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#1 2007-12-15 17:41:46
- Jai Ganesh
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- Registered: 2005-06-28
- Posts: 46,148
Golden Section and Golden Ratio.
Golden Section : The division of a line into two segments so that the ratio of the whole line to the larger segment equals the ratio of larger segment to the smaller one; often called the golden section. This ratio was once considered by some as having an inherent aesthetic value.
Golden Ratio : A ratio, observed especially in the fine arts, between the two dimensions of a plane figure or the two divisions of a line such that the smaller is to the larger as the larger is to the sum of the two, a ratio of roughly three to five.
Golden Section, in mathematics, division of a line segment into two segments such that the ratio of the original segment to the larger division is equal to the ratio of the larger division to the smaller division. If c is the original segment, b is the larger division, and a is the smaller division, then c=a+b and c/b=b/a. Thus, b is the geometric mean of a and c; the ratio is known as the Divine Proportion. The Golden Rectangle, whose length and width are the segments of a line divided according to the Golden Section, occupies an important position in painting, sculpture, and architecture, because its proportions have long been considered the most attractive to the eye. The constructions of regular polygons of 5, 10, and 15 sides depend on the division of a line by the Golden Section. The numerical ratio of the greater segment of the line to the shorter segment as determined by the Golden Section is symbolized by the Greek letter phi and has the approximate value 1.618. It occurs in many widely varying areas of mathematics. For example, in the Fibonacci sequence (the sequence of numbers formed by adding successive members to find the next member0, 1, 1, 2, 3, 5, 8, 13,...), the values of the ratios 1, 2/1, 3/2, 5/3, 8/5, 13/8,... approach the value of the Golden Section.
Golden Section
A mathematical proportion where the ratio between a small section and a larger section is equal to the ratio between the larger section and both sections put together. Used by many 20th century composers, especially Bela Bartok, to determine the point of climax for a given work.
It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.
Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.
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#2 2007-12-15 18:22:22
- MathsIsFun
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- Registered: 2005-01-21
- Posts: 7,711
Re: Golden Section and Golden Ratio.
I did two pages:
"The physicists defer only to mathematicians, and the mathematicians defer only to God ..." - Leon M. Lederman
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#3 2007-12-15 18:35:02
- Jai Ganesh
- Administrator
- Registered: 2005-06-28
- Posts: 46,148
Re: Golden Section and Golden Ratio.
Very good pages, MathsIsFun!
Some of the interesting properties of phi are explained.
It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.
Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.
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#4 2007-12-15 23:15:34
- MathsIsFun
- Administrator
- Registered: 2005-01-21
- Posts: 7,711
Re: Golden Section and Golden Ratio.
I could do more I suppose ... any favorites?
"The physicists defer only to mathematicians, and the mathematicians defer only to God ..." - Leon M. Lederman
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