What is the golden section? Thanks.

Divide a line segment into two parts so that the ratio of one part to the full length is equal to the ratio of the other part to this part. The ratio is [5^(1/2)-1]/2, and the approximate value of the first three digits is 0.618. Because the shape designed according to this ratio is very beautiful, it is called the golden section, also known as the ratio between Chinese and foreign parts. This is a very interesting number. We use 0.618 to approximate it. Through simple calculations, we can find:

1/0.618=1.618

(1-0.618)/0.618=0.618

The role of this value is not only reflected in art fields such as painting, sculpture, music, architecture, etc., but also plays an important role in management, engineering design, etc.

Let us first start with a sequence. Its first few numbers are: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144... this The name of the sequence is "Fibonacci Sequence" and these numbers are called "Fibonacci Sequence". The characteristic is that except for the first two numbers (which have a value of 1), each number is the sum of the two preceding numbers.

What is the relationship between the Fibonacci sequence and the golden section? Research has found that the ratio of two adjacent Fibonacci numbers gradually approaches the golden ratio as the sequence number increases. That is, f(n)/f(n-1)-→0.618…. Since Fibonacci numbers are all integers, the quotient of dividing two integers is a rational number, so it only gradually approaches the irrational number of the golden ratio. But when we continue to calculate the later larger Fibonacci numbers, we will find that the ratio of two adjacent numbers is indeed very close to the golden ratio.

Not only is this the "Fibonacci number" starting from 1,1,2,3,5...., just pick two integers, and then follow the rules of the Fibonacci number As the arrangement continues, the ratio between the two numbers will gradually approach the golden ratio.

A very illustrative example is the five-pointed star/regular pentagon. Five-pointed stars are very beautiful. There are five in our country’s national flag. Many other countries also use five-pointed stars in their national flags. Why is this? Because the length relationship between all line segments that can be found in the five-pointed star is consistent with the golden ratio. All triangles that appear after the diagonals of a regular pentagon are connected are golden section triangles.

The golden section triangle also has a special feature. All triangles can use four congruent triangles to generate a triangle similar to itself, but the golden section triangle is the only one that can use 5 triangles. Instead of 4 triangles that are congruent with itself, we generate a triangle that is similar to itself.

Since the top angle of the five-pointed star is 36 degrees, it can also be concluded that the value of the golden section is 2Sin18.

The golden section is approximately equal to 0.618:1

It refers to dividing a line segment into two parts, so that the ratio of the length of the original line segment to the longer part is the golden section. point. There are two such points on the line segment.

Using the two golden points on the line segment, you can make a regular five-pointed star or a regular pentagon.

More than 2,000 years ago, Eudoxus, the third greatest mathematician of the School of Athens in ancient Greece, first proposed the golden section. The so-called golden section refers to dividing a line segment of length L into two parts, so that the ratio of one part to the whole is equal to the ratio of the other part to that part. The simplest way to calculate the golden section is to calculate the ratio of the last two numbers in the Fibonacci sequence 1, 1, 2, 3, 5, 8, 13, 21,... 2/3, 3/5, 5/8 , 8/13, 13/21,... Approximate values.

The golden section was introduced to Europe through the Arabs before and after the Renaissance, and was welcomed by Europeans. They called it the "golden method". A European mathematician in the 17th century even called it "The most valuable algorithm among all algorithms". This algorithm is called the "three-rate method" or the "three-number rule" in India, which is what we often call the proportional method now.

In fact, our country also has records about the "golden section". Although it is not as early as ancient Greece, it was independently created by ancient Chinese mathematicians and was later introduced to India. After research. The European proportional algorithm originated from my country and was introduced to Europe from Arabia through India, rather than being introduced directly from ancient Greece.

Because it has aesthetic value in plastic arts, in the length and width design of arts and crafts and daily necessities, the use of this ratio can arouse people's sense of beauty. It is also widely used in real life, such as buildings. The golden section is scientifically used for the ratio of some line segments in the stage. The announcer on the stage does not stand in the center of the stage, but on one side of the stage. The position of standing at the golden section point of the length of the stage is the most beautiful and the sound is The best spread. Even in the plant world, there are places where 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 rules of the golden section. In many scientific experiments, a 0.618 method is commonly used to select a plan, that is, the optimization method, which allows us to rationally arrange a smaller number of tests to find reasonable Western and suitable process conditions. It is precisely because it has extensive and important applications in architecture, literature and art, industrial and agricultural production, and scientific experiments that people call it the "golden section" preciously.

The Golden Section is a mathematical proportional relationship.

The golden section has strict proportion, artistry and harmony, and contains rich aesthetic value. When applied, it is generally taken to be 0.618, just like the pi ratio is taken to be 3.14 when applied.

The ratio of the length to width of the Golden Rectangle is the golden ratio. In other words, the long side of the rectangle is 1.618 times the short side. The golden ratio and golden rectangle can bring beauty and pleasure to the picture. It can be found in many works of art as well as in nature. The Temple of Passa in Athens, Greece, is a good example, and Leonardo da Vinci's "Vitruvian Man" conforms to the golden rectangle. The face of "Mona Lisa" also conforms to the golden rectangle, and "The Last Supper" also uses this proportional layout.

Edit this paragraph to discover history

Since the Pythagoreans in ancient Greece in the 6th century BC studied the construction of regular pentagons and regular decagons, modern mathematics Experts concluded that the Pythagoreans had already touched or even mastered the golden section at that time.

In the 4th century BC, the ancient Greek mathematician Eudoxus was the first to systematically study this problem and establish the theory of proportion.

When Euclid wrote "Elements" around 300 BC, he absorbed the research results of Eudoxus and further systematically discussed the golden section, becoming the earliest treatise on the golden section.

After the Middle Ages, the golden section was cloaked in mystery. Several Italian painters, Pacioli, called the middle ratio a sacred ratio and wrote books specifically about it. German astronomer Kepler called the golden section the divine section.

It was not until the 19th century that the name golden section gradually became popular. The golden section has many interesting properties, and its practical applications are also widespread. The most famous example is the golden section method or 0.618 method in optimization, which was first proposed by the American mathematician Kiefer in 1953 and popularized in China in the 1970s.

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a b

a:b=(a+b):a

Usually This value is represented by the Greek letter Ф.

The wonderful thing about the golden section is that its ratio is the same as its reciprocal. For example: the reciprocal of 1.618 is 0.618, and 1.618:1 is the same as 1:0.618.