# Golden section

##### Article

October 28, 2021

The golden ratio or gold ratio is a proportionality that often appears in both nature and art, creating a natural balance between symmetry and asymmetry. Gold engraving proportions can be found in many ancient buildings, medieval and renaissance works of art. The ancient Pythagoreans (Pythagoras and his followers), who argued that reality rests on mathematical foundations, believed in discovering one of the fundamental laws of existence in the golden ratio, as this ratio can also be recognized in nature (e.g., the human body or snail vase). Forms containing the proportions of the golden ratio still have great aesthetic value, and are used in many fields (such as typography or photography). The irrational number Φ ≈ 1,618 (Greek large fí), which numerically expresses the gold ratio, has several interesting mathematical properties. Mathematical definition Two parts (a and b, a> b) are proportioned to each other according to the golden ratio, if the whole (a + b) is proportional to the larger part (a) as the larger part (a) to the smaller part (b): the + b the the b {\ displaystyle {a + b \ over a} {a \ over b}} .That is, the larger part is equal to the geometric mean of the sum and the smaller part: the 2 ( the + b ) ⋅ b {\ displaystyle a ^ {2} (a + b) \ cdot b} .The wording equivalent to the above is that the major part is proportional to the smaller part as the smaller part is the difference between the two parts: the b b the - b , {\ displaystyle {a \ over b} {b \ over a-b},} say: b 2 the ⋅ ( the - b ) {\ displaystyle b ^ {2} a \ cdot (a-b)} .