Gold Number

Plato, “Conversations, Timeo”

The origin of the gold number is ancient. One cannot know from when man knows (apika, harean pentagram, makila without lifting, which he could imagine since he was conscious). The Egyptians already knew him, but it was Euclid who defined him:

“A right is divided into the proportion between the heads and the bisector when the proportion between the greater and the greater is the same between the latter and the lesser.”

For example, in Figure 1, the AA part is the gold segment or AB segment section,

AA' = A'BAB
= AA'

The proportion is met.

If the reader does not believe that he checks a rule. Regardless of the AB segment,

AA'

A'B

the quotient given by the gold section always takes the same value:

d = 0.618033988

It is an irrational algebraic number d whose concrete expression is:

This number surprises us in Antiquity, in the Renaissance and today.

We will mention two curious properties:

If we add 1 to d or turn it around, we get the same value:

1 + ? = 1 / ? edo1

+ 0,618033... Subtracting = 1,618033 = 1 / 0,618033...1-i

? or calculating ?2, we ^will obtain the same value:1 – ?

= ?2 edo1 – 0,618033...

= 0.391966... = (0,618033...)2

Since ancient times the methods of division of a segment in golden proportion (1. irud. ). The proportion of gold is also found in many geometric figures.

Figure . A classic way of calculating the gold section. The golden section of the segment AB is at point A’, i.e. AA’ = 0.618033.. x AB. In
addition, AB = =0.618033... x A”B, i.e. AB A”B is the golden section of the segment.

The best known is the pentagram (or five-vertex star), magical symbol of the Pythagoreans and alchemists (2. irud. ). Although the proportion of gold was given great importance in the Renaissance, both in theory and in practice, it was not first used in the Renaissance. On the contrary, in the pyramid of Keops (3. irud. ), It was already used in the Parthenon of Athens, Cathedral of Chartres, etc.

Figure . According to the Greek geographer Herodotus, who described the Egyptian pyramids, the height h of the pyramid of Keops is half of the base b and half of the apotopic x. That is,

It is also found in the human body itself. According to Leonardo’s “canon”, the most beautiful and harmonious proportions of the body are obtained when they are in golden proportion. Leonardo put the “brilliant man” in a circle whose center is the same and whose radius is in a golden proportion with the height of man (4. irud. ).

Figure . Known canon of Leonardo da Vinci. Inside it has a golden section or “divine proportion”. The radius of the circle is the number of gold on the square side and therefore the height and width of man.

Renaissance scientists adapted the value of the universal law of harmony to the golden number. They were not wrong, XVII. As Kepler demonstrated in the 20th century. The golden proportions appear in the distances between planets. The gold number indicates when a specific year corresponds to a lunar cycle of 19 years.

The golden section is also found in nature and its phenomena. Often there are flowers and plants with golden proportions. Among the animals there are also examples: the star of the sea, the shell of the animated fossil Nautilus (5. irud. ). The carbon atom, i.e., diamonds and the basic compound of living things, also has a gold structure.

Figure . Golden spiral. In nature there are spirals with golden section. The most obvious case is the Nautilus shell. The spiral of the image is obtained by superimposing the quadrants of the circles, whose rays grow in golden proportion.