Golden ratio

"In Euclid's Elements we meet the concept which later plays a significant role in the development of science. The concept is called the "division of a line in extreme and mean ratio" (DEMR). ...the concept occurs in two forms. The first is formulated in Proposition 11 of Book II. ...why did Euclid introduce different forms... which we can find in Books II, VI and XIII? ...Only three types of regular polygons can be faces of the s: the equilateral triangle... the square... and the regular pentagon. In order to construct the Platonic solids... we must build the two-dimensional faces... It is for this purpose that Euclid introduced the ... (Proposition II.11)... By using the "golden" isosceles triangle...we can construct the regular pentagon... Then only one step remains to construct the ... which for Plato is one of the most important regular polyhedra symbolizing the universal harmony in his cosmology."

"In Western Culture, starting from Phidias and the Parthenon, the Golden Section and the Golden Number are present, consciously or unconsciously, in very famous works. In the Renaissance, after the rediscovery of Fibonacci, it was a symbol of aesthetic perfection to be used in architecture and art with, among others, Leonardo da Vinci (1542-1519) and Albrecht Dürer (1471-1528). The Golden Number is in many geometric figures making them Golden. We have it among other things in the octagonal architecture of Castel del Monte. The Golden Ratio enters the pentagon which is Golden because the side of the star and the side of the pentagon are in the ratio of 38% and 62%, as required by the Golden Number."