Mathematicians and artists have throughout modern history been fascinated by a special proportion known as the golden ratio, or golden section as it is also called. The special feature of this proportion is that if you divide a line into a larger and smaller section, A and B, then A is to B as the whole line is to A. Numerically it is about 1: 1.618. The golden ratio has been used by many architects as a basis for their buildings, as well as by many painters and even musicians.
No other number in the history of mathematics has inspired thinkers of all disciplines like the golden ratio. It has inspired men for at least 2.400 years since Pythagoras and Euklid in ancient Greece. Other outstanding thinkers, who have pondered the golden ratio, are Leonardo of Pisa, Johannes Kepler and present day physicist Roger Penrose. It has fascinated biologists, artists, musicians, architects, psychologists and occultists alike. The 12'th century mathematician Fibonacci came upon what is today known as the Fibonacci sequence: 1, 1, 2, 3, 5, 8, 13, 21, etc. in which each new number is the sum of the two preceding. The further you take this sequence, the closer it comes to the golden ratio. The pentagram is a peculiar figure in that all its line segments stand in a golden ratio relationship with some other segment of the pentagram.
The golden ratio is also known as Phi in honor of the great Greek sculptor Phidias, from about 400 BC, who used the golden ratio extensively in his sculptures. The golden ratio has also been known as the divine proportion since 1509, when Luca Pacioli published a three volume book on the golden ratio entitled "De Divina Proportione". Pacioli saw religious significance in the proportion, hence the title of his book. The book was a major influence on artists and architects for hundreds of years
The modern Swiss architect Le Corbusier is famous for his use of the golden ratio. He saw the ratio and the Fibonacci sequence as representing a mathematical order of the universe, and he described them as: "rhythms apparent to the eye and clear in their relations with one another. And these rhythms are at the very root of human activities. They resound in man by an organic inevitability, the same fine inevitability which causes the tracing out of the Golden Section by children, old men, savages and the learned."
Painters, such as the 17'th century master Vermeer, have used the golden ratio extensively, so did a modern master like Salvador Dali. Dali adored Vermeer, by the way. Composers have also used the golden ratio and the Fibonacci sequence. The modern composer Bartok, for example, based the xylophone progression in "Music for Strings, Percussion and Celeste" on the Fibonacci sequence 1, 2, 3, 5, 8, 5, 3, 2, 1. Similarly Satie and Debussy are known to have used the golden ratio as a basis for some of their compositions.
The golden ratio can be found throughout nature. The arrangement of branches along the stems of plants, for instance, often follows the golden ratio.
No other number in the history of mathematics has inspired thinkers of all disciplines like the golden ratio. It has inspired men for at least 2.400 years since Pythagoras and Euklid in ancient Greece. Other outstanding thinkers, who have pondered the golden ratio, are Leonardo of Pisa, Johannes Kepler and present day physicist Roger Penrose. It has fascinated biologists, artists, musicians, architects, psychologists and occultists alike. The 12'th century mathematician Fibonacci came upon what is today known as the Fibonacci sequence: 1, 1, 2, 3, 5, 8, 13, 21, etc. in which each new number is the sum of the two preceding. The further you take this sequence, the closer it comes to the golden ratio. The pentagram is a peculiar figure in that all its line segments stand in a golden ratio relationship with some other segment of the pentagram.
The golden ratio is also known as Phi in honor of the great Greek sculptor Phidias, from about 400 BC, who used the golden ratio extensively in his sculptures. The golden ratio has also been known as the divine proportion since 1509, when Luca Pacioli published a three volume book on the golden ratio entitled "De Divina Proportione". Pacioli saw religious significance in the proportion, hence the title of his book. The book was a major influence on artists and architects for hundreds of years
The modern Swiss architect Le Corbusier is famous for his use of the golden ratio. He saw the ratio and the Fibonacci sequence as representing a mathematical order of the universe, and he described them as: "rhythms apparent to the eye and clear in their relations with one another. And these rhythms are at the very root of human activities. They resound in man by an organic inevitability, the same fine inevitability which causes the tracing out of the Golden Section by children, old men, savages and the learned."
Painters, such as the 17'th century master Vermeer, have used the golden ratio extensively, so did a modern master like Salvador Dali. Dali adored Vermeer, by the way. Composers have also used the golden ratio and the Fibonacci sequence. The modern composer Bartok, for example, based the xylophone progression in "Music for Strings, Percussion and Celeste" on the Fibonacci sequence 1, 2, 3, 5, 8, 5, 3, 2, 1. Similarly Satie and Debussy are known to have used the golden ratio as a basis for some of their compositions.
The golden ratio can be found throughout nature. The arrangement of branches along the stems of plants, for instance, often follows the golden ratio.
About the Author:
Power Retouche, the plugin company, has a dedicated Photoshop plugin for drawing the Golden Section, Golden Spiral, etc. in Photoshop.
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