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.
Second, the historical discovery history of the golden section:
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.
How to discover legends:
In the 6th century BC, PInthagoras, an ancient Greek mathematician and philosopher, passed by a blacksmith's shop one day and was attracted by the crisp and pleasant sound of striking the iron. He stopped to listen carefully and intuitively concluded that the voice was "secret"! He walked into the workshop and carefully measured the dimensions of the anvil and the hammer, and found that the ratio between them was close to 1:o.6 18. After returning home, he took a wooden stick and asked his students to carve a mark on it, which not only made the distance between the two ends of the stick unequal, but also made people look satisfied. After many experiments, a very consistent result is obtained, that is, the ratio of the whole section AB to the long section cB is equal to the ratio of the long section CB to the short section CA when the stick AB is divided by the point C. Later, Pythagoras found that putting a shorter line segment on a longer line segment also produces the same proportion: so infinity (see Figure 5-5- 1).
After calculation, the ratio of long segment (assuming a) to short segment (assuming b) is 1:o.6 18, and its ratio is L 6 18. The formula can be used.
a :b=(a+b):a
Expression, and there is a mathematical relationship. At this time, the square of the long segment length is exactly equal to the product of the whole stick and the short segment length, that is, A = (A+B) B) B.
This magical proportional relationship was later praised by Plato, a famous ancient Greek philosopher and aesthetician, as the "golden section law", which was called "Huang Jinlv" and "golden ratio" for short. It is appropriate to use the word "gold" here to describe the importance of this law. What's even more amazing is that 1 divided by 1.6 18 is just equal to O.66548. 1 divided by 1.5 18 is not equal to O, 5 1 8 ...
Equal to 0.618 (accurate to 0.00 1). Therefore, it is correct to say that the golden ratio is 1.6 18 (long segment: short segment) or 0.6 18 (short segment: long segment). Mathematicians also found that 2:3 or 3:5 or 5:8 is an approximation of the golden ratio, and the sum of numerator and denominator is a new denominator. 13/21,21/34.34/55, 55/88 ... The larger the number, the closer the ratio of its numerator to denominator is to O.6 18, which is mathematically called "Fibonacci series". According to the law of this series, the golden ratio of "area" can be calculated from the golden ratio of "line segment". According to this series, le corbusier, a modern architect, invented the "golden ruler" (building standard ruler, slightly increased by 1.6 times). The medieval mathematician Kepler called the golden section law and Pythagorean theorem "two treasures in geometry". /kloc-Pachouri, a Venetian mathematician in the 10th and 9th centuries, praised the golden section law as a "godsend proportion".
Third, what are the records in the history of the discovery of the golden section law? The golden section law was discovered a long time ago.
Pythagoras, an ancient Greek mathematician in the 6th century BC, made a thorough and detailed study on how to choose a point C on the line segment S, and finally discovered the world-famous golden section law. But where should point C be located? To solve this problem, we can first set the length of the line segment to 1, the length from point C to point X, and the length from point C to point S to (1-x), so that1:x-x' (1-x) 751can be solved. C = (y-y) minus the negative value, we get J5. 12-2=0.6 18.
"0.6 18" is the only point that satisfies the golden section law and is called the golden section point.
Fourth, the example of the golden section divides a line segment into two parts, so that the proportion of one part to the total length is equal to the proportion 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 approximate it with 0.6 18, and we can find it by simple calculation:1/0.618 =1.618 (1-0.618). 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 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, 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 Fibonacci sequence 1, 1, 2,3,5,8,13,21. The ratio of the last two digits is 2/3, 3/5, 4/8, 8/ 13, 13/2 1.
Probably. Around the Renaissance, the golden section was introduced to Europe by * * * people 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.
The European proportional algorithm originated in China and was introduced into Europe from 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.
The Pythagorean school in ancient Greece studied the drawing methods of regular pentagons and regular decagons in the 6th century BC, so modern mathematicians concluded that Pythagoras school had contacted 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.
|。
. . Answer.
..| + - + - + - | | | .| | | .| 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. The former 1024 bits are: 0.6180339884820 458683435 638177203091798057286041899.
5. The historical anecdote related to the golden section, also known as Huang Jinlv, refers to the existence of 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 the ratio is 1: 0.6 18 or10.618. 0.6 18 is recognized as the most beautiful scale figure. The above ratio is the ratio that can most arouse people's aesthetic feeling, so it is called the golden section.
Story: Most people think that the origin of the golden ratio comes from Pythagoras. It is said that in ancient Greece, Pythagoras was walking in the street one day. Before passing the blacksmith's shop, he heard the sound of striking the iron, so he stopped to listen. He found that the blacksmith had a regular rhythm when he was striking iron, and the proportion of this sound was expressed mathematically by Pythagoras. It has been applied in many fields, and later many people devoted themselves to it. Kepler called it "sacred division", and some people called it "golden section". Pythagoras' law appeared only 1000 years after the completion of the pyramid, which shows that it existed very early. I just don't know the answer.
Sixth, although I didn't write the discovery history of the golden section myself, I hope this can be useful to you!
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.
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.
|。 . . . Answer. . ..|
+ - + - + -
| | | .
| | | .
| 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 (root number 5- 1)/2.
In fact, the so-called golden section means that the above section satisfies b/(a-b)=a/b, that is, a 2-ab-b 2 = 0, and b/a= (root number 5- 1)/2 can be calculated.
Do the golden section of known line segments.
More than 2000 years ago, the ancient Greek Plato scholar Odoksos first drew the golden section of known line segments with straightedge and straightedge. His approach is as follows:
1. Let the known line segment be AB, the crossing point B be BC⊥AB, and BC = AB/2;
2. even AC;
3. Make an arc with C as the center, CB as the radius, and AC and D intersect;
4. Make an arc with A as the center and AD as the radius, and intersect AB at P, then point P is the golden section of AB.
Proof: From Pythagorean Theorem, we know that AC= root number (AB 2+AC 2) = root number 5/2*AB.
AD=AC-DC= radical number 5/2*AB-AB/2= (radical number 5- 1)/2*AB.
AP=AD= (radical number 5- 1)/2*AB
AP:AB= (radical number 5- 1)/2
Point P is the golden section of AB.
7. Interesting story about the golden section On some plants, the angle between two adjacent petioles is137 28', which is exactly the angle between the 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. The leaves of plants, with different shapes and vitality, bring 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, four to five layers ... are all at this angle. Botanists have calculated that this angle is the best angle for lighting and ventilation of leaves. How delicate the arrangement of leaves is! 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 ingenious and magical arrangement of leaves actually hides the ratio of 0.6 18.
Medicine is inextricably linked with 0.6 18, which can explain why people feel most comfortable in the environment of 22-24℃. Because the product of the body temperature of 37℃ and 0.6 18 is 22.8℃, and the metabolism, circadian rhythm and physiological function of the 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.
Human body temperature is 37 degrees, and room temperature is 23 degrees, which is the most comfortable temperature for human beings, while 23÷37≈0.622 is very close to 0.6 18.
The ideal weight calculation is very close to the height *( 1-0.6 18).
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 is137 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 are particularly fond of 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.
Eight, the example of the golden section The golden section divides a line segment into two parts, so that the proportion of one part to the total length is equal to the proportion 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 approximate it with 0.6 18, and we can find it by simple calculation:1/0.618 =1.618 (1-0.618). 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 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, 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 Fibonacci sequence 1, 1, 2,3,5,8,13,21. The ratio of the last two digits is 2/3, 3/5, 4/8, 8/ 13, 13/2 1.
Probably. Around the Renaissance, the golden section was introduced to Europe by * * * people 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.
The European proportional algorithm originated in China and was introduced into Europe from 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.
The Pythagorean school in ancient Greece studied the drawing methods of regular pentagons and regular decagons in the 6th century BC, so modern mathematicians concluded that Pythagoras school had contacted 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.
|。
. . Answer.
..| + - + - + - | | | .| | | .| 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. The former 1024 bits are: 0.6180339884820 458683435 638177203091798057286041899.
Nine, an example of the golden section A very 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 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.