Knowledge about 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. 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.

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.

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, 4/8, 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 1.6 18 in application, just as pi is 3. 14 in application.

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.

|..........a...........|

+ - + - + -

| | | .

| | | .

| B | A | b

| | | .

| | | .

| | | .

+ - + - + -

|......b......|..a-b...|

This value is usually expressed in Greek letters.

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 the root number 5+ 1/2.

The golden section number is irrational, and the first 1024 bits are:

1.6 180339887 4989484820 4586834365 638 1 177203 09 17980576

2862 135448 6227052604 628 1890244 9707207204 18939 1 1374

8475408807 538689 1752 1266338622 2353693 179 3 180060766

7263544333 8908659593 9582905638 32266 13 199 2829026788

0675208766 89250 17 1 16 9620703222 10432 16269 5486262963

136 14438 14 975870 1220 3408058879 5445474924 6 185695364

86444924 10 4432077 134 4947049565 8467885098 743394422 1

2544877066 47809 15884 607499887 1 24007652 17 0575 179788

34 16625624 9407589069 70400028 12 1042762 177 1 1 17778053

153 17 14 10 1 1704666599 1466979873 176 1356006 70874807 10

13 17952368 942752 1948 4353056783 0022878569 9782977834

7845878228 9 1 10976250 0302696 156 1700250464 3382437764

86 1028383 1 2683303724 292675263 1 1653392473 167 1 1 12 1 15

88 186385 13 3 162038400 5222 16579 1 2866752946 549068 1 13 1

7 159934323 5973494985 0904094762 1322298 10 1 726 1070596

1 164562990 98 16290555 2085247903 52406020 17 2799747 175

3427775927 786256 1943 20827505 13 12 18 156285 5 122248093

947 1234 145 1702237358 05772786 16 0086883829 5230459264

78780 17889 92 19902707 7690389532 1968 1986 15 1437803 149

974 1 106926 0886742962 2675756052 3 172777520 3536 139362

1076738937 6455606060 5922 ...