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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 strategic campaign

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, it was estimated by military experts that if the equipment and personnel of the Republican 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.

Napoleon the Great was defeated by the golden section?

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.

The Parthenon in ancient Greece is a world-famous perfect building with an aspect ratio of 0.6 18. Architects found that the palace designed according to this ratio is more magnificent and beautiful; If you design a villa, it will be more comfortable and beautiful. Even if a door and window is designed as a golden rectangle, it will be more harmonious and pleasing to the eye. The application of the golden section in life was highly praised by the medieval mathematician Kepler (157 1- 1630). He said: geometry has two treasures: one is Pythagorean theorem and the other is golden section. The golden section was discovered by Pythagoras, an ancient Greek mathematician in the 6th century BC. Later, Plato, an ancient Greek aesthete, called it the golden section. So, what is the golden section? Finding a point on a known line segment and dividing one part of the line segment by this point is the median ratio of the whole line segment to another part, which is the golden section problem. As shown in the following figure 1, architectural monuments and "golden ratio" scientists and artists generally believe that Huang Jinlv is the law that architectural art must follow. Therefore, ancient architects and sculptors skillfully used the golden ratio to create magnificent architectural masterpieces and artistic treasures: the Great Pyramid of Khufu, which was built in 3000 BC, and its original height and base length were about 1: 1.6, and the solemn Parthenon in Athens, which was built in the fifth century BC. The building was built in the prosperous era of ancient Greek mathematics, and its beauty is based on strict mathematical laws. If we draw a rectangle around the temple, we find that its length is about 1.6 times as wide. This rectangle is called a golden rectangle. Toronto TV Tower in Canada is one of the tallest buildings in the world today, with a height of 553.3m, and its seven-story working hall is 340m high, with a ratio of 340:553≈0.6 15. Coincidentally, historical buildings in these three different periods all adopted the golden ratio. After measuring a large number of human bodies, the Italian mathematician Fibonacci learned that the ratio of the length above the navel to the height of the human body was close to 0.6 18, and the ratio of a few people was equal to 0.6 18, which was called "standard beauty". Therefore, when artists create artistic human bodies, they all create with the golden ratio as the standard. For example, Apollo, the sun god, and Venus, the goddess in ancient Greek mythology, are completely in line with the golden ratio. The golden section of human body includes four aspects, namely 18 "golden section point", such as umbilical cord is the boundary point from the top of the head to the sole of the foot, Adam's apple is the boundary point from the top of the head to the umbilical cord, and brow point is the boundary point from the hairline to the chin. 15 "golden rectangle", such as trunk contour, head contour, face contour, lip contour, etc. The six "golden indexes", such as the nose and lip index, refer to the ratio of the width of the nose wing to the length of the oral fissure, the lip index refers to the ratio of the length of the oral fissure to the distance between the outer canthus, and the lip height index refers to the ratio of the red height of the lower lip in the facial midline. Three "golden triangles", such as the anterior triangle of the external nose, the lateral triangle of the external nose, and the triangle from the root of the nose to both sides. In addition, in recent years, domestic scholars have found that the related "golden section" data, such as the length-width ratio of anterior teeth, the ratio of eyebrow spacing to inner canthus spacing, are close to the proportional relationship of "golden section". Experts believe that the discovery of these data not only shows that the human body is the most beautiful object in the world, but also provides a scientific basis for the development of cosmetic medicine and the creation and restoration of human beauty and appearance beauty in clinic. The ancient Greeks believed that beauty was the language of God. They found a mathematical evidence that the golden section was a measure of God. Euclid, a genius in geometry, went further: he found that the secret of natural beauty lies in ingenious and harmonious mathematical proportions, which are mostly close to 1 to 0.5438+08. Medical experts have also observed that when the lower limit of brainwave frequency is 8 Hz, while the upper limit is 12.9 Hz, and the ratio of the upper and lower limits is close to 0.6 18, it is the most enjoyable moment of body and mind. When the temperature is around the golden section of normal body temperature-23℃, it is the most moderate temperature of human body and mind. The most abundant substance in human body is water, and the water content of adult human body accounts for 0.6 18 of body weight. The Golden Section in Nature shows that the contours of various plants do not grow randomly, but follow certain mathematical laws. Looking down from the top of the plant stem, it is found that the included angle between the upper and lower adjacent leaves is about 137.5. This angle is most conducive to the lighting and ventilation of the blades. And137.5: (360-137.5) ≈ 0.618.

In the animal kingdom, graceful animals such as horses, mules, lions, tigers, leopards and dogs. As far as the length and width of body parts are concerned, it is generally close to the golden section. The ratio of butterfly body length to wing length is also close to 0.6 18. The five-pointed star in mathematical golden section geometry is a figure with many golden section points, and its five sides are divided into the golden ratio, which is the most symmetrical ratio. The origin of the five-pointed star shape is very early, and now the earliest five-pointed star pattern is found in Maruk (present-day Iraq) in the lower reaches of the Euphrates River. A clay tablet made around 3200 BC. The "stars" on the national flags of many countries in the world are drawn as five-pointed stars. The golden section diagram is related to the drawing of regular pentagons, regular decagons and pentagonal stars, especially the need to draw pentagonal stars. In addition to the five-pointed star, there are golden triangles, golden ellipses, golden hyperbolas and so on. Fibonacci was a famous European mathematician in13rd century. He is Italian. His book abacus, published in 1202, introduced oriental mathematics to Europeans. The revised edition of this book 1228 introduces a "rabbit problem". This problem needs to calculate how many pairs of rabbits can be bred after one year. Suppose a pair of rabbits can give birth to a pair of rabbits every month, and rabbits can give birth to new rabbits in the second month after birth, so that there is one pair at first, two pairs after one month, three pairs after two months and five pairs after three months ..... The number of rabbits per month is arranged in series: 1, 2, 3, 5, 8, 18. European physicist and astronomer J. Kepler (KLOC-0/571-1630) once said: "There are two treasure houses in geometry: one is Pythagorean theorem (we call it Pythagorean theorem); The other is the golden section. The front can be compared with gold mines, and the back can be compared with precious diamond mines. " Of course, the golden section is everywhere in real life, and there may be many mysteries of the golden section waiting for us to explore, discover and use. Most ancient Greek sculptures defined the proportion of human body as 7 heads long, and later determined it as 8 heads long. At the same time, the golden section in geometry is considered as the proportion of beauty applied to artistic creation. For example, the model work of the Greek sculpture "The Spear Holder" has created an image of a young soldier with strong physique and steady movements, which embodies the author's exploration and application of the "golden section" with the most harmonious proportion of the human body. Fifth, the golden section in art In addition to modeling, the principle of color matching in painting is also a wonderful principle of obtaining beauty through proportion. When two primary colors are combined, an intermediate color will be produced. For example, red and yellow produce orange, and this orange can take on different tones according to the different proportions of red and yellow. Of course, the two primary colors used to blend a halftone are not equal, and the usual blending equivalents are: yellow 3- red 5- cyan 8, that is, yellow 3+ red 5= orange 8, or yellow 3+ cyan 8= green 1 1, cyan 5+ red 8= purple 13. In fact, this harmonic quantity conforms to Fibonacci sequence, that is, to the golden section theorem, so the color it adjusts is more appropriate and natural, and looks very beautiful. As for the mixture of two intermediate colors, three primary colors, intermediate colors and black, primary colors and black, and primary colors and their complementary colors, although the proportion is complicated, it is not difficult to reach a satisfactory level as long as we find out their respective proportions in line with the golden section. The golden section can also be found in beautiful music and poetry. It is said that in the 6th century BC, Pythagoras, an ancient Greek mathematician and philosopher, passed by a blacksmith's shop one day and was attracted by the crisp and pleasant sound of striking the iron. He intuitively decided that the voice was "secret". He walked into the workshop and carefully measured the dimensions of the anvil and the hammer, and found that the ratio between them was close to 1: 0.6 18. After returning home, he brought a wooden stick and asked the students to carve a mark on it. The position of this mark should not only make the distance between the two ends of the stick unequal, but also make people look satisfied. After many experiments, a very consistent result is obtained, that is, the ratio of the whole section AB to the long section CB is equal to the ratio of the long section CB to the short section CA when the stick AB is divided by the point C. Pythagoras later discovered that putting a shorter line segment on a longer line segment would also produce the same proportion. This story shows that the earliest invention of the golden section seems to be related to sound. Sixth, the role of the golden section in other aspects "golden section ratio" is also widely used in daily life. For example, according to extensive investigation, the ratio of short sides to long sides of all pleasing rectangles, including TV screens, desks, books, doors and windows, is mostly 0.6 18. Even the length-width ratio of the matchbox and the national flag is 0.6 18. At the concert, the best position of the announcer on the stage is 0.618 of the stage width; In order to get the best timbre of erhu, its "weight" should be placed at 0.6 18 of the chord length. The most interesting thing is that the "golden number" of 0.6 18 can also be skillfully used in the consumption field to obtain the effect of "good quality and low price". According to experts, in the case that the same commodity has multiple varieties and multiple values, subtracting the high-grade price from the low-grade price and multiplying it by 0.6 18 is the optimal price for selecting the commodity. 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. The golden ratio of the five-pointed star shape is also very intriguing. The "stars" on the national flags of nearly 40 countries in the world (such as China, the United States, North Korea, Turkey and Cuba) are all five-pointed stars. The golden section law also provides a basis for the establishment of direct optimization methods. Optimization method is a method to solve the optimization problem, that is, how to make the output the highest, the quality the best and the consumption the least. The solutions to mathematical optimization problems can be roughly divided into two categories: indirect optimization methods and direct optimization methods. Indirect optimization method is to express the research object with mathematical equation, and then find the optimal solution by mathematical method. But in many cases, the indirect optimization method cannot be used because the object itself is not handled clearly, so people try to find the optimal solution through a lot of experiments. How to arrange experiments quickly and economically and get the optimal solution is a direct optimization method. If the experimental point is set at about 0.6 18 of the interval, the number of experiments will be greatly reduced. Experimental statistics show that for a factor problem, using "0.6 18 method" to do 16 experiments can achieve the effect of "dichotomy" to do 2500 experiments. 1953, the "0.6 18 method" proposed by Kiefer in the United States was widely used, especially in engineering design, with the best effect. In the field of furniture and interior decoration, Italian Tommasala furniture successfully applied the "golden section" to production, realizing the overall harmonious beauty. In Tommasala's exhibition hall, you can see the aspect ratio of the ground cabinet. The aspect ratio of the small door on the basement is golden section, and a pair of drawers are designed under the opposite door. The ratio of the length of the drawer to the height of the cupboard door and the length-width ratio of the whole wardrobe also conform to the golden section law. This big golden section sets a small golden section, which makes the whole piece of furniture look symmetrical, harmonious and elegant everywhere. The complete set of furniture composed of single furniture designed by golden section is in harmony with its overall coordination and ornamental performance.