The golden ratio is a proportion that has been considered the most perfect and harmonious since ancient times. It forms the basis of many ancient structures, from statues to temples, and is very common in nature. At the same time, this proportion is expressed in surprisingly elegant mathematical constructions.
The golden proportion is defined as follows: it is such a division of a segment into two parts that the smaller part refers to the larger one in the same way as the larger part refers to the entire segment.
If the length of the entire segment is taken as 1, and the length of the greater part is taken as x, then the sought proportion will be expressed by the equation:
(1 - x) / x = x / 1.
Multiplying both sides of the proportion by x and transferring the terms, we get the quadratic equation:
x ^ 2 + x - 1 = 0.
The equation has two real roots, of which we are naturally only interested in the positive. It is equal to (√5 - 1) / 2, which is approximately equal to 0, 618. This number expresses the golden ratio. In mathematics, it is most often denoted by the letter φ.
The number φ has a number of remarkable mathematical properties. For example, even from the original equation it is seen that 1 / φ = φ + 1. Indeed, 1 / (0.618) = 1.618.
Another way to calculate the golden ratio is to use an infinite fraction. Starting from any arbitrary x, you can sequentially construct a fraction:
1 / (x + 1)
1 / (1 / (x + 1) + 1)
1 / (1 / (1 / (x + 1) + 1) +1)
To facilitate calculations, this fraction can be represented as an iterative procedure, in which to calculate the next step, you need to add one to the result of the previous step and divide one by the resulting number. In other words:
x0 = x
x (n + 1) = 1 / (xn + 1).
This process converges, and its limit is φ + 1.
If we replace the calculation of the reciprocal with the extraction of the square root, that is, to carry out an iterative loop:
x0 = x
x (n + 1) = √ (xn + 1), then the result will remain unchanged: regardless of the initially chosen x, the iterations converge to the value φ + 1.
Geometrically, the golden ratio can be constructed using a regular pentagon. If we draw two intersecting diagonals in it, then each of them will divide the other strictly in the golden ratio. This observation, according to legend, belongs to Pythagoras, who was so shocked by the found pattern that he considered the correct five-pointed star (pentagram) to be a sacred divine symbol.
The reasons why it is the golden ratio that seems to a person the most harmonious are unknown. However, experiments have repeatedly confirmed that the subjects who were instructed to divide the segment into two unequal parts most beautifully do it in proportions very close to the golden ratio.