Plant leaves are diverse and full of vitality, bringing a beautiful green world to nature. Although the shape of leaves varies from species to species, their arrangement on the stem (called phyllotaxis) is very regular.

You look down from the top of the plant stem. After careful observation, you find that the angle between the two adjacent leaves in the upper and lower layers is approximately 137.5°. If only one leaf is drawn for each layer to represent it, the angle difference between the two adjacent leaves on the first and second layers is about 137.5°, and then the second to third layers, the third to fourth layers, the fourth to fifth layers... two The angles between the leaves are all at this angle. Botanists have calculated that this angle is optimal for lighting and ventilation of the leaves. How delicately arranged the leaves are!

What "password" is hidden in the 137.5° angle between the leaves? We know that a week is 360°,

360°-137.5°=222.5°

137.5° :222.5° 222≈0.618.

Look, this is the "password"! There is 0.618 hidden in the exquisite and magical arrangement of the leaves.

The growth of petals and branches on the trunk of some plants also conforms to this rule.

In the mid-19th century, the German psychologist Fechner once conducted an ingenious experiment. He held a "Rectangle Exhibition" where he displayed various rectangles he had carefully crafted and asked visitors to vote for the rectangle they thought was the most beautiful. As a result, the following four rectangles were selected:

The ratio of rectangular length × width to length

1 8×5 5:8=0.625

2 13×8 8∶13=0.615

3 21×13 13∶21=0.619

4 34×21 21:34=0.618

Interestingly, the obtained The ratios of the length and width of the four rectangles are close to 0.618.

What surprises people today is that the human body itself is closely related to 0.618. Leonardo da Vinci, an Italian painter who was very knowledgeable about human anatomy, discovered that the human belly button is located at 0.618 of the body length. Scientists also found that people will feel most comfortable when the external ambient temperature is 0.618 times the human body temperature.

Are these all accidental coincidences? No! It is one of the laws reflected in the objective world. Mathematicians discovered:

If a line segment AB is divided into two parts AC and CB using point C, if

AB∶AC=AC∶CE,

That is__

√5 -1

When AB=1, AC=-------≈0.618.

2

Since the 0.618 obtained in this way has many extremely precious properties, people cherish it as the golden number, call point C the golden section point, and call this Divided into the golden section.

The golden number 0.618 is now more and more recognized and used by people.

The ancient Greek Parthea Temple has become a world-famous perfect building because the ratio of height to width is 0.618. Architects have found that if they design palaces in this proportion, they will be more majestic and magnificent; if they design villas, they will be more comfortable and beautiful. Even a door or window designed as a golden rectangle will look more coordinated and pleasing to the eye.

In the elegant art palace, the footprints of the golden number have naturally been left. Painters have found that designing the ratio of leg length to height according to 0.618:1 will produce the most beautiful human figure. However, for today's women, the length below the waist only accounts for 0.58 of the height on average. Therefore, the ancient Greek Venus statues and The images of the sun god Apollo all create artistic beauty by deliberately extending their legs so that the ratio to their height is 0.618. No wonder many girls are willing to wear high heels, and ballet dancers tiptoe from time to time when dancing.

Musicians have found that when playing the erhu, the ratio of the "Qianjin" strings is 0.618:1, the tone produced is the most harmonious and pleasant.

......

As long as you pay attention, you can find the footprints of the golden number, the "messenger" of beauty, everywhere. The 0.618 method used in scientific experiments and industrial and agricultural production optimization methods can also bring us huge economic benefits! The golden number 0.618 is really a gorgeous treasure that benefits mankind!

The ancient Greek city of Athens There is a shrine made of marble, and the goddess in the center of the temple hall is carved of ivory and gold. The goddess's body is graceful and graceful, attractive. Expert research found that the ratio of the distance from the heel to the navel of her body to the entire height is exactly 0.618. Not only did the goddess Athena have such a beautiful figure, but so did the body proportions of many other Greek goddesses. The ratio of 0.618 can be found in the familiar statues of Milos "Venus", the image of the sun god Apollo, the Sea Maiden-Oman and other ancient statues.

Around 1483, in an unfinished oil painting painted by Leonardo da Vinci, the black line surrounding the body of Saint Jerome was a rectangle of golden section. At that time, Leonardo seemed to intend to use this The ratio of the golden section. "Review" is an oil painting by French Impressionist painter Schelet, and its bar structure ratio is exactly 0.618. The British painter Feilakman's famous book "Greek Myths and Legends" contains 96 pictures of beauties. The beauties in every painting are extremely charming and graceful. If you carefully measure their proportions, they are similar to Athena.

China's oldest guqin is full of the magic of the golden section. There are two ponds on the back of the instrument, with a dragon on the left and a phoenix on the right. There are three key points that control the pronunciation of the strings: Zhen, Fuzhang, and Fengcun. The Qin has five strings, the sound has octaves, and the Qin section is the emblem. "One third of the length of the piano loses one, and another third gains one, and then increases and decreases in turn." There are thirteen emblems on the whole string. Arrange these together, two pools, three buttons, five strings, eight tones, and thirteen emblems. What a wonderful arrangement, it is the Fibonacci odd number, and the ratio of two adjacent Fibonacci odd numbers is getting closer and closer to the golden ratio. Is it intentional or a coincidence? It seems that the ancient Chinese people’s understanding and application of the golden ratio are closely related to There are indeed many similarities in the West.

Architecture: As early as the fifth century BC, Greek architects knew that the ratio of 0.618 was a harmonious, balanced structure. The pyramids of civilized China and Egypt are shaped like square pyramids and come in various sizes. But the ratio of the side length to the height of the base of these pyramids is close to 0.618. In ancient times, the height and width of some temples were also built according to the ratio of golden numbers. They believed that such a rectangular shape was more beautiful. The golden rule is a rule that architectural art must follow. In terms of architectural modeling, people can build pavilions or design platforms at the golden section of the tower to make the straight and monotonous tower body colorful.

We have already had a preliminary appreciation of the great buildings of ancient times, now let us see the mystery of modern golden numbers.

Pythagoras has a famous saying: All beautiful things have the same characteristics, that is, the harmony between parts and parts and the whole.

We have done a survey and found that there are two main types of TV programs on the market, one with a width/length of 3:4, and the other with a width/length of 9:16. Both ratios are very close to 0.618, which is because the golden rectangle is the most beautiful.

Golden numbers are also used in chemical pharmaceuticals. For example, if drugs are currently being synthesized, it is unknown which temperature between 0 and 100°C will yield the highest synthesis rate and best efficacy. Obviously, it is impractical to test each temperature one by one. It would be much simpler if you use golden numbers. In mathematics, since he is the ancestor of the golden number, he has a lot of knowledge in this area. In fact, there is also a lot of knowledge about golden numbers in biology.

Our research has found that there are some laws about the golden number of the human body.

The law of the golden section was discovered by the ancient Greek mathematician Pythagoras in the sixth century BC. Later, the ancient Greek esthetician Plato called this the golden section. This is actually a numerical proportional relationship, that is, dividing a line into two parts. At this time, the ratio of the long segment to the short segment is exactly equal to the ratio of the entire line to the long segment. The numerical ratio is 1.618:1 or 1:0.618, which is also This means that the square of the long segment is equal to the product of the total length and the short segment. 0.618, with strict proportion, artistry and harmony, it contains rich aesthetic value. Why do people instinctively feel the beauty of such proportions? In fact, this is closely related to human evolution and normal development of the human body. According to research, during the evolution from apes to humans, the skull and leg bones have changed the most in terms of bones. The shape of the body has changed the least because it is close to gold, while the rectangular shape has changed the least. There are many proportions in the human body structure that are close to 0.618, making the human body beautiful in the hundreds of thousands. It has been fixed in years of historical accumulation. Human beings are most familiar with themselves, and they will inevitably regard the beauty of the human body as the highest aesthetic standard. By extension, from objects to people, from people to objects, and by extension, any object that is similar to the human body will like it and feel beautiful. Therefore, the golden section law, as an important formal beauty law, has become a classic aesthetic law passed down from generation to generation, and it continues to this day! In recent years, when studying the relationship between the golden section and the human body, it was discovered that there are 14 "golden points" (the ratio of the short segment to the long segment of the object is 0.618) and 12 "golden rectangles" (the ratio of the width to the length is 0.618) in the human body structure. rectangle) and 2 "golden indices" (the proportional relationship between the two objects is 0.618). Golden points: (1) Navel: the dividing point between the top of the head and the sole of the foot; (2) Throat: the dividing point between the top of the head and the sole of the foot; (3), (4) Knee joints: the dividing point between the navel and the sole of the foot; (5), (6) Elbow joint: the dividing point between the shoulder joint and the tip of the middle finger; (7), (8) Nipple: the dividing point on the longitudinal axis of the trunk nipple; (9) Glabellar point: the upper 1/3 of the distance between the hairline and the base of the chin and the dividing point between the middle and lower 2/3; (10) Subnasal point: the dividing point between the lower 1/3 and the upper and middle 2/3 of the distance between the hairline and the base of the chin; (11) Lip point: the distance between the base of the nose and the base of the chin The dividing point between the upper 1/3 and the middle and lower 2/3; (12) The center point of the mental labial groove: the dividing point between the lower 1/3 and the upper and middle 2/3 of the distance between the base of the nose and the base of the chin; (13) The left corner of the mouth: The dividing point between the left 1/3 and the right 2/3 of the horizontal line of the oral cleft; (14) The right corner point of the oral cavity: the dividing point between the right 1/3 and the left 2/3 of the horizontal line of the oral cleft.

The Golden Section of the Facial Golden Rectangle with Three Courts and Five Eyes: (1) Body contour: The average of shoulder width and hip width is width, and the height from shoulder to hip bottom is length; (2) Facial contour: Face width at eye level It is wide, and the distance from the hairline to the base of the chin is long; (3) Nose contour: the nose wing is wide, and the distance from the root of the nose to the base of the nose is long; (4) Lip contour: the distance between the upper and lower lip peaks in the resting state is wide, and the corner of the mouth is The distance is length; (5), (6) hand outline: the transverse diameter of the hand is width, and the average of the five fingers together is length; (7), (8), (9), (10), (11) , (12) Outline of the maxillary incisors, lateral incisors, and canines (three on the left and right): the largest mesiodistal diameter is wide, and the gingival diameter is long.

The golden index: (1) The nasolabial index, which reflects the relationship between the nose and mouth: the ratio of the width of the nose to the distance between the corners of the mouth is close to the golden number; (2) The eye-lip index, which reflects the relationship between the eyes and mouth: the ratio between the distance between the corners of the mouth and the distance between the eyes The ratio of the lateral canthus distance is close to the golden number. 0.618, as one of the standard scales of human bodybuilding, is beyond reproach, but its "fuzzy characteristics" cannot be ignored. Like other aesthetic parameters, it has a range of allowable changes, which is affected by race, region, and individual differences. restrictions.

(2) Proportional relationship uses numbers to express the beauty of the human body and compares it based on a certain benchmark. The method of using a certain part of the same human body as a benchmark to determine its proportional relationship with the human body is called the same body method (see middle picture). Divided into three groups: the coefficient method, which often refers to the index of head height and body length. In the human body, there are five sitting and seven standing, that is, the height when sitting is five times the height of the head and when standing is 7 or 7.5 times; the percentage method, which divides the body length into Treated as 100%, the proportion of each part of the body; dichotomy: that is, the human body is divided into two parts, large and small, the large part is from the feet to the navel, and the small part is from the navel to the top of the head. A standard facial shape with a coordinated length-to-width ratio, consistent with three stops and five eyes (see the picture on the right). The three stops refer to the length of the face. The distance from the hairline to the chin is divided into three equal parts, that is, the distance from the hairline to the eyebrows, the eyebrows to the tip of the nose, and the tip of the nose to the chin are each divided into one equal part, each called a stop. ***Three stops; Five Eyes refers to the width of the face. The length of the frontal projection between the ears is the length of the five eye fissures. Except for the eye fissures, the inner distance is the length of one eye fissure, from the outer canthus angles on both sides to the ears. Each eye has a cleft length, which is called five eyes.

(3) Angle relationship Observed from different angles, the reflected human body shapes are also different. The side angle theory proposed by Belt, Campen and others reflects the beauty of the human body through angles. Among them, Campen's theory is a method of measuring the inclination angle of the forehead when viewed from the side based on the straight line connecting the subnasal point and the ear hole point. In this way, the complex three-dimensional head can be described with a simple contour line - It is called the side qualitative analysis method. Use the straight line connecting the nose tip point and the submental point to observe the protrusion of the lips and evaluate the beauty and ugliness of the lower face. The tip of the nose, the red front edge of the lower lip, and the submental point are on the same straight line, which is called the Ricketts aesthetic plane and is a sign of beauty.

Our research has found that plants, like animals, also contain golden numbers.

The golden number is represented by the Greek letter Φ