How can an ordinary rectangle have differences in beauty? I believe many people will have such doubts when they see this question. In fact, a psychologist once conducted an experiment. He carefully designed many rectangles of various sizes and asked people to choose the most beautiful ones. As a result, four rectangles received the most votes. They appeared to have well-proportioned and symmetrical sides, giving people a sense of comfort. After measurement, the ratios of their side lengths were 8 to 5, 13 to 8, 21 to 13, and 34 to 21.
The ratios of the long and short sides of these rectangles are all adjacent terms of the Fibonacci sequence, which is very close to the golden ratio. In mathematics, a rectangle whose length-to-width ratio is equal to the golden ratio is called a golden rectangle. Rectangles that are too square or too flat obviously don't bring much visual beauty. People generally believe that rectangles with a length-to-width ratio equal to the golden ratio are the most beautiful of all rectangles.
If a square with a width as its side is cut from a golden rectangle, the remaining part will still be a golden rectangle. This process continues, resulting in a series of smaller and smaller golden rectangles, called golden rectangle sets. If we draw a quarter-circle in each square when cutting out the squares, we can obtain a curve very close to the logarithmic spiral.
In addition to golden rectangles, in geometry, an isosceles triangle whose leg length to base length ratio is the golden ratio is called a golden triangle (that is, an isosceles triangle with a top angle of 36º); an ellipse whose long semi-axis a to short semi-axis b ratio is the golden ratio is called a golden ellipse; a hyperbola whose semi-focal length to real semi-axis a ratio is the golden ratio is called a golden hyperbola. All these pleasing figures have one thing in common: they are closely related to the golden ratio.