What is the "golden ratio"

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.

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. The ratio is [5 (1/2)- 1]/2, and the approximation 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 value is not only reflected in the fields of painting, sculpture, music and architecture, 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.

This Fibonacci number not only starts from 1, 1, 2, 3, 5 ... Like this, if you choose two integers at will and sort by Fibonacci number, the ratio between the two numbers will gradually approach 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 ratio of 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, 5/8, 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.

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 top 50 digits are:

0.6 180339887 4989484820 4586834365 638 1 177203 09 17980576

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.

The relationship between golden section and human beings

The golden section is closely related to people. The latitude range of the earth's surface is 0-90 degrees. If divided into the golden section, 34.38-55.62 is the golden zone of the earth. No matter from the aspects of average temperature, annual sunshine hours, annual precipitation and relative humidity, it is the most suitable area for human life. Coincidentally, this region covers almost all the developed countries in the world.

Medicine is inextricably linked with 0.6 18, which can explain why people feel most comfortable in the environment of 22 to 24 degrees Celsius. Because human body temperature is 37℃, the product of 0.6 18 is 22.8℃, and the metabolism, circadian rhythm and physiological function of human body are in the best state at this temperature. Scientists also found that when the external environment temperature is 0.6 18 times of human body temperature, people will feel most comfortable. Modern medical research also shows that 0.6 18 is closely related to the way of keeping in good health, and the dynamic-static relationship is 0.6 18, which is the best way of keeping in good health. Medical analysis also found that people who eat 60% to 70% full will hardly have stomach problems.

In the elegant art hall, there are naturally golden footprints. The painters found that the ratio of leg length to height was 0.6 18: 1, and the figure they drew was the most beautiful. However, nowadays, the average length of women below the waist only accounts for 0.58 of their height. Therefore, the statue of Venus in ancient Greece and the image of Apollo, the sun god, deliberately put their legs out to make the ratio with their height reach 0.666. Thereby creating artistic beauty. No wonder many girls are willing to wear high heels, while ballerinas stand on tiptoe from time to time when dancing. Musicians found that when the chord ratio of "forward" is 0.618:1,the timbre played is the most harmonious and pleasant.

The leaves of plants are full of vitality, bringing a beautiful green world to nature. Although the shape of leaves varies from species to species, its arrangement order on the stem (called leaf order) is very regular. The growth of petals and branches on the trunk of some plants also conforms to this law. You look down from the top of the plant stem, and after careful observation, you find that the included angle between the upper and lower adjacent leaves is about 137.5. If only one leaf is drawn on each layer, the angle difference between two adjacent leaves on the first layer and the second layer is about 137.5, and the next two to three layers, three to four layers and four to five layers all form this angle. Botanists' calculations show that this angle is beneficial to the lighting of leaves. What is the "password" hidden in the angle of 137.5 between leaves? We know that a week is 360,360-137.5 = 222.5,137.5: 222.5 ≈ 0.618. Look, this is the "password"! The exquisite and magical arrangement of leaves actually hides 0.6 18.