trunk length to hip circumference; The ratio of lower limb length to upper limb length is about 0.6 18. And the closer you get to this value, the more symmetrical and perfect your whole body will be. It is most suitable for people to live at the ambient temperature of 22℃-24℃, because the normal body temperature is 36℃-37℃, and the product of this body temperature and 0.6 18 is exactly 22.4℃-22.8℃, and at this ambient temperature, the physiological function, life rhythm and other metabolic levels of the human body are in the best state. For another example, nutrition emphasizes that 60% of coarse grains and 40% of flour and rice in a staple food are eaten together, which is beneficial to the digestion and absorption of the stomach and avoids gastrointestinal diseases. This can also be included in the 0.6 18 law of diet. Anti-aging includes physiological anti-aging and psychological anti-aging, which is more important? Studies have proved that the physiological anti-aging is four and the psychological anti-aging is six, which also conforms to the golden section law. Fully mobilizing and reasonably coordinating psychological and physiological forces to delay aging can achieve the best effect of prolonging life. Reasonable daily life also conforms to the division of 0.6 18. Of the 24 hours, 2/3 are for work and life, and 1/3 is for rest and sleep. On the relationship between motion and stillness, is it "life lies in motion" or "life lies in stillness"? From the dialectical point of view and a lot of life practice, it is proved that the relationship between activity and static is the same as the proportion of work and rest on the same day. Four points of activity and six points of activity are the best health care methods. Static and dynamic: From a dialectical point of view, the static and dynamic relationship is 0.6 18, and roughly four shifts and six divisions are better health preservation methods. Diet: medical experts also found that people who eat 60% to 70% full have almost no stomach problems; The proper diet is six portions of coarse grains and four portions of refined food. According to the golden section law, the best season for getting married is 0.6 18 in one year 12 months, from the end of July to the end of August. Medical research shows that autumn is the best golden season for human immunity. Because there are the most lymphocytes in human blood from July to August, a large number of lymphatic factor resistant to various microorganisms can be produced. At this time, people's immunity is strong. Less small huxing attracts many young people who have just entered the society as owners with its "low total price, low down payment and low monthly payment". Although the market demand for small huxing is very enthusiastic, there are also investment risks. How to invest in small houses? An interesting "golden section theory" is popular in the market. Because the ratio of working time to family time just constitutes a golden section, that is, 0.6 18 to 0.382, experts believe that the most valuable position may be the golden section between work and community. Small units should be carefully calculated because of their small size. In the process of getting hotter and hotter, there is a trend in the market, that is, huxing. But absolutely small does not meet the normal needs of residents, and it will definitely not be a trend. The new consumption or investment trend shows that there is also a golden ratio in small apartment area. Between 30-80 square meters, there is a golden ratio, just over 50 square meters. Therefore, the sales volume of small-sized units of more than 50 square meters in the market exceeds other specifications. The space is mainly bedroom and living room, and 30 square meters can't subdivide any functional area at all, so it is difficult to meet the high-quality home life. More than 50 square meters is the smallest area of functional golden section, which can be divided into 30 square meters of main space and 20 square meters of supporting space to solve various functions such as independent kitchen and bathroom, balcony and storage. Therefore, the small apartment selected according to the golden section theory should not only save the apartment area and reduce the total investment, but also meet the aesthetic and functional needs of space and ensure the quality of life and home interest of residents.

The golden ratio existed in the objective world before it was discovered, and people had a clear understanding of it after unveiling this mystery. When people re-observe nature according to this law, they are surprised to find that this law can be seen in many beautiful things in nature, such as leaves, flowers, snowflakes and five-pointed stars of plants ... Many animals and insects, especially the human body, have a rich golden ratio relationship. When people know this natural law, it is widely used in human life. Since then, in our living environment, it can be seen everywhere, such as building doors and windows, cabinets and desks; Books, newspapers and magazines that we often contact; Modern movie screen. Television screens and many household appliances are made up of similar numbers. It is applied as a classic law in the history of art. Many artists have created many immortal masterpieces with it.

The golden section can be said to have a natural connection with the composition of photographic pictures. For example, the film window ratio of a camera is 135, and the camera is 24X36, which is a ratio of 2:3, which is typical. The 4.5X6 of 120 camera is about 3: 5. Although 6x6 is a box, most of the post-production is cut into rectangles. As long as we open the album and have a look, we will find that most of the painting forms are similar to this ratio. This may be influenced by tradition and form people's aesthetic habits. In addition, it is true that due to its pleasant nature, sometimes people do not notice this ratio in time, but deliberately use it, but often unconsciously enter this law. This also shows that the golden section itself has a beautiful nature. In photography practice, the golden section rule is applied, mainly in the application of golden section points, lines and surfaces. The golden section, in panoramic composition, is mostly the position of the main performance object or visual center, while in mid-shot and close-shot composition, it is mostly the position of the main part of the scene. In portrait composition, people's eyes are often treated near the golden section. Gold thread is often used as the location of horizon, horizontal line and skyline.

Dream is a reproducible three-part musical form, which consists of three parts: A, B and A'. Each paragraph consists of two 4-bar phrases of equal length. The whole song is divided into 6 sentences and 24 bars. Theoretically, the golden section should be in 14 (240. 18 = 14.83), which coincides with the climax of the whole song. Some music accords with the golden ratio from the whole to each part, and the six phrases of this song are divided into negative phases in their second bar (short before and long after); The three parts of this song, A, B and A, are divided in the second paragraph (long before and short after) of their respective phrases, thus forming a vivid situation of multi-layer compound division of the music from the whole to each part, making the content and form of the music more perfect. The sonata form and complex trilogy form in large and medium-sized forms are a three-part structure, while others, such as variations, Rondo and some freestyle, all have three factors to varying degrees. The golden ratio principle is also reflected in these large and medium-sized music to varying degrees. Generally speaking, the larger the scale of the musical form, the position of the golden section will be behind the central part or the expanded part, or even postponed to the beginning of the reproduction part, so as to obtain a stronger artistic effect. The first movement of Mozart's sonata in D major is 160, which reappears in the 99th section, just falling on the golden section (1600+08 = 98.88). According to the statistics of American mathematician Qiao Ba, 94% of Mozart's piano sonatas meet the golden ratio, which is amazing. We may not find out whether Mozart consciously made his music conform to the golden section, or whether it was just a purely intuitive coincidence. However, another American musician thinks. "You know, Mozart, who created these immortal works, is also a genius who likes digital games. Mozart understood the golden section and used it consciously. " The second movement of Beethoven's pathetique sonata op13 is adagio and Rondo, with 73 bars. Theoretically, the golden section should be in 45 bars, and the climax of the whole song is formed in 43 bars. With the change of mode and tonality, the climax is basically consistent with the golden section. Chopin's serenade in D flat major is a trilogy. Excluding the prelude 76 bars, it is theoretically calculated that the golden section should be in 46 bars, and the reproduction part is just in 46 bars, which is the climax of the whole song. This is really great. Let's give another example of large-scale symphony music. The great Russian composer remus Kosakov wrote in the fourth movement of his symphony suite Arabian Nights that Sinbad's ship ran irretrievably into a cliff with a bronze knight in the rough waves. In the deafening sound of the whole band, the band struck a sonorous gong, which extended for six bars. With its sound, at the climax of the whole song, that is, the "golden point", the tragic effect caused by the fatal blow of the big gong is amazing.

Huang Jinlv has always been dyed with magnificent and mysterious colors, and is called the most wonderful form proportion of "natural rationality". The beauty of numbers exists in every corner of the world. For our eyes, especially for those who study music, "beauty is everywhere, not the lack of beauty, but the lack of discovery."

"0.6 18" has always had an indissoluble bond with military development, and it often happens by chance. Whether it is the beautiful wheel of the Parthenon in ancient Greece or the Terracotta Warriors and Horses in ancient China, the relationship between the vertical line and the horizontal line completely conforms to the ratio of 1: 0. 18. Genghis Khan's Mongolian cavalry swept across Eurasia, which was amazing. Through research, it is found that the combat formation of Mongolian cavalry is very different from the traditional western phalanx. In his five-column formation, the heavy cavalry and light cavalry are 2∶3, the heavy cavalry with helmet vest is 2, and the fast and flexible light cavalry is 3, which coincides with the golden section law. Europeans first consciously applied the golden section law to religion and art, and its military application began in the Black Powder Period. At that time, muskets replaced spears. Maurice, the Dutch general who took the lead in mixing musketeers and spearmen, failed to break through the shackles of traditional formations. After King Gustav of Sweden adjusted this formation, the Swedish army became the most effective army in Europe at that time. His method is to add 96 musketeers to the original 2 16 musketeers and 198 musketeers of General Morris. This change conforms to the influence law of scientific and technological development and weapons and equipment progress on tactical development, highlights the role of firearms in combat, and makes it cross the watershed of the era of cold weapons and hot weapons. The ratio of 198+96 musketeers to 2 16 spearmen once again shows us the magical effect of the golden section law. 1865438+In June 2002, Napoleon attacked Russia; In September, he entered Moscow after the Battle of Borodino. Napoleon didn't realize at this time that genius and luck were disappearing from him bit by bit, and the peak and turning point of his career came at the same time. A month later, the French army withdrew from Moscow in the heavy snow. After three months of victorious marching and two months of prosperity and decline, from the time axis, Napoleon just stepped on the golden section.

In another June after 130, Nazi Germany launched the "Barbarossa" plan against the Soviet Union. For more than two years, the Germans kept the offensive momentum until August 1943, when the "Castle" operation ended, the Germans turned to the defensive and never launched a campaign-scale attack on the Soviet Union again. The Battle of Stalingrad, recognized by all war historians as the turning point of the Soviet Patriotic War, took place in1July after the war broke out, which was the golden point of the 26-month timeline of the German army's decline from prosperity. During the Gulf War, the US military repeatedly extended the air raid time by 38 days. It was not until 38% of Iraq's 4,280 tanks, 32% of 2,280 armored vehicles and 47% of the 3 100 artillery pieces in the theater were destroyed, that is, Iraq's military strength was weakened to the golden section, that the "desert saber" was pulled out to attack Saddam. The purpose of this war was achieved in only 65,438+000 hours on the ground.

Through some scattered examples in the war, the shadow of "0.6 18" is faintly visible, swaying and wandering. If viewed in isolation, it seems to be accidental coincidence, but if too many accidents follow the same trajectory, it becomes a law, which is particularly worthy of in-depth study.

Once I accidentally played ball with my classmates on the playground and measured Newton's nose. The distance between his nostrils and the ratio to the bridge of his nose are close to 0.6 18. After that, the noses of several people were measured, and the results were all in line with the golden section. In the following life, we became very sensitive to 0.6 18. After students' speculation and practice, we found that the aspect ratio of Duo Mi Le ancient cards, the proportion of butterfly body parts and the aspect ratio of beautiful petals also conform to this law. Inquiring a lot of relevant information, such as the pyramids of Egypt, is the best application of this law.

Imagine how to make a very ordinary thin rubber band make a "Doraemon" sound. Tighten, fix and stir, which is "1", and then measure its length and do a geometric problem in Grade Three-divide this "line segment" into golden sections, and you can measure the longer one of the two line segments obtained by division, which is about 0.6 18 times the length of the original line segment. Pinch this point and pluck the longer "string" to make a "2"; Then divide the longer line into golden sections and find "3", and so on, you can also find "4, 5, 6, 7".

Have you ever seen Toronto, a famous Canadian city by the lake Ontario, where the clear water flows gently on TV? In this modern city with rows of tall buildings, the most striking building is the towering Toronto TV Tower, which is magnificent and straight into the sky. Interestingly, the flat-topped castle embedded in the middle and upper part of the tower is located at 0.6 18 times of the total length of the tower, which is the golden section of the tower height. It makes the thin TV tower look harmonious, elegant and unique. Toronto TV Tower is called "the king of towers", and this wonderful "0. 18" has played a decisive role. Similarly, the second floor of the world-famous French "father of the Eiffel Tower" is located in the prime location of the tower, which adds infinite charm to the tower.

Magnificent architecture is indispensable for "0.6 18", especially in art. On the stage, the actors are not standing in the middle or on the edge of the stage, but standing at 0.6 18 times the total length of the stage. At this point, the audience looks very comfortable. The "golden ratio"-0.618-can be found in Milos' famous statues such as Venus, Athena and Amanda the Sea Girl, so the works have reached a beautiful fairyland.

Leonardo da Vinci's Mona Lisa and Raphael's The Gentle and Handsome Virgin both used this ratio intentionally or unintentionally. Because many parts of the human body follow the golden ratio. It is recognized that the most perfect face shape-"goose egg" shape, the ratio of face width to face length is about 0.6 18. If we calculate the graceful figure of ballerinas who want to live forever, we can know that the ratio of their leg length to their body length is also around 0.6 18, which constitutes the beauty of the human body.

An erhu player in China found in his long playing career that if the "weight" of the erhu is placed somewhere on the strings, the timbre will be unparalleled. Verified by mathematicians, this is exactly the golden section of the string, 0.6 18! The golden ratio is working miracles! ?

Accidental? No, around people, there are masterpieces of 0.6 18 everywhere: people always make desktops, doors and windows into rectangles with an aspect ratio of 0.6 18. Mathematically, 0.6 18 is even more amazing. 0.6 18, the proportion of beauty, beautiful color and beautiful melody are widely reflected in people's daily life and are closely related to people. 0.6 18, a wonderful number! It has created countless beautiful scenery and unified people's aesthetics.

The joking 0.6 18 created many "coincidences". In the whole world, 0.6 18 shining like gold is everywhere! People have been creating the golden section intentionally or unintentionally. As long as you pay a little attention, you can find how close it is to our life! Mathematics is very close to us, and it is applied all the time!

We should first feel and appreciate the beauty in mathematics learning. Mathematical beauty is different from other beauty, it is unique and inherent. This kind of beauty, as Russell, a famous British philosopher and mathematical logician, said: "Mathematics, if viewed correctly, has not only truth, but also supreme beauty, just like the beauty of sculpture, which is a kind of cold and serious beauty. This beauty is not the weak aspect that attracts our nature. This beauty is not as gorgeous as painting or music. It can be pure and sublime, and it can reach the perfect state that only great art can express. " The teacher often tells us the beauty of mathematics in class. Through the study of advanced mathematics, I gradually realized the true meaning of the beauty of mathematics. This feeling is strange, subtle, understandable but difficult to express. Mathematics is so fascinating to me ... as long as we are good at observing and thinking and combine what we have learned with life, we will feel the fun of mathematics. Mathematical knowledge is everywhere in life.