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. It has the special characteristic that if you divide a line into two sections, a larger A and a smaller B, according to the golden ratio, then A is to B as A+B is to A. Numerically it is about 1: 1.618. Artists and architects, even musicians, have used the golden ratio as a basis for their paintings and buildings, or music.
No other number in the history of mathematics has inspired thinkers of all disciplines like the golden ratio. It has fascinated for at least 2.400 years since Pythagoras and Euklid in ancient Greece. Amongst the prominent thinkers, who have pondered the golden ratio, we can mention Leonardo of Pisa, Johannes Kepler and the 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: 0, 1, 1, 2, 3, 5, 8, 13, 21, etc. in which each new number is the sum of the two preceding. As you continue this sequence, it will accurately reach 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.
In honor of Phidias, the great Greek sculptor from about 400 BC who used the golden proportion extensively in his sculptures, the golden proportion is now commonly known as Phi, the first letter of Phidias' name. 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. For hundreds of years the book had a major influence on artists and architects.
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 fascinated for at least 2.400 years since Pythagoras and Euklid in ancient Greece. Amongst the prominent thinkers, who have pondered the golden ratio, we can mention Leonardo of Pisa, Johannes Kepler and the 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: 0, 1, 1, 2, 3, 5, 8, 13, 21, etc. in which each new number is the sum of the two preceding. As you continue this sequence, it will accurately reach 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.
In honor of Phidias, the great Greek sculptor from about 400 BC who used the golden proportion extensively in his sculptures, the golden proportion is now commonly known as Phi, the first letter of Phidias' name. 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. For hundreds of years the book had a major influence on artists and architects.
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.
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