What is the use of the golden section?

The golden section is approximately equal to 0.6 18: 1.

Refers to the point where a line segment is divided into two parts, so that the ratio of the length of the original line segment to the longer part 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.

Around the Renaissance, the golden section was introduced to Europe by Arabs and was welcomed by Europeans. 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.

Divide 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 use 0.6 18 to approximate it, and we can find it by simple calculation:

1/0.6 18= 1.6 18

( 1-0.6 18)/0.6 18=0.6 18

This kind of value is not only reflected in painting, sculpture, music, architecture and other artistic fields, but also plays an important role in management and engineering design.

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.

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 a length of L into two parts, so that the ratio of one part to the whole is equal to 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.

Around the Renaissance, the golden section was introduced to Europe by Arabs and was welcomed by Europeans. 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 1.6 18 in application, just as pi is 3. 14 in application.

Discover history

Since the Pythagorean school in ancient Greece studied the drawing methods of regular pentagons and regular decagons in the 6th century BC, modern mathematicians have come to the conclusion that Pythagoras school had touched 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.

|..........a...........|

+ - + - + -

| | | .

| | | .

| B | A | b

| | | .

| | | .

| | | .

+ - + - + -

|......b......|..a-b...|

This value is usually expressed in Greek letters.

The wonder of the golden section is that its proportion is the same as its reciprocal. For example, the reciprocal of 1.6 18 is 0.6 18, while1.618 is the same as 1:0.6 18.

The exact value is the root number 5+ 1/2.

The golden section number is irrational, and the first 1024 bits are:

1.6 180339887 4989484820 4586834365 638 1 177203 09 17980576

2862 135448 6227052604 628 1890244 9707207204 18939 1 1374

8475408807 538689 1752 1266338622 2353693 179 3 180060766

7263544333 8908659593 9582905638 32266 13 199 2829026788

0675208766 89250 17 1 16 9620703222 10432 16269 5486262963

136 14438 14 975870 1220 3408058879 5445474924 6 185695364

86444924 10 4432077 134 4947049565 8467885098 743394422 1

2544877066 47809 15884 607499887 1 24007652 17 0575 179788

34 16625624 9407589069 70400028 12 1042762 177 1 1 17778053

153 17 14 10 1 1704666599 1466979873 176 1356006 70874807 10

13 17952368 942752 1948 4353056783 0022878569 9782977834

7845878228 9 1 10976250 0302696 156 1700250464 3382437764

86 1028383 1 2683303724 292675263 1 1653392473 167 1 1 12 1 15

88 186385 13 3 162038400 5222 16579 1 2866752946 549068 1 13 1

7 159934323 5973494985 0904094762 1322298 10 1 726 1070596

1 164562990 98 16290555 2085247903 52406020 17 2799747 175

3427775927 786256 1943 20827505 13 12 18 156285 5 122248093

947 1234 145 1702237358 05772786 16 0086883829 5230459264

78780 17889 92 19902707 7690389532 1968 1986 15 1437803 149

974 1 106926 0886742962 2675756052 3 172777520 3536 139362

1076738937 6455606060 5922 ...

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 very 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 music forms are a three-part structure, while others, such as variations, Rondo and some free forms, 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 height of the tower. 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.