the film
Geometric ideas and concepts

Golden Ratio,
Divine Proportion,
or Golden Section

from Numbers, the Universal Language by Denis Guedj

Harmony is expressed in numbers. Whether in pictorial or architectural space, or in the realm of music, many have tried to express harmony in the language of number. In Classical thought the beautiful was said to be lodged in a wonderful number called the golden number, represented by the Greek letter φ = (1 - squareroot(5))/2, one of two roots of the equation x*x - x - 1 = 0, ordinarily used in its decimal value of 1.618 (bluefish adds that the other root is its negative reciprocal, -0.6180339).

The golden number turns up everywhere, from the architecture of the Egyptian pyramids and Greek temples to compositions by Raphael, Leonard da Vinci, Poussin, Cezanne, and Le Corbusier. The structure of Rogier van der Weyden's great altarpiece The Descent from the Cross c. 1435, comprises golden-number geometry, linked to a Christian symbolism in which Christ's body represents the perfect harmony of parts.

A golden section expresses perfect, harmonious proportion: it is the division ("section") of a line or the proportion of a geometrical figure such that the smaller dimension is to the greater as the greater is to the whole.

a:b :: b:(a+b)

1.6180339... = 1 + 1/ (1+ 1/ ( (1+ 1/ (1+ 1/ (1+ 1/ (1+ 1/ ... ) ) ) ) )

1.6180339... = squareroot(1 + squareroot(1 + squareroot(1 + squareroot( ... ) ) ) )

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