The Golden Section and the Phi Ratio (1.618)
by Robert Levy - designer of The Golden Divider for Arts
(adapted by dynamicsymmetryart.com)
Man has always tried to measure and quantify the world surrounding him. To do this, he has used the measurements and proportions of his own body. Until the French Revolution of 1789 (birth of the metric system), the whole world measured their environment in hand spans, palms, handbreadths, feet, and cubits.
These five measures have the following particularity: the sum of two adjacent measures is equivalent to the following measure (hand span + palm = handbreadth; handbreadth + foot = cubit) and the relation between two adjacent measures is constant, and equivalent to the number 1.618: hand span x 1.618 = palm, palm x 1.618 = handbreadth, handbreadth x 1.618 = foot and foot x 1.618 = cubit. The cubit (or Egyptian royal cubit) was equivalent to 52.9 cm (after the reform under the 26th dynasty of the Pharaohs).
While Euclid (300 BC) already spoke of this relation of two lengths in his Elements, this proportion was named Divine Proportion by Luca Pacioli, a mathematician and contemporary of Leonardo da Vinci (1509), much later the Golden Section by the German philosopher and mathematician Adolf Zeising around 1850 and finally the Golden Ratio in 1932, by the Romanian diplomat Matila Ghyka. Undoubtedly first used in geometry, the Golden Ratio, (called φ or Phi in honor of the Greek architect of the Parthenon, Phidias) represents a constant relation between two sizes of the same nature, such as lengths, surfaces, volumes, or numbers.
Phi is an irrational number and signifies (1+√5)/2, i.e. a value approximating to 1.618. For hundreds, even thousands of years, the Golden Ratio has served to define ideal proportions between two geometric or mathematical entities. It is on the border of these two domains and symbolizes their joining. It represents a benchmark for harmonious proportions, and while it has been used (or revealed) primarily in architecture and painting, it is also found in fields as diverse as science, physics, nature, music, finance, and acoustics.
Until the 19th century, it is almost certain that the Golden Ratio was used in history in a deductive (conscious) manner, but practically no writing substantiates this. Its deliberate use has nevertheless remained secret and been transmitted from generation to generation by certain trades such as architects, the ‘compagnons’ (apprentice craftsmen) or the great painters: some even called it the Ratio of the Initiated.
Nowadays the Ratio is no longer as secret as it once was in history. But it remains no less mythical, and still retains an air of mysticism and mystery for some people. Its conscious application in domains such as architecture, painting, sculpture, industrial aesthetics, crafts, interior design and decoration, landscaping, marketing, and many other fields is well established. As the Golden Ratio represents (for many) the ideal of harmony in proportions, many of the buildings constructed around us contain these ratios, and a large number of advertising logos are designed on this principle as well.
Painting above from the Myron Barnstone drawing DVD series demonstrating the Phi rectangle.