The history of the golden section

Since the Pythagorean school in ancient Greece studied the drawing methods of regular pentagons and regular decagons in the 6th century BC, modern mathematicians have come to the conclusion that Pythagoras school had touched and even mastered the golden section at that time.

In the 4th century BC, eudoxus, an ancient Greek mathematician, first studied this problem systematically and established the theory of proportion.

When Euclid wrote The Elements of Geometry around 300 BC, he absorbed eudoxus's research results and further systematically discussed the golden section, which became the earliest treatise on the golden section.

After the Middle Ages, the golden section was cloaked in mystery. Several Italians, pacioli, called the ratio between China and the destination sacred and wrote books on it.

German astronomer Kepler called the golden section sacred.

It was not until the19th century that the name golden section gradually became popular.

The golden section number has many interesting properties and is widely used by human beings.

The most famous example is the golden section method or 0.6 18 method in optimization, which was first proposed by American mathematician Kiefer in 1953 and popularized in China in 1970s.

How to discover legends:

In the 6th century BC, PInthagoras, an ancient Greek mathematician and philosopher, passed by a blacksmith's shop one day and was attracted by the crisp and pleasant sound of striking the iron. He stopped to listen carefully and intuitively concluded that the voice was "secret"! He walked into the workshop and carefully measured the dimensions of the anvil and the hammer, and found that the ratio between them was close to 1: O.6 18. After returning home, he took a wooden stick and asked his students to carve a mark on it, which not only made the distance between the two ends of the stick unequal, but also made people look satisfied.

After many experiments, a very consistent result is obtained, that is, the ratio of the whole section AB to the long section cB is equal to the ratio of the long section CB to the short section CA when the stick AB is divided by the point C. Later, Pythagoras found that putting a shorter line segment on a longer line segment also produces the same proportion: so infinity (see Figure 5-5- 1).

After calculation, the ratio of the long segment (assuming a) to the short segment (assuming b) is 1: O.6 18, and the ratio is L6 18. The formula can be used.

a :b=(a+b):a

Expression, and there is a mathematical relationship. At this time, the square of the long segment length is exactly equal to the product of the whole stick and the short segment length, that is, A = (A+B) B) B.

This magical proportional relationship was later praised by Plato, a famous ancient Greek philosopher and aesthetician, as the "golden section law", which was called "Huang Jinlv" and "golden ratio" for short. It is appropriate to use the word "gold" here to describe the importance of this law. What's even more amazing is that 1 divided by 65438+o.6 18 is just equal to O.66548. 1 divided by 1.5 18 is not equal to O, 5 1 8 ...

Equal to 0.618 (accurate to 0.00 1).

Therefore, it is correct to say that the golden ratio is 1.6 18 (long segment: short segment) or 0.6 18 (short segment: long segment). Mathematicians also found that 2: 3 or 3: 5 or 5: 8 is an approximation of the golden ratio, and the sum of numerator and denominator is a new denominator. 13/21,21/34.34/55, 55/88 ... The larger the number, the closer the ratio of its numerator to denominator is to O.6 18, which is mathematically called "Fibonacci series".

According to the law of this series, the golden ratio of "area" can be calculated from the golden ratio of "line segment". According to this series, le corbusier, a modern architect, invented the "golden ruler" (building standard ruler, slightly increased by 1.6 times).

The medieval mathematician Kepler called the golden section law and Pythagorean theorem "two treasures in geometry".

/kloc-Pachouri, a Venetian mathematician in the 10th and 9th centuries, praised the golden section law as a "godsend proportion".