Many biological forms comply with the golden ratio. For example, the three knuckles on each finger of a human being are about one to six times as large as the next one from the bottom to the top. The golden ratio is found everywhere from seashells to flower petals to buildings. The five-pointed star is the symbol of the Pythagoreans, and its long and short lines all conform to the golden ratio. The golden ratio is also expressed in beautiful poetic music, including Bach's Fugue in D minor and the refrain from the Russian poet Voznesensky's "Goya".
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Dan Brown [U.S.] A teaching dialogue about gold cutting in the book "The Da Vinci Code" can help everyone understand gold cutting better:
They came to the emergency stairway, and Sophie opened it carefully Door. There were no sirens, just an alarm net connected to the door leading outside the Louvre. Sophie led Langdon along the zigzag stairs to the first floor. They quickened their pace.
As Langdon hurriedly followed Sophie, he asked, "When your grandfather talked about the pentagram, did he mention goddess worship or resentment against the Catholic Church?"
p>Sophie shook her head. "I prefer to analyze it from a mathematical perspective - things like the golden section, PHI, Fibonacci sequence."
Langdon was surprised: "Your grandfather taught you PHI?"
"Of course, the golden section." She said a little shyly, "Actually, he once joked that I was half of the golden section...that's because of the way my name is spelled."
Langdon thought for a moment and muttered: "so-PHI-e."
As Langdon went downstairs, he thought about PHI again. He began to realize that Saunière's clues were more holistic than he had imagined.
Leonardo da Vinci... Fibonacci sequence... pentagram.
Incredibly, all of this is tied together by a concept from art history, and Langdon often spends several class hours explaining this very basic concept. PHI
He suddenly had an illusion, as if he was back at Harvard again, standing on the podium in the classroom and explaining "Symbols in Art", and wrote his favorite number on the blackboard: 1.618.
Langdon turned to the many eager students in the audience and asked, "Who can tell me what number this is?"
A big math guy sitting in the back row The department student raised his hand: "That's PHI." He pronounced it "fei."
"Well said, Stiller." Langdon said, "Everyone knows about PHI."
Stiller smiled and added: "Don't compare it to PI. π) is confused. We math people like to say: PHI has one more H, but it is much better than PI!" Langdon laughed, but the others didn't understand what he meant.
Stiller sat down with a thud.
Langdon continued: "PHI, 1.618 has an extremely important position in art. Who can tell me why?"
"Because it is very beautiful?" Steve Le tried to save face.
Everyone burst into laughter.
Langdon said: "Actually, Stiller is right again. PHI is often considered the most beautiful number in the world."
The laughter stopped abruptly. Stiller was smug.
Langdon put the picture on the slide projector and explained that PHI originated from the Fibonacci sequence—this sequence is very famous not only because the sum of two adjacent items in the sequence is equal to One term, and because the quotient obtained by dividing adjacent two terms is approximately equal to 1.618, which is PHI.
Langdon continued to explain that from a mathematical point of view, the origin of PHI is quite mysterious, but what is even more puzzling is that it also plays an extremely important role in the composition of nature. Plants, animals, and even humans have qualities that are strikingly similar to this ratio.
Langdon turned off the lights in the classroom and said: "PHI is everywhere in nature. This is obviously not a coincidence, so the ancestors estimated that PHI was predetermined by the Creator. Early scientists called 1.618 The golden section."
"Wait a minute," said a girl sitting in the front row, "I am a biology major, and I have never seen the golden section in nature."
"No?" Langdon grinned, "Have you ever studied male and female bees in a hive?" "Of course. There are always more female bees than male bees."
"Yes. Did you know that if you count the male and female bees in any hive in the world, you will get the same ratio? "Yes, it's PHI."
The girl was stunned. "That's impossible."
"Possible!" Langdon retorted. He smiled and showed a slideshow of spiral shells. "Do you know this?"
"Nautilus," the student replied. "A mollusk that regulates its buoyancy by sucking air into its shell."
"That's right.
Can you guess the ratio of the diameter of each rib on its body to the diameter of the adjacent ribs? "
The girl looked at the concentric arcs on the spiral nautilus and couldn't tell the exact answer. Langdon nodded and said, "PHI. golden section. 1.618. "
The girl showed a surprised expression.
Langdon then showed the next slide - a close-up of a sunflower. "Sunflower seeds are arranged in opposite arcs on the flower disk. Can you guess the ratio of diameters between two adjacent circles? ”
“PHI? "Someone said.
"You guessed it right. ” Langdon began running through the slideshow—the spiral shape of a pine cone, the arrangement of leaves on a plant stem, the segmentation of an insect—all of which fit perfectly into the golden section.
“It’s incredible. ! "Someone shouted.
"Yes, but what does this have to do with art? "Another person said.
"Ah! Good question. " Langdon said, showing another slide - Leonardo da Vinci's famous male nude painting "Vitruvian Man." The painting was painted on a piece of parchment that was slightly faded. Yellowing. The title of the painting is taken from the name of the outstanding Roman architect Mark Vitruvius, who praised the golden section in his book "Architecture"
"No one. Understands the delicate structure of the human body better than Leonardo da Vinci. In fact, Leonardo da Vinci once exhumed human corpses to measure the exact proportions of human skeletal structures, and he was the first person to claim that the structural proportions of the human body perfectly match the golden ratio. "
Everyone here cast doubtful glances at Langdon.
"Don't believe it? Langdon said, "Next time you take a shower, bring a measuring tape." " Several students on the football team snickered.
"It's not just you jocks who are starting to get restless," Langdon prompted. "All of you, boys and girls, give it a try. Measure your height and divide it by the distance from your belly button to the ground. Guess what the result is. ”
“It can’t be PHI! "A sports student said in a skeptical tone.
"That's PHI," Langdon replied. "Exactly 1.618. Want to see another example? Measure the distance from your shoulder to your fingertips, then divide it by the distance from your elbow to your fingertips to get your PHI. Divide the distance from the hip to the ground by the distance from the knee to the ground to get PHI. Look at the finger joints, the toes, the segments of the spine, and you can get PHI from them. Friends, each of us is a creature inseparable from the golden section. "Although the lights in the classroom were turned off, Langdon could see that everyone was shocked. A warm current surged into his heart, which was why he loved teaching. "Friends, as you can see, there are many Complex nature has hidden rules. When the ancients discovered PHI, they were sure they had stumbled upon the proportions of God's creations, and because of this they worshiped the natural world. The work of God can be confirmed in nature, and there is still a pagan organization to this day - the Cult of the Earth Mother. Many of us, like the pagans, praise nature, but we don't realize it. For example, our celebration of May Day is a good example of this. May Day is a festival that celebrates spring. People use it to celebrate the recovery of the earth and its gifts to mankind. From the beginning, the mysterious qualities of the Golden Section have been established. People can only act according to the rules of nature, and art is an attempt by people to imitate the beauty of the Creator's creation, so this semester we will see many examples of the golden section in works of art. ”
For the next half hour, Langdon showed the students slides of works by Michelangelo, Albrecht Dürer, Leonardo da Vinci, and many other artists. , these artists consciously and strictly followed the golden ratio when designing and creating their works. Langdon revealed to everyone the golden ratio used in the architectural design of the Parthenon in Greece, the Pyramids of Egypt, and even the United Nations Building in New York. Pointing out that PHI has also been used in Mozart's sonatas, Beethoven's Fifth Symphony, and the creations of Bartok, Debussy, Schubert and other musicians, Langdon also told everyone that even Stradivarius. The golden section was also used to determine the exact location of the f-shaped hole in the construction of his famous violin.
As he walked toward the blackboard, Langdon said, "Let's go back to the symbolism. "He drew a five-pointed star composed of five straight lines on the blackboard. "This is the most symbolic shape you will learn this semester. The pentacle - known to the ancients as the pentacle - is considered sacred and magical in many cultures. Can anyone tell me why this is? "
Stiller - the math student - raised his hand again. "Because if you draw a five-pointed star, then those line segments will automatically divide themselves according to the ratio of the golden section. Cut into several paragraphs. "
Langdon nodded at the young man, feeling proud of him. "Good answer. The ratios of the line segments in the five-pointed star are consistent with the golden ratio, making it the primary representative of the golden ratio. It is for this reason that the pentacle has always been used as a symbol of beauty and perfection, and is associated with goddesses and the divine feminine. ”
The girls in the class are all smiling.