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
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.
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, people first found that the rectangle with the aspect ratio of 1: 0.6 18 is very harmonious. The solemn Parthenon in Athens, built in the 5th century BC, has an aspect ratio of 1: 1.6. This proportion has also been strictly used in artistic creation, especially in classical painting during the Renaissance. Ruda? The composition of Finch's Vitruvian Man, dawit's Sabine Woman and Miller's Gleaning are all arranged in strict accordance with the golden section. In China's ancient painting theory, the approximate proportion of mountains, trees, horses and people in landscape painting is actually based on the golden section. The design of guqin "loses one or three points, gains one or three points, disguised", and the whole string has thirteen emblems. Put these together, two pools, three buckles, five strings, eight tones and thirteen emblems, which is the beautiful Fibonacci sequence of 1.438+08.
The golden section has good use value and aesthetic characteristics in our real life. If we can make full use of it, we will achieve good results. Please look at a few examples.
When writing, if I am in the pen length of 0.6 18, I can save energy and speed up writing.
In winter, when the classroom temperature with heating equipment is adjusted to 23℃, the ratio with the body temperature of 37℃ is just close to 0.6 18, and students feel comfortable.
When giving a speech, the best position for a teacher to stand should be 0.6 18 of the width of the platform. At this time, the teacher's expression can be fully displayed, and the sound effect can be brought into full play to achieve the best effect.
Venus sculpture is extremely beautiful because the proportion of each part is 0.6 18.
The tallest building in the world, the pavilion of Toronto TV Tower in Canada and the platform of Eiffel Tower in Paris, France, all fall at 0.6 18 of the height of the whole tower, which is even more powerful.
The first pyramid in Kisha, Egypt is146m high, and the ratio of its base length to 230m is 0.6 18, so it is magnificent. Magnificent.
In recent years, people use 0.6 18 for shopping. In the case of many varieties and high prices of the same commodity, the most expensive cost is too high to bear economically. Buy the lowest, but be afraid that the quality is too poor to meet the requirements. The following formula can help you get the most suitable price: (highest price-lowest price) × 0.6 18 ten lowest prices.
Except for tea, most of the good tea producing areas in China are located in the golden section of latitude, which is about 30 degrees north latitude. Especially the best "Qi Hong" in black tea is also produced in this latitude. A place related to 30 degrees north latitude. There are also many beautiful scenery, including grotesque rocks and peaks, Huangshan Mountain, Lushan Mountain, Jiuzhaigou and Yuan Xian, and three freshwater lakes in China engulf the Yangtze River.
The golden section "0.6 18", a wonderful value, is not only a common ratio in aesthetic modeling, but also a dietary parameter. It can make the proportion of cereals, vegetables, high-quality protein and alkaline foods in people's diet reach the golden ratio.
The digestive tract of human body is 9 meters long. Its 6 1.8% is about 5.5 meters, which is the length of the small intestine that undertakes the task of digestion and absorption. Humans, who are mainly cereals, are omnivores and are most suitable for digesting the complex diet of vegetarianism. When the heat supply of carbohydrates (mainly starch in cereals) accounts for 6 1.8% of the total heat, the human body's demand for heat energy can be best met. Therefore, experts suggest that people should eat a grain-based diet.
"0.6 18" has always had an indissoluble bond with military development, and it often happens by chance. Genghis Khan's Mongolian cavalry swept across Eurasia, which was amazing. Through research, it is found that the combat formation of Mongolian cavalry is very different from the traditional western phalanx. In his five-column formation, the heavy cavalry and light cavalry are 2∶ 3, the heavy cavalry in helmet vest is 2, and the fast and flexible light cavalry is 3, which just conforms to the golden section law. Let us once again see the magical function of the golden section law.
During the Spring and Autumn Period, Jin Ligong led a great army to attack Zheng and fought a decisive battle with the Chu army supporting Zheng in Yanling. Gong Li took the advice of the rebellion of the Chu army and used some China troops to attack Zuo Jun of the Chu army. The other part attacked the army of the Chu army, and assembled soldiers from the upper army, the lower army, the new army and the public to attack the right army of the Chu army. The attack point of the Jin army happened to be chosen in the golden location of the Chu army's battle formation, hitting the other side in the key, crushing the enemy in one fell swoop and quickly achieving the purpose of the war.
Through some scattered examples in the war, the shadow of "0.6 18" is faintly visible. If viewed in isolation, it seems to be accidental coincidence, but if too many accidents follow the same trajectory, it becomes a law, which is particularly worthy of in-depth study.
Knowing the ingenious and harmonious beauty of the golden ratio, we need to ponder and speculate on how to apply it in practice. There is a story about a stonemason who skillfully used the golden section to make a big-headed Buddha statue, which also reminds us that the visual experience of the golden ratio should be corrected by "visual error" For example, I still remember that the design of the first whole station project was divided into screens in strict accordance with 0.6 18, but it was not as perfect as expected. Considering the effect of color block contrast, it is necessary to appropriately increase the area of light-colored areas. For example, when participating in a competition, players tend to get high scores at 0.6 18, and have "golden points" for a long time (or a long distance) and so on.
Is all this accidental? No, around people, there are masterpieces of 0.6 18 everywhere: people always make desktops, doors and windows into rectangles with an aspect ratio of 0.6 18. Mathematically, 0.6 18 is even more amazing. 0.6 18, the proportion of beauty, beautiful color and beautiful melody are widely reflected in people's daily life and are closely related to people. 0.6 18, a wonderful number! It has created countless beautiful scenery and unified people's aesthetics. The joking 0.6 18 created many "coincidences". In the whole world, 0.6 18 shining like gold is everywhere! People have been creating the golden section intentionally or unintentionally. As long as you pay a little attention, you can find how close it is to our life! Mathematics is very close to us, and it is applied all the time!
We should first feel and appreciate the beauty in mathematics learning. Mathematical beauty is different from other beauty, it is unique and inherent. This kind of beauty, as Russell, a famous British philosopher and mathematical logician, said: "Mathematics, if viewed correctly, has not only truth, but also supreme beauty, just like the beauty of sculpture, which is a kind of cold and serious beauty. This beauty is not the weak aspect that attracts our nature. This beauty is not as gorgeous as painting or music. It can be pure and sublime, and it can reach the perfect state that only great art can express. " The teacher often tells us the beauty of mathematics in class. Through the study of mathematics, I gradually realized the true meaning of the beauty of mathematics. This feeling is strange, subtle, understandable but difficult to express. Mathematics is so fascinating to me ... as long as we are good at observing and thinking and combine what we have learned with life, we will feel the fun of mathematics. Mathematical knowledge is everywhere in life.