The Greeks knew since ancient times the
so-called golden section of a segment,
which is nothing but its division into two parts, so that the longer divided by
the shortest is the same as the length of the total segment divided by the
longest. Consider, for example, segment AB. Its golden division is given by the
point X if and only if AX/XB = AB/AX.
A X B
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| Leonardo: the Vitruvian Man |
For the Greeks, as for many great painters, the
golden ratio or golden section divides a segment in the most aesthetically
attractive way. The Italian mathematician Lucas Paccioli, who called this ratio
the divine
proportion, influenced Leonardo da Vinci and Albrecht Durer. In the
twentieth century, neo-Impressionist painters like Seurat have used the golden
section to define the dimensions of some of their compositions. Architects like
Le Corbusier used the golden ratio to design their works. And many books
published in the sixteenth to eighteenth centuries had the dimensions of a
golden rectangle. The golden ratio has also been used by musicians such as Erik
Satie and Debussy, and provided some mystics with food for thought.
The golden ratio has curious properties. For
example, you can build a golden rectangle whose height is the golden section of
its base. If you take from this rectangle the square whose side is equal to its
height, the smaller rectangle is also golden. This effect can be repeated
indefinitely from the new rectangle.
