What happened to the golden ratio in mathematics?

The golden section, also known as Huang Jinlv, means that there is a certain mathematical proportional relationship between the parts of things, that is, the whole is divided into two parts, and the ratio of the larger part to the smaller part is equal to the ratio of the whole to the larger part, and its ratio is 1: 0.6 18 or10/. The above ratio is the ratio that can most arouse people's aesthetic feeling, so it is called the golden section.

A detailed introduction to 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. This is a golden section method.

The ratio is (√5- 1)/2, and the approximate value of the first three digits is 0.6 18. Because the shape designed according to this ratio is very beautiful and soft, it is called golden section, also called Chinese-foreign ratio. This is a very interesting number. We approximate it with 0.6 18, and we can find it by simple calculation:1/0.618 =1.618 (1-0.618). Let's start with a series. The first two numbers are 1 and 1, and each number after it is the sum of the first two numbers. For example: 1, 1, 2, 3, 5, 8, 13, 2 1, 34, 55, 89, 144 ... This series is called Fibonacci series, and these numbers are called Fibonacci series.

Fibonacci sequence and golden section

What is the relationship between Fibonacci sequence and golden section? It is found that the ratio of two adjacent Fibonacci numbers is another golden section method with the increase of serial number.

And gradually tend to the golden ratio. 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.

Golden section triangle (regular triangle)

All triangles that appear after the diagonal of a regular pentagon is full are golden section triangles. The golden triangle has another particularity. All triangles can generate triangles similar to themselves with four congruent triangles, but the golden section triangle is the only triangle that can generate triangles similar to itself with five congruent triangles instead of four congruent 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 equal to about 0.6 18: 1, which means that a line segment is divided into two parts, so that the ratio of the long part to the long part of the original line segment is the golden section. There are two such points on the line segment. Using two golden points on the line segment, we can make a regular pentagram, a regular pentagon and so on.

Golden section and plastic arts

It has aesthetic value in plastic arts. In the design of the length and width of arts and crafts and daily necessities, this ratio can be used to make use of the golden ratio of the Forbidden City.

People's sense of beauty 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, and 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.

Golden section periphery

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. The golden ratio and the golden rectangle can bring aesthetic feeling to the picture, which is pleasant. It can be found in many artistic and natural works. The Parthenon in Athens, Greece is a good example. Leonardo da Vinci's Vitruvian Man fits the golden rectangle. Mona Lisa's face in Mona Lisa also conforms to the golden rectangle, and The Last Supper also applies this proportional layout.

Edit the history of this golden section.

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. _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ ABBA: b = (a+b): A is usually represented by the Greek letter Ф. 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 is an infinite cyclic decimal. The first 2000 digits are: 0.6180339887 49894820 458683435 6381177203 096538+07980538+03538. 5438+08939 1 1374 8475408807 538689 1752 1266338622 2353693 179 3 1 80060766 7263544333 8908659593 9582905638 3226665438+ 03 199 2829026788 0675208766 89250 17 1 16 9620703222 1 0432 16269 5486262963 136 14438 14 975870 12 20 3408058879 5445474924 6 185695364 86444924 1 0 4432077 134 4947049565 8467885098 743394422 1 2544877066 47809 15884 60749988 7 1 24007652 17 0575 1 79788 34 16625624 9407589069 70400028 12 1042762 177 1 1 17778053 15365 438+07 14 / kloc-0/0 1 1704666599 1466979873 176 1356006 70874807 10 13 17952368 942752 1 948 4353056 783 0022878569 9782977834 7845878228 9 1 10976250 0302696 156 1700250464 3382437764 86 1028383 / kloc-0/ 2683303724 292675263654 38+0 1653392473 167 1 1 12 1 15 88 186385 13 3 / kloc-0/62038400 5222 16579 1 2866752946 54906865 438+0 13 1 7 159934323 5973494985 0904094762 1322298 / kloc-0/0 1 726 1070596 1 164562990 98 16290555 20852 47903 52406020 17 2799747 1 75 3427775927 786256 1943 20827505 13 12 18 156285 5 122248093 947 1234 1 45 1 702237358 05772786 16 0086883829 5230459264 78780 17889 92 19902707 7690389532 1968 1986 15 1 43780365438 +049 974 1 106926 0886742962 2675756052 3 172777520 3536 139362 1076738937 6455606060 592 1 658946 675955 1900 40055 59089 5022953094 23 12482355 2 122 124 1 54 4400647034 0565734797 6639723949 4994658457 8873039623 0903750339 938562 1024 23690 285 138 6804 145779 95698 1 2244 5747 178034 173 1264532 204 1639723 2 134044449 4873023 154 1767689 375 2 1 03068737 880344 1700 9395440962 7955898678 7232095 124 2689355730 9704509595 68440 17555 1988 192 1 80 206405 2905 5 189349475 9260073485 2282 10 1088 1946445442 223 1889 13 1 9294689622 00230 / kloc-0/4437 7026992300 7803085266 5438+0 1807545 192 88770502 10 9684249362 7 135925 187 6077788466 5836 1 50238 9 13493333 1 223 1053392 32 136243 19 2637289 1 06 7050339928 2265263556 2090297986 4247275977 25655086 15 4875435748 2647 18 14 14 5 12700 0602 3890 1 62077 7322449943 5308899909 50 1680328 1 12 19432048 1964387675 8633 147985 7 19 1 1 3978 1 5397807476 1507722 1 17 5082694586 3932045652 0989698555 678 14 10696 8372884058 746 1 03378 1 0544439094 36 8358358 1 38 1 13 1 1689 9385557697 5484 149 144 534 1509 / kloc-0/29 54070050 19 4775486 163 0754226465 438+07 2939468036 73 1980586 1 8339 183285 99 13039607 20 / kloc-0/4455950 4497792 120 76 12478564 59 16 160837 05 94987860 06970 18940 9886400764 436 1709334 172709 19 14 33650 137

European part

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 (long part) to the whole is equal to the other part (short 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, 5/8, 8/655. 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.

Asian part

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.

Edit this life application

Peripheral application

Interestingly, this kind of figure can be seen everywhere in nature and people's lives: the navel of a person is the beauty of the golden section of the whole human body.

Golden section, the human knee is the golden section from navel to 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.

Application of architectural art

The golden section is considered as the most ideal proportion in architecture and art, and architects have a special preference for the number of 0.6 18 … 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 … In addition, the golden section is used in the design of ancient Greek temples. 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.

Mathematical application

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.