Golden Section

Divide a line segment into two parts so that the ratio of one part to the full length is equal to the ratio of the other part to this part. The ratio is an irrational number, and the approximate value of the first three digits is 0.618. Because the shape designed according to this ratio is very beautiful, it is called the golden section, also known as the ratio between Chinese and foreign parts. This is a very interesting number. We use 0.618 to approximate it and we can find it through simple calculation:

1/0.618=1.618

(1-0.618)/0.618=0.618

The role of this value is not only reflected in art fields such as painting, sculpture, music, architecture, etc., but also plays an important role in management, engineering design, etc.

Let us first start with a sequence. Its first few numbers are: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144... this The name of the sequence is "Fibonacci Sequence", and these numbers are called "Fibonacci numbers". The characteristic is that except for the first two numbers (which have a value of 1), each number is the sum of the two preceding numbers.

What is the relationship between the Fibonacci sequence and the golden section? Research has found that the ratio of two adjacent Fibonacci numbers gradually approaches the golden ratio as the sequence number increases. That is, f(n)/f(n-1)-→0.618…. Since Fibonacci numbers are all integers, the quotient of dividing two integers is a rational number, so it only gradually approaches the irrational number of the golden ratio. But when we continue to calculate the later larger Fibonacci numbers, we will find that the ratio of two adjacent numbers is indeed very close to the golden ratio.

A very illustrative example is the five-pointed star/regular pentagon. Five-pointed stars are very beautiful. There are five in our country’s national flag. Many other countries also use five-pointed stars in their national flags. Why is this? Because the length relationship between all line segments that can be found in the five-pointed star is consistent with the golden ratio. All triangles that appear after the diagonals of a regular pentagon are connected are golden section triangles.

Since the top angle of the five-pointed star is 36 degrees, it can also be concluded that the value of the golden section is 2Sin18.

The golden section is approximately equal to 0.618:1

It refers to dividing a line segment into two parts, so that the ratio of the length of the original line segment to the longer part is the golden section. point. There are two such points on the line segment.

Using the two golden points on the line segment, you can make a regular five-pointed star or a regular pentagon.

More than 2,000 years ago, Eudoxus, the third greatest mathematician of the School of Athens in ancient Greece, first proposed the golden section. The so-called golden section refers to dividing a line segment of length L into two parts, so that the ratio of one part to the whole is equal to the ratio of the other part to that part. The simplest way to calculate the golden section is to calculate the ratio of the last two numbers in the Fibonacci sequence 1, 1, 2, 3, 5, 8, 13, 21... 2/3, 3/5, 4/8 ,8/13,13/21,... Approximate values.

The golden section was introduced to Europe through the Arabs before and after the Renaissance, and was welcomed by Europeans. They called it the "golden method". A European mathematician in the 17th century even called it "The most valuable algorithm among all algorithms". This algorithm is called the "three-rate method" or the "three-number rule" in India, which is what we often call the proportional method now.

In fact, our country also has records about the "golden section". Although it is not as early as ancient Greece, it was independently created by ancient Chinese mathematicians and was later introduced to India. After research. The European proportional algorithm originated from my country and was introduced to Europe from Arabia through India, rather than being introduced directly from ancient Greece.

Because it has aesthetic value in plastic arts, in the length and width design of arts and crafts and daily necessities, the use of this ratio can arouse people's sense of beauty. It is also widely used in real life, such as buildings. The golden section is scientifically used for the ratio of some line segments in the stage. The announcer on the stage does not stand in the center of the stage, but on one side of the stage. The position of standing at the golden section point of the length of the stage is the most beautiful and the sound is The best spread. Even in the plant world, there are places where 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 rules of the golden section.

In many scientific experiments, a 0.618 method is commonly used to select a plan, that is, the optimization method, which allows us to rationally arrange a smaller number of tests to find reasonable western and suitable process conditions. It is precisely because it has extensive and important applications in architecture, literature and art, industrial and agricultural production, and scientific experiments that people preciously call it the "golden section".

The Golden Section is a mathematical proportional relationship. The golden section has strict proportion, artistry and harmony, and contains rich aesthetic value. When applied, it is generally taken to be 0.618, just like the pi ratio is taken to be 3.14 when applied.

The ratio of the length to width of the Golden Rectangle is the golden ratio. In other words, the long side of the rectangle is 1.618 times the short side. The golden ratio and the golden rectangle can bring beauty to the picture and make people happy. Pleasure. It can be found in many works of art as well as in nature. The Temple of Passa in Athens, Greece, is a good example. His "Vitruvian Man" conforms to the golden rectangle. The face of "Mona Lisa" also In line with the golden rectangle, "The Last Supper" also applies this proportional layout.

Discovering history

Because the Pythagoreans in ancient Greece in the 6th century BC studied orthodoxy The drawing of pentagons and regular decagons, so modern mathematicians infer that the Pythagoreans had touched or even mastered the golden section at that time.

In the 4th century BC, the ancient Greek mathematician Eudoxus was the first to systematically study this problem and establish the theory of proportion.

When Euclid wrote "Elements" around 300 BC, he absorbed the research results of Eudoxus and further systematically discussed the golden section, becoming the earliest treatise on the golden section.

After the Middle Ages, the golden section was cloaked in mystery. Several Italian painters, Pacioli, called the middle ratio a sacred ratio and wrote books specifically about it. German astronomer Kepler called the golden section the divine section.

It was not until the 19th century that the name golden section gradually became popular. The golden section has many interesting properties, and its practical applications are also widespread. The most famous example is the golden section method or 0.618 method in optimization, which was first proposed by the American mathematician Kiefer in 1953 and popularized in China in the 1970s.

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Usually Use Greek letters to represent this value.

The wonderful thing about the golden section is that its ratio is the same as its reciprocal. For example: the reciprocal of 1.618 is 0.618, and 1.618:1 is the same as 1:0.618.

The exact value is (√5-1)/2

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

0.6180339887 4989484820 4586834365 6381177203 0917980576 < /p>

2862135448 6227052604 6281890244 9707207204 1893911374

8475408807 5386891752 1266338622 2353693179 3180060766

7263544333 8908659593 9582905638 3226613199 2829026788

0675208766 8925017116 9620703222 1043216269 5486262963

1361443814 9758701220 3408058879 5445474924 6185695364

8644492410 4432077134 4947049565 8467885098 7433944221

2544877066 4780915884 6074998871 2400765217 0575179788

3416625624 9407589069 7040002812 1042762177 1117778053

1317952368 9427521948 4353056783 0022878569 9782977834

7845878228 9110976250 0302696156 1700250464 3382437764

8610283831 2683303724 2926752631 1653392473 1671112115

8818638513 3162038400 5222165791 2866752946 5490681131

7159934323 5973494985 0904094762 1322298101 7261070596

1164562990 9816290555 2085247903 5240602017 2799747175

3427775927 7862561943 2082750513 1218156285 5122248093

94 71234145 1702237358 0577278616 0086883829 5230459264

7878017889 9219902707 7690389532 1968198615 1437803149

9741106926 0886742962 2675756052 3172777520 3536139362

1076738937 6455606060 5922...

Life Application< /p>

What’s interesting is that this number can be seen everywhere in nature and people’s lives: people’s navel is the golden section of the total length of the human body, and people’s knees are the golden section from the navel to the heel. The width-to-length ratio of most doors and windows is also 0.618...; on some plant stems, the angle between two adjacent petioles is 137 degrees 28', which is exactly the angle between the two radii that divides the circumference into 1:0.618... According to research, this angle has the best ventilation and lighting effects for plants.

Architects are particularly fond of mathematics 0.168..., whether it is the pyramids of ancient Egypt, the Notre Dame Cathedral in Paris, or the Eiffel Tower in France in the recent century, there are data related to 0.168.... People have also discovered that the themes of some famous paintings, sculptures, and photography are mostly located at 0.168... of the picture. Artists believe that placing the bridge of a string instrument at 0.168... of the strings can make the sound softer and sweeter.

The number 0.168... is of more concern to mathematicians. Its appearance not only solves many mathematical problems (such as: dividing the circumference into ten and five equal parts; finding the sine and cosine of angles of 18 degrees and 36 degrees) value, etc.), and also makes the optimization method possible. The optimization method is a method for solving optimization problems. For example, when making steel, it is necessary to add a certain chemical element to increase the strength of the steel. Assume that the amount of a certain chemical element to be added to each ton of steel is between 1,000 and 2,000 grams. In order to find the most appropriate amount, it is necessary to The test was conducted in the range of 1000g and 2000g. Usually, the midpoint of the interval (i.e. 1500 grams) is taken for the test. Then compare the test results with the experimental results at 1000 grams and 2000 grams respectively, select the two points with higher intensity as the new interval, then take the midpoint of the new interval for testing, and then compare the endpoints, and so on until you obtain Optimal results. This experimental method is called the bisection method. However, this method is not the fastest experimental method. If the experimental point is taken at 0.618 of the interval, the number of experiments will be greatly reduced. This method of taking 0.618 of the interval as the test point is the one-dimensional optimization method, also called the 0.618 method. Practice has proved that for a factor problem, using the "0.618 method" for 16 tests can achieve the same effect as the "bisection method" for 2500 tests. Therefore, the great painter Leonardo da Vinci called 0.618... the golden number.

0.618 and War: Was Napoleon the Great defeated by the Golden Section?

0.618 is an extremely fascinating and mysterious number, and it also has a very beautiful name - the golden section. It was invented by Pythagoras, a famous ancient Greek philosopher and mathematician, more than 2500 years ago. discovered before. Throughout the ages, this number has been regarded as the golden rule of science and aesthetics by future generations. In the history of art, almost all outstanding works have unanimously verified this famous golden ratio. Whether it is the Parthenon in ancient Greece or the Terracotta Warriors and Horses in ancient China, there is an unexpected difference between their vertical and horizontal lines. Completely consistent with the ratio of 1 to 0.618.

Perhaps, we already know a lot about the performance of 0.618 in science and art, but have you heard that 0.618 is also related to the tragic and cruel battlefields where artillery fire, smoke, and flesh and blood are everywhere? With an indissoluble bond, it also shows its huge and mysterious power in the military?

0.618 and weapons and equipment

In the cold weapon age, although people did not know the concept of the golden ratio at all, when people made swords, broadswords, spears and other weapons, gold The law of division ratio has also been reflected everywhere, because weapons made according to such proportions will be easier to use.

When the rifle that fired bullets was first manufactured, the length ratio of its handle to the gun body was very unscientific and reasonable, making it very inconvenient to hold and aim. In 1918, a corporal of the American Expeditionary Force named Alvin York modified this rifle. The ratio of the improved gun body and gun handle exactly matched the 0.618 ratio.

In fact, from the curvature of the sharp saber edge, to the apex of bullets, artillery shells, and ballistic missiles flying along the trajectory; from the optimal bombing height and angle when the aircraft enters the dive bombing state, to the tank shell design When considering the optimal bullet avoidance slope, we can easily find that the golden ratio is everywhere.

In artillery shooting, if a certain indirect artillery has a maximum range of 12 kilometers and a minimum range of 4 kilometers, its optimal shooting distance is about 9 kilometers, which is 2/3 of the maximum range. Very close to 0.618. When deploying for combat, if it is an offensive battle, the artillery position should be placed at 1/3 times the maximum firing range from one's own front line. If it is a defensive battle, the artillery position should be placed 2/3 times the maximum firing range away from one's own front line.

0.618 and tactical formation

In some wars that occurred very early in our country’s history, the law of 0.618 was followed.

During the Spring and Autumn Period and the Warring States Period, Jin Ligong led his army to attack Zheng, and fought a decisive battle with the Chu army who was aiding Zheng in Yanling. Duke Li followed the advice of the Chu rebel Miao Benhuang and made Chu's right army the main attack point. Therefore, he used one part of the central army to attack the Chu army's left army; he used the other part to attack the Chu army's middle army. The army, the new army and the soldiers of the public clan attacked Chu's right army. The choice of its main attack point is right at the golden section.

The series of battles commanded by Genghis Khan should be considered the most outstanding military operation that embodies the golden section law in war. For hundreds of years, people have been puzzled as to why Genghis Khan's Mongolian cavalry could sweep across the Eurasian continent like a hurricane sweeping down fallen leaves. This is because they only relied on the nomads' fierceness, cruelty, cunning, good riding and archery, and the mobility of the cavalry. The reasons are not enough to provide a completely convincing explanation for this. Maybe there are other more important reasons? After careful study, I discovered the great role of the golden section law. The battle formation of the Mongolian cavalry is very different from the traditional Western phalanx. In its 5-row formation, the ratio of heavy cavalry with human helmets and vests to fast and agile light cavalry is 2:3. This is another golden section. ! You can't help but admire the genius of the horseback military strategist. It would be strange if an army led by such a genius commander didn't dominate the world and be invincible.

The Battle of Abela between Macedonia and Persia is a relatively successful example of Europeans using 0.618 in war. In this battle, Alexander the Great of Macedonia chose the attack point of his army on the left wing and central junction of the army of King Darius of Persia. Coincidentally, this part happened to be the "golden point" of the entire battle line, so although the Persian army was dozens of times larger than Alexander's troops, Alexander relied on his strategic wisdom to defeat the Persian army. The profound impact of this war is still clearly visible today. In the Gulf War, multinational forces used a similar formation to defeat the Iraqi army.

When two armies are fighting, if one of them loses more than 1/3 of its troops and weapons, it will be difficult to fight the other side. Because of this, in modern high-tech wars, major military countries with high-tech weapons and equipment adopt long-term air strikes to completely destroy more than 1/3 of the opponent's troops and weapons, and then launch ground attacks. Let's take the Gulf War as an example. Before the war, military experts estimated that if the equipment and personnel of the National Guard and the National Guard suffered losses of 30% or more through aerial bombing, they would basically lose their combat effectiveness. In order to bring the Iraqi army's losses to this critical point, the U.S. and British coalition forces repeatedly extended the bombing time, lasting for 38 days, until they destroyed 38% of the 428 tanks, 32% of the 2,280 armored vehicles, and 3,100 artillery pieces in the theater. 47%. At this time, the strength of the Iraqi army dropped to about 60%. This is the critical point when the army loses its combat effectiveness. That is to say, after weakening the Iraqi military strength to the golden section, the US and British coalition forces drew out the "Desert Saber" and slashed at Saddam. It only took 100 hours of ground combat to achieve the war goal. In this war known as "Desert Storm", General Schwarzkopf, who created the miracle of only killing more than a hundred people in a battle, was not a master, but his luck was almost the same as all others. As good as a master of the military arts. In fact, what really matters is not luck, but that the commander-in-chief of a modern army accidentally involved 0.618 in the planning of the war. In other words, he was more or less blessed by the law of the golden section.

In addition, in modern wars, the armies of many countries often carry out specific offensive tasks in echelons. The strength of the first echelon accounts for about 2/3 of the total strength, and the strength of the second echelon About 1/3. In the first echelon, the troops invested in the main attack direction are usually 2/3 of the total strength of the first echelon, and the assist direction is 1/3. In a defensive battle, the strength of the first line of defense is usually 2/3 of the total number, and the strength and weapons of the second line of defense is usually 1/3 of the total number.

0.618 and strategic battles

0.618 is not only reflected in weapons and battlefield formations at one time and place, but also in macroscopic wars with vast areas and long time spans. be fully demonstrated.

The heroic Emperor Napoleon may never have thought that his fate would be closely linked to 0.618.

In June 1812, it was the coolest and most pleasant summer season in Moscow. After the Battle of Borodino failed to eliminate the effective strength of the Russian army, Napoleon led his army into Moscow at this time. At this time, he was very ambitious and arrogant. He did not realize that genius and luck were disappearing from him bit by bit at this time, and the peak and turning point of his life's career were coming at the same time. Later, the French army evacuated Moscow in despair amid heavy snowfall and howling cold wind. Three months of victorious march and two months of prosperity and decline. Looking at the timeline, when the French emperor looked down at the city of Moscow through the blazing flames, he was stepping on the golden section.

On June 22, 1941, Nazi Germany launched the "Barbarossa" plan against the Soviet Union and implemented a blitzkrieg. In a very short period of time, it quickly occupied the vast territory of the Soviet Union and continued to advance deeper into the country. For more than two years, the German army maintained its offensive momentum. Until August 1943, when Operation Barbarossa ended, the German army turned to the defensive and was no longer able to launch a so-called offensive against the Soviet army. Offensive for campaign operations. The Battle of Stalingrad, recognized by all war historians as the turning point of the Soviet Patriotic War, took place 17 months after the outbreak of the war. It was the golden section of the 26-month timeline of the German army's rise and fall.

We often hear the term "golden section". Of course, "golden section" does not refer to how to divide gold. It is a metaphor, which means that the ratio of division is as precious as gold. So what is this ratio? is 0.618. People call the dividing point of this ratio the golden section point, and 0.618 is called the golden number. And people think that if it meets this proportion, it will look more beautiful, better-looking, and more coordinated. In life, there are many applications of the "golden section".

The most perfect human body: the distance from the navel to the soles of the feet/the distance from the top of the head to the soles of the feet=0.618

The most beautiful face: the distance from the eyebrows to the neck/the distance from the top of the head to the neck=0.618

Proof method:

Suppose the length of a line segment AB is a, point C is on the golden section point close to point B and AC is b

AC/ AB=BC/AC

b^2=a*(a-b)

b^2=a^2-ab

a^-ab+(1 /4)b^2=(5/4)*b^2

(a-b/2)^2=(5/4)b^2

a-b/2= (root 5/2)*b

a-b/2=(root 5)b/2

a=b/2+(root 5)b/2 < /p>

a=b(root 5+1)/2

a/b=(root 5+1)/2