There is a very useful number in economic life and scientific research-0.618, which determines an optimization method. Using it, people save a lot of time, money and material resources. When people explore its origin, they find that it is actually the product of pure mathematical thinking! How can the product of pure mathematical thinking be so realistic? This is a beautiful riddle contained in this number. Eudoxus, eudoxus's "Comparison between China and Foreign Countries", was a Greek mathematician in the 4th century BC. He studied many proportional problems and founded the proportional theory. In the process of studying proportion, a question was raised: can a line segment be divided into two unequal parts, so that the longer part is the median proportion of the original line segment and the shorter part? He found through research that a known line segment can be divided into two segments, so that the ratio of long line segment to short line segment is equal to the ratio of complete line segment to long line segment, that is, long line segment is the median of the ratio of complete line segment to short line segment. If it is known that the line segment is ab, and point C divides ab into ac, bc, AC > BC, and AC2 = ab CB, then the specific division method of point C is: connecting ad, drawing an arc with D as the center, bd as the radius, ad intersecting with E, drawing an arc with A as the center, ae as the radius, and ab intersecting with C, then point C is the required division point. Therefore, eudoxus called this comparison "Chinese-foreign comparison". In the history of mathematics, eudoxus first put forward the comparison between China and foreign countries, but the Greeks found it earlier. The mysterious Pythagorean school once took the five-pointed star as its symbol, and the drawing of the five-pointed star included the comparison between China and foreign countries. The Parthenon in Athens is a masterpiece of ancient Greece. The aspect ratio of this temple, which was built in the 5th century BC, coincides with that of China and foreign countries. The comparison between China and foreign countries was later called the "golden section" by the world. Although eudoxus was the first person to study the golden section systematically, when and why did it come into being? The origin of the golden section People think that the drawing of the golden section is related to the drawing of regular pentagons, regular decagons and pentagons, especially the drawing of pentagons. The five-pointed star shape is a very intriguing pattern, and the "stars" on the national flags of many countries in the world are all drawn in pentagons. At present, nearly 40 countries (such as China, the United States, North Korea, Turkey, Cuba, etc. There is a five-pointed star on the national flag. Why is it the Pentagon and not the other corners? Maybe it's an old habit. The origin of the pentagram is very early. The earliest pentagram pattern found now is a clay tablet made in Maruk (present-day Iraq) in the lower reaches of the Euphrates River around 3200 BC. The Pythagorean school in ancient Greece used the five-pointed star as their badge or symbol, which was called "health". It can be considered that Pythagoras is familiar with the practice of the five-pointed star, which shows that he has mastered the golden section. It is generally believed that the golden section was discovered by Pythagoras in the 6th century BC. The earliest record of systematically discussing the golden section is Euclid's Elements of Geometry. The fourth volume of this book tells the problem of using the golden section to make pentagons and decagons. In the second volume, the section 1 1 describes the calculation method of the golden section in detail, in which it is written: "Cut the line segment ab with point H according to the middle ratio, so that ab: ah = ah: HB". It is called "mid-end ratio" in the Elements of Geometry. Until the Renaissance, people rediscovered ancient Greek mathematics and found that this proportion existed widely in the natural structure of many figures, so they highly praised the wonderful nature and use of the middle ratio. Italian mathematician pacioli called the ratio of China to the finish line "sacred ratio"; Kepler, a German astronomer, called the ratio of China to the terminal "proportional division", and thought that Pythagorean theorem was "like gold" and the ratio of China to the terminal was "gem". The first person to use the name "golden section" in his works was the German mathematician M. Ohm, who was the younger brother of G.S. Ohm who discovered Ohm's law. In his book Pure Elementary Mathematics (second edition, 1835), he used the German word "der goldene schnitt" to express the comparison between China and the United States. After that, this title gradually became popular. The golden section and the "rabbit problem" Fibonacci was a famous mathematician in Europe in the13rd 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. The Fibonacci series is called "Fibonacci series", and its structure starts from the third term, each term is the sum of the first two terms, that is, fn=fn- 1+fn-2(n≥3), and fn stands for the nth term. If the golden section number is represented by G, these ratios are getting closer and closer to G, in fact, G is the limit. This interesting property is very strange: problems from two completely different mathematical fields have the same result. The magical connection between the two makes the golden section more mysterious and charming. The Enlightenment of the Golden Section With the development of society, people find that the golden section is widely used in nature and society. For example, there are two optimization methods related to the golden section. One is the "0.6 18 method" pointed out at the beginning of this paper, which is an optimization method proposed by American mathematician Kiefer in 1953. Since 1970, it has been popularized in China and achieved good economic benefits. In modern optimization theory, it enables us to find suitable technological conditions and reasonable formula with less experiments. Although G is an irrational number and 0. 168 is an approximate value, it is accurate enough in practice. The second is the fractional method, which takes the approximate value of G, but it is not 0.6 18, but the asymptotic fraction of G's continued fraction expansion, that is, using the fraction of a Fibonacci sequence. The application of golden section also shows a law of mathematical development. It shows that it is very important to study and develop mathematical theory. The development of pure theory may not have a direct effect on practice, but the natural laws it reveals will certainly guide people's social practice. Therefore, on the one hand, we should find mathematical methods to solve problems, and on the other hand, we should open up application fields for pure mathematical theory. In addition, the phenomenon of attaching importance to the mystery of "golden section" also exists. For example, the relationship between golden section and "beauty", some people say that the rectangle with two sides obtained by golden section (that is, the rectangle with the ratio of two sides =g) is the most beautiful. There is no sufficient basis for this, and experts also denied this conclusion when doing social surveys. So the conclusion that the golden rectangle is the most beautiful is uncertain. Many conjectures derived from this are naturally unreliable. For another example, the length ratio of various parts of the human body (such as from the top of the head to the navel, from the navel to the heel) is the most beautiful in the golden section ratio; The proportion of all parts of the building is the most beautiful if it conforms to the golden ratio, and so on. Most of these statements are far-fetched. It is also a misunderstanding that the chord length ratio of musical instruments is equal to the golden ratio, and the sound played is harmonious and pleasant. In fact, the chord length of harmony music must be simple, and the golden ratio is an irrational number! The so-called golden section is such a division: an interior point divides a line segment into short parts and long parts, so that their lengths satisfy such a relationship: short: long = long: complete. In this proportional formula, "short" and "long" refer to the length of short line segment and long line segment divided by the inner point respectively, while "full" refers to the length of the whole line segment, that is, full = short+long. It is said that the ancient Greek mathematician Odysseus first studied the golden section. This is why it is called the golden section, because it has many wonderful properties and applications. For example, the shape of rectangular objects (such as windows and books) whose aspect ratio meets the golden ratio will make people feel beautiful and pleasing to the eye. In the Middle Ages, the golden section, as a symbol of beauty, almost penetrated into all parts of architecture and art. For example, it is said that the length of the upper body and lower body of human sculpture is the most symmetrical and beautiful if it conforms to the golden ratio. References:

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