Explain the golden ratio in art?

The golden section is an ancient mathematical method. \

Its magical functions and magic power have not been clearly explained in mathematics so far, but in practice it is found that it often plays an unexpected role.

Here is to explain how to get the golden section line, and according to the golden section line to guide the next operation of buying and selling stocks.

There are two kinds of golden section: one-point golden section and two-point golden section.

Here's the method: there are two factors in drawing a single point (one is the golden number, the other is the highest point or the lowest point)

The first step in drawing the golden section is to remember some special numbers:

0. 19 1 0.382 0.6 18 0.809

1. 19 1 1.382 1.6 18 1.809

2. 19 1 2.382 2.6 18 2.809

Among these figures, 0.382, 0.6 18, 1.382, 1.6 18 are the most important, and the stock price is likely to generate support and pressure at the golden section generated by these four figures.

Step two, find a point. This point is the highest point when the rising market turns around, or the lowest point when the falling market turns around. Of course, we know that the high and low points here refer to a certain range and are local. As long as we can confirm that a trend (whether up or down) has been tied up or temporarily ended, the turning point of this trend can be used as the golden section point. Once this point is selected, we can draw the golden section line.

When the rising market begins to reverse, we are extremely concerned about where this decline will be supported. The golden section provides the following price points. They are multiplied by several special figures listed above, and then multiplied by the peak price of this rise. Assuming that the peak of this rise is 10 yuan, then

8.09= 10×0.809

6. 18= 10×0.6 18

3.82= 10×0.382

1.9 1= 10×0. 19 1

These prices are very likely to be the support, among which 6. 18 and 3.82 are the most likely.

In the same way, when the falling market starts to turn around, we are concerned about where the rising market will be under pressure. The position provided by the gold thread is the reserve price of this decline multiplied by the special figure above. Assuming that the price of Luogu is 10 yuan, then

1 1.9 1= 10× 1. 19 1 2 1.9 1= 10×2. 19 1

13.82= 10× 1.382 23.82= 10×2.382

16. 18= 10× 1.6 18 26. 18= 10×2.6 18

18.09= 10× 1.809 28.09= 10×2.809

20= 10×2

It is likely to become the pressure level in the future. Among them, 13.82 and 16. 18 and 20 yuan are the easiest pressure lines, and those exceeding 20 are rarely used.

In addition, there is another usage of the golden section, that is, the two-point golden section.

Select the highest point and lowest point (local), take this interval as the whole length, and then make the golden section line on this basis to calculate the rebound height and reverberation height. This golden section line is actually a special case of percentage line.

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 (√5- 1)/2.

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

0.6 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 ...

golden ratio

The golden ratio is an irrational number, defined as (√5- 1)/2.

It is used in a wide range of fields, such as mathematics, physics, architecture, art and even music.

The unique nature of the golden ratio was first applied to dividing straight lines. If the total length of a straight line is the denominator of the golden ratio plus the unit length of the molecule, if we divide it into two halves, the long half is the unit length of the molecule and the short half is the unit length of the mother and child, then the ratio of the long line length to the short line length is the golden ratio.

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.

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 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 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, and the proportional layout is also applicable.

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

| | | .

| | | .

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+ - + - + -

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

This value is usually expressed in Greek letters.

Life application

Interestingly, this number can be seen everywhere in nature and people's lives: the navel is the golden section of the whole human body, and the knee is the golden section from the navel to the heel. The aspect ratio of most doors and windows is also 0.618. On some plants, the included angle between two adjacent petioles is 137 degrees 28', which is exactly the included angle between two radii that divide the circumference into 1: 0.6 18. According to research, this angle has the best effect on ventilation and lighting of the factory building.

Architects have a special preference for 0.6 18… in mathematics. No matter the pyramids in ancient Egypt, Notre Dame de Paris, or the Eiffel Tower in France in recent centuries, there are data related to 0.6 18 … It is also found that the themes of some famous paintings, sculptures and photos are mostly at 0.6 18…. The artist thinks that placing the bridge of a stringed instrument at the position of 0.6 18 can make the sound softer and sweeter.

The number 0.6 18 ... is more concerned by mathematicians. Its appearance not only solves many mathematical problems (such as dividing the circumference into ten parts and dividing the circumference into five parts; Find the sine and cosine values of 18 degrees and 36 degrees. ), it also makes the optimization method possible. Optimization method is a method to solve the optimization problem. If it is necessary to add a chemical element to increase the strength of steel during steelmaking, it is assumed that the amount of a chemical element added per ton of steel is between1000-2000g. In order to find the most suitable dosage, it needs to be tested between 1000 g and 2000 g. Usually take the midpoint of the interval (i.e. 1500g) for testing. Then compared with the experimental results of 1000g and 2000g respectively, two points with higher intensity were selected as new intervals, and then the midpoint of the new interval was taken for experiments, and the endpoints were compared in turn until the most ideal results were obtained. This experimental method is called dichotomy. However, this method is not the fastest experimental method. If the experimental point is 0.6 18 of the interval, the number of experiments will be greatly reduced. This method of taking 0.6 18 of the interval as the test point is a one-dimensional optimization method, also known as 0.6 18 method. Practice has proved that for the problem of one factor, using "0.6 18 method" to do 16 experiments can complete the effect of "dichotomy" to do 2500 experiments. So Da Vinci, the great painter, called 0.618 ... the golden number.

0.6 18 and the war: Napoleon the Great lost to the golden section?

0.6 18 is an extremely fascinating and mysterious number, and it also has a very nice name-the golden section law, which was discovered by Pythagoras, a famous ancient Greek philosopher and mathematician, more than 2,500 years ago. 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 excellent works have verified this famous golden section law. Whether it is the Parthenon in ancient Greece or the Terracotta Warriors in ancient China, the ratio of vertical line to horizontal line is exactly 1 to 0.6 18.

Perhaps, we have learned a lot about the performance of 0.6 18 in science and art, but have you ever heard that 0.6 18 has an indissoluble bond with the fierce and cruel battlefield of gunfire and bloodshed, and also shows its great and mysterious power in the military?

0.6 18 and weapons and equipment

In the era of cold weapons, although people don't know the concept of the golden ratio at all, when people make weapons such as swords, broadswords and spears, the law of the golden ratio has already been reflected everywhere, because weapons made according to this ratio will be more handy to use.

When the rifle for firing bullets was first manufactured, the ratio of the length of the handle to the length of the gun body was unscientific and unreasonable, which was very inconvenient for grasping and aiming. 19 18, a corporal named alvin york of the American Expeditionary Force reformed this rifle, and the ratio of the reformed gun body to the handle was exactly 0.6 18.

In fact, from the sharp edge radian to the apex of bullets, shells and ballistic missiles flying along the track; It is not difficult to find the golden ratio everywhere, from the best dropping height and angle of the plane entering the dive bombing state to the best bomb avoidance slope when designing the tank shell.

In artillery firing, if the maximum range of an indirect gun is 12 km and the minimum range is 4 km, its optimal firing distance is about 9 km, which is 2/3 of the maximum range and very close to 0.6 18. In battle deployment, if it is an offensive battle, the position of artillery position is generally 1/3 times of the maximum range from its own front, and if it is a defensive battle, the position of artillery position should be 2/3 times of the maximum range from its own front.

0.6 18 and tactical arrangement

Some wars that happened very early in the history of our country all followed the law of 0.6 18. During the Spring and Autumn Period and the Warring States Period, Jin Ligong led an army to attack Zheng and fought a decisive battle with the Chu army supporting Zheng in Yanling. Gong Li followed the advice of Miao Benhuang, a traitor from Chu, and took the right-wing army of Chu as the main attack point, so he attacked Zuo Jun, a part of China's army. Attack the Chu army with another department, and gather the soldiers of the upper army, the lower army, the new army and the public to attack the Chu right army. The choice of its main attack point is just at the golden section.

A series of wars commanded by Genghis Khan should be the first military action that embodies the golden section law in the war. For hundreds of years, people have been puzzled why Genghis Khan's Mongolian cavalry swept across Eurasia like a hurricane, because the nomadic people's bravery, cruelty, cunning, good riding and shooting, cavalry mobility and other reasons are not enough to make a completely convincing explanation. Maybe there are other more important reasons? After careful study, we found the great function of the golden section law. The combat formation of Mongolian cavalry is very different from the traditional western phalanx. In its five-row formation, the ratio of heavy cavalry wearing helmets and vests to quick and agile light cavalry is 2:3, which is another golden section! You can't help but admire the genius of the horseback strategist. Strangely, the army led by such a talented commander is not invincible in all directions.

The Battle of Abela between Macedon and Persia is a successful example of Europeans using 0.6 18 in the war. In this battle, Alexander the Great of Macedonia chose the attack point of his army at the left-middle junction of the army of King Darius of Persia. Coincidentally, this part is also the "golden point" of the whole front, so although the Persian army is dozens of times more than Alexander's military forces, Alexander defeated the Persian army with his own strategic wisdom. The far-reaching impact of this war is still clearly visible today. In the Gulf War, the multinational forces used similar disposal methods to defeat the Iraqi army.

When two armies are at war, if one of them loses more troops and weapons than 1/3, it will be difficult to fight with the other. Because of this, in modern high-tech wars, military powers with high-tech weapons and equipment take a long-term air strike, first completely destroying the other side's troops and weapons above 1/3, and then launching ground attacks. Let's take the Gulf War as an example. Before the war, according to military experts' estimation, if the equipment and personnel of * * * and the National Guard were lost by air strikes by 30% or more, they would basically lose their combat effectiveness. In order to make the Iraqi army's losses reach this critical point, the US-British coalition forces repeatedly extended the bombing time by 38 days until they destroyed 38% of 428 tanks, 32% of 2,280 armored vehicles and 47% of 3 100 guns in the theater. At this time, the Iraqi army's strength dropped to about 60%, which was the critical point for the army to lose its combat effectiveness. That is, after Iraq's military strength was weakened to the golden section, American talents pulled out the "desert saber" and cut it at Saddam. It only took 100 hours of ground combat to achieve the purpose of the war. In this war known as "Desert Storm", General schwarzkopf, who created a miracle that only 100 people were killed in a great war, was not a master, but his luck was almost as good as that of all military art masters. In fact, what really matters is not luck, but the commander-in-chief who leads a modern army intentionally or unintentionally involved 0.6 18 in the war planning, which means that he was more or less blessed by the golden section law.

In addition, in modern wars, multinational armies often carry out specific offensive tasks by echelon. The strength of the first echelon accounts for about 2/3 of the total strength, and that of the second echelon accounts for about 1/3. In the first echelon, the troops devoted to the main attack direction are usually 2/3 of the total strength of the first echelon, and the auxiliary direction is 1/3. In defensive operations, the strength of the first line of defense is usually 2/3 of the total, and the strength and weapons of the second line of defense are usually 1/3 of the total.

0.6 18 and strategic campaign

0.6 18 is not only reflected in the weapons and battlefield layout at a time and place, but also fully displayed in the macro-war with a vast territory and a long time span.

Napoleon the Great, a lean man, never thought that his fate would be closely linked with 0. 18. June, 18 12, is the coolest and pleasant summer in Moscow. After the battle of Borokino, which failed to destroy the Russian army, Napoleon led the army into Moscow at this time. At this time, he is full of ambition and arrogance. He didn't realize that genius and luck were disappearing from him at this time, and the peak and turning point of his career came at the same time. Later, the French army withdrew from Moscow in frustration in the heavy snow and howling cold wind. Three months of triumph, two months of climax and decline, from the time axis, when the French emperor overlooked Moscow through the flame, his foot just stepped on the golden section.

194 1 On June 22nd, Nazi Germany launched the "Barbarossa" plan against the Soviet Union and conducted a blitzkrieg. In a very short period of time, it quickly occupied the vast territory of the Soviet Union and continued to advance further to China. For more than two years, the Germans kept the momentum of attack, until the "Barbarossa" operation ended in August 1943, and the Germans turned to the defensive, and they were no longer able to launch an attack that could be called a battle against the Soviets. 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 rise and fall of the German army.

We often hear the word "golden section". Of course, "golden section" does not mean how to divide gold. This is a visual statement that the proportion of points is as precious as gold. So what's the ratio? It is 0.6 18. People call the dividing point of this ratio the golden section point and 0.6 18 the golden section number. And people think that if it meets this ratio, it will look more beautiful, more beautiful and more harmonious. In life, "golden section" has many applications.

The most perfect human body: the distance from navel to sole/the distance from top of head to sole =0.6 18.

The most beautiful face: the distance from eyebrow to neck/the distance from the top of head to neck =0.6 18.

Golden section and chemistry

There is also a golden section in chemistry.

After adding 1* 10-7 NAOH into the water, the concentration of H+ at this time turned out to be (√5- 1)/2* 10-7.

Exploration of golden section