Golden ratio division, I want a complete proof and the calculated irrational number (example)

a line segment is divided 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 .618. Because the shape designed according to this ratio is very beautiful, it is called golden section, also known as Chinese-foreign ratio. This is a very interesting number, and we approximate it by .618. Through simple calculation, we can find that:

1/.618 = 1.618

(1-.618)/.618 = .618

This number not only plays a role in artistic fields such as painting, sculpture, music and architecture.

let's start with a series. The first few numbers are: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144 ... The name of this series is called Fibonacci series, and these numbers are called Fibonacci numbers. The characteristic is that each number is the sum of the two previous 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 ratio with the increase of serial numbers. That is, f (n)/f (n-1)-→ .618. Because Fibonacci numbers are all integers, and the quotient of division of two integers is rational, it is only gradually approaching the irrational number of golden ratio. But when we continue to calculate the larger Fibonacci number, we will find that the ratio of adjacent two numbers is really very close to the golden ratio.

a telling example is the five-pointed star/regular pentagon. Five-pointed stars are very beautiful. There are five on the national flag of our country, and many countries also use five-pointed stars in their national flags. Why? Because the length relationship between all the line segments that can be found in the five-pointed star is in line with the golden 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 2Sin18.

the golden section point is equal to about .618: 1

, which means that 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 point. There are two such points on the line segment.

Using the two golden points on the line segment, a regular pentagram and a regular pentagon can be made.

more than p>2 years ago, the third largest mathematician of the Athens School in ancient Greece, Odox Sass, first proposed the golden section. The so-called golden section refers to dividing a line segment with a length of L into two parts, so that the ratio of one part to the whole is equal to that of the other part. The simplest way to calculate the golden section is to calculate the approximate values of the ratio of the last two numbers of Fibonacci series 1, 1, 2, 3, 5, 8, 13, 21, ... 2/3,3/5,4/8,8/13,13/21, ...

The golden section was introduced to Europe by Arabs around the Renaissance, and was welcomed by Europeans. They called it "the golden method", and a mathematician in 17th century Europe 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 not as early as ancient Greece, it was independently created by ancient Chinese mathematicians and later introduced to India. After textual research. The European proportional algorithm originated in China and was introduced to Europe from Arabia through 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 the length and width of arts and crafts and daily necessities, and it is also widely used in real life. The ratio of some line segments in buildings adopts the golden section scientifically, and the announcer on the stage is not standing in the center of the stage, but on the side of the stage. The position standing at the golden section of the stage length is the most beautiful and the sound spreads 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, the .618 method 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 applications 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 strictly proportional, artistic and harmonious, and contains rich aesthetic value. Generally, it is .618 in application, just as pi is 3.14 in application.

the ratio of length to width of a Golden Rectangle is the golden section ratio, in other words, the long side of a rectangle is 1.618 times of the short side. The golden section ratio and the golden rectangle can bring aesthetic feeling 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 his <: Vitruvian > Conforms to the golden rectangle. <; Mona Lisa > The face also conforms to the golden rectangle, <; Last supper > The proportional layout was also applied.

Discovery History

Since the Pythagorean school in ancient Greece studied the drawing of regular pentagons and regular decagons in the 6th century BC, modern mathematicians concluded that the Pythagorean school had touched or even mastered the golden section at that time.

in the 4th century BC, eudoxus, an ancient Greek mathematician, was the first to study this problem systematically and established the proportional theory.

When Euclid wrote The Elements of Geometry around 3 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, and several Italian pacioli called the ratio of China to the end sacred, and wrote books on it. German astronomer Kepler called the golden section sacred.

it was not until the 19th century that the name "golden section" gradually became popular. The golden section number has many interesting properties, and it is also widely used by human beings. The most famous example is the golden section method or .618 method in optimization, which was first put forward by American mathematician Kiefer in 1953 and popularized in China in 197s.

|..........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 1.618 is .618, and 1.618:1 is the same as 1:.618.

The exact value is (√5-1)/2

The golden section number is irrational. The first 124 bits are:

.669887.45882.4586561723 91798576

.45987.45998985 52 1266338622 2353693179 3186766

7263544333 898659593 958295638 3226613199 282926788

67528766 892517116 96273222 143216269 5486262963

1361443814 97587122 34858879 5445474924 6185695364

864449241 443277134 494749565 846788598 7433944221

254487766 478915884 67499887 1 24765217 575179788

3416625624 94758969 742812 142762177 111777853

153171411 174666599 1466979873 17613566 7874871

13179523 68 9427521948 435356783 22878569 9782977834

7845878228 91197625 32696156 1725464 3382437764

861283831 268333724 2926752631 1653392473 1671112115

8818638513 3162384 5222165791 2866752946 549681131

7159934323 5973494985 9494762 132229811 72617596

116456299 98162955 5 28524793 5246217 2799747175

3427775927 7862561943 28275513 1218156285 512224893

9471234145 172237358 577278616 86883829 523459264

787817889 92199277 769389532 1968198615 143783149

974116926 886742962 267575652 317277752 3536139362

176738937 6455666 5922...

life application

interestingly, this number can be seen everywhere in nature and people's lives: people's navel is the golden section of the total length of the human body, and people's knees are the golden section from navel to heel. The ratio of width to length of most doors and windows is also .618 ...; On some stems, 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: .618 .... According to the research, this angle has the best effect on plant ventilation and lighting.

Architects are particularly fond of .168… 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 .168…. It is also found that the themes of some famous paintings, sculptures and photographs are mostly at .168… in the picture. Artists believe that placing the bridge of a stringed instrument at .168… of the string can make the sound softer and sweeter.

The number .168 ... is more concerned by mathematicians. Its appearance not only solves many mathematical problems (such as dividing the circle into ten parts and dividing the circle into five parts; Find the sine and cosine values of angles of 18 degrees and 36 degrees, etc.), and make the optimization method possible. Optimization method is a method to solve optimization problems. 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 to be added in each ton of steel is between 1 and 2 grams. In order to obtain the most appropriate addition amount, tests need to be carried out in the interval between 1 grams and 2 grams. Usually take the midpoint of the interval (that is, 15 grams) for the test. Then compare the test results with those at 1g and 2g respectively, select two points with higher intensity as new intervals, then take the midpoint of the new interval for experiment, compare the endpoints, and go on in turn until the most ideal results are obtained. This experimental method is called bisection method. However, this method is not the fastest experimental method. If the experimental point is .618 in the interval, the number of experiments will be greatly reduced. This method of taking .618 of the interval as the test point is a one-dimensional optimization method, also known as .618 method. Practice has proved that for the problem of one factor, 16 experiments with ".618 method" can complete the effect of 25 experiments with "bisection method". Therefore, the great painter Leonardo da Vinci called .618… the golden number.

.618 and war: Napoleon the Great was defeated by the golden section?

.618 is an extremely fascinating and mysterious number, and it also has a very beautiful name-the golden section law, which was discovered by Pythagoras, a famous ancient Greek philosopher and mathematician, more than 2,5 years ago. Throughout the ages, this number has been regarded as the golden rule of science and aesthetics by future generations. In the history of art, almost all outstanding works have verified this famous golden section law. Whether it is the Parthenon in ancient Greece or the Terracotta Warriors and Horses in ancient China, the ratio between the vertical line and the horizontal line is exactly 1: .618.

Perhaps, we have learned a lot about the performance of .618 in science and art, but have you ever heard that .618 is also inextricably linked to the fierce and cruel battlefield where gunfire, smoke and blood are everywhere, and it also shows its great and mysterious power in the military?

.618 and weapons and equipment

In the era of cold weapons, although people don't know the concept of golden ratio at all, the law of golden ratio has already been reflected everywhere when people make weapons such as swords, broadswords and spears, because weapons made according to this ratio will be more handy to use.

When the rifle for firing bullets was just manufactured, the ratio of the length of the handle to the body of the rifle was unscientific and unreasonable, which was not convenient for grasping and aiming. In 1918, a corporal of the American Expeditionary Force named alvin york reformed this rifle, and the ratio of the improved gun body to the gun handle was exactly .