To divide a line segment in the most aesthetically pleasing and meaningful way, where would you choose the division point on this line segment? Among the countless points on this line segment, there is one very special point that divides the line segment into two parts, where the ratio of the longer part to the shorter part is exactly equal to the ratio of the entire line segment to the longer part. This ratio is the golden ratio, also known as the golden section, often represented by φ. The understanding of the golden ratio has a long history. As early as the 6th century BC, the Pythagorean School in ancient Greece studied regular pentagons and decagons, and the five-pointed star was even the secret symbol of this school. These figures are closely related to the golden ratio. Euclid's Elements, considered the "first book of mathematics", published in the 3rd century BC, was the first book to mention the golden ratio that has survived to this day.
For a long period of history, the concept of the golden ratio dominated Western architectural aesthetics. The Parthenon temple in ancient Greece, from its appearance, has a width-to-height ratio close to the golden ratio. Experienced announcers often do not stand in the center of the stage when announcing, but at the golden section of the stage, leaving a more harmonious image for the audience. When taking photos, placing people in the center or too far on the edge is not the best choice. The best position is near the golden section. Even the five-pointed star on the solemn and beautiful Five-Star Red Flag also contains the golden ratio: each side of the five-pointed star is exactly divided by the two sides that intersect with it in the golden ratio.
Surprisingly, the human body itself is closely related to the golden ratio. Painters and sculptors of the human body believe that the ratio of a person's height to the height of their navel is close to the golden ratio, and the ratio of the navel height to the knee height is also close to the golden ratio. This is probably a criterion for judging a standard figure! If you don't believe it, you can measure it yourself.
Interestingly, it was the famous Italian scientist, artist, and engineer Leonardo da Vinci who first gave this ratio relationship the nickname "golden". In his masterpiece The Vitruvian Man, the basis of various human body proportions is closely related to the golden ratio. The name of this painting is derived from the name of the outstanding Roman architect Vitruvius, who praised the human body proportion and the golden ratio in his book De Architectura.
Over the years, a large amount of literature has been accumulated around the golden ratio, and many people believe that it is the foundation for understanding all morphology (including human anatomy), art, architecture, and music. Who would have thought that this seemingly simple line segment division would have such a profound impact in a wide range of fields such as mathematics, technology, and art!
Of course, in the fields of art and music, personal subjective feelings play an important role. It would be too rigid to insist that it must strictly conform to the golden ratio principles. Being able to conform to it "within an acceptable range" is already quite good. Therefore, we say that the golden ratio is everywhere, which is a fact. But we also can't understand it too mechanically, thinking that even the number of decimal places after the decimal point should be completely consistent, which would only lead to ridiculous results and be detrimental to creative thinking. Being able to roughly conform to the golden ratio principles is enough to make people admire it.