Golden ratio: Difference between revisions

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{{Image|Altes Rathaus Leipzig.jpg|right|350px|The old [[townhall]] in [[Leipzig]]. The tower is positioned  between the left (a) and right (b) sections so that <math>\scriptstyle \frac{a}{b}</math> equals the golden ratio.}}
{{Image|Altes Rathaus Leipzig.jpg|right|350px|The old [[townhall]] in [[Leipzig]]. The tower is positioned  between the left (a) and right (b) sections so that <math>\scriptstyle \frac{a}{b}</math> equals the golden ratio.}}


The '''golden ratio''' is a mathematical proportion that is important in the arts and interesting to mathematicians. In architecture and painting, some works have been proportioned to approximate the golden ratio ever since antiquity. It has been claimed that the golden ratio is found in the proportions of architectural structures ranging from the  [[Great Pyramid of Giza]]<ref>[http://www.dartmouth.edu/~matc/math5.geometry/unit2/unit2.html]''The Golden Ratio & Squaring the Circle in the Great Pyramid'', Paul Calter</ref>, to the [[Parthenon]]<ref>[http://community.middlebury.edu/~harris/Humanities/TheGoldenMean.html] The Golden Mean, William Harris</ref>, to [[Chartres Cathedral]]<ref>[http://www.johnjames.com.au/chartres-shorthistory.shtml] Chartres Cathedral, John James</ref>. In the 20th century, the golden section was applied by the French architect [[Le Corbusier]], who used the proportion in the design of virtually all his architectural works, and in his his painting, and who wrote a series of books on the subject called ''The Modulor''. The ratio itself is also known as the '''mean and extreme ratio''', the '''golden section''', the '''golden mean''', and the '''divine proportion'''.  
The '''golden ratio''' is a mathematical proportion that is important in the arts and interesting to mathematicians. In architecture and painting, some works have been proportioned to approximate the golden ratio ever since antiquity. It has been claimed that the golden ratio is found in the proportions of architectural structures ranging from the  [[Great Pyramid of Giza]]<ref>[http://www.dartmouth.edu/~matc/math5.geometry/unit2/unit2.html]''The Golden Ratio & Squaring the Circle in the Great Pyramid'', Paul Calter</ref>, to the [[Parthenon]]<ref>[http://community.middlebury.edu/~harris/Humanities/TheGoldenMean.html] ''The Golden Mean'', William Harris</ref>, to [[Chartres Cathedral]]<ref>[http://www.johnjames.com.au/chartres-shorthistory.shtml] ''Chartres Cathedral'', John James</ref>. In the 20th century, the golden section was applied by the French architect [[Le Corbusier]]<ref>[http://www.archsociety.com/e107_plugins/content/content.php?content.24] ''Working With Corbusier'', Jerzy Soltan</ref>, who used the proportion in the design of virtually all his architectural works, and in his his painting, and who wrote a series of books on the subject called ''The Modulor''. The ratio itself is also known as the '''mean and extreme ratio''', the '''golden section''', the '''golden mean''', and the '''divine proportion'''.  


The earliest existing description is found in the [[Euclid's Elements|Elements]] of [[Euclid]] (Book 5, definition 3): ''A straight line is said to have been cut in extreme and mean ratio when, as the whole line is to the greater segment, so is the greater to the less.''     
The earliest existing description is found in the [[Euclid's Elements|Elements]] of [[Euclid]] (Book 5, definition 3): ''A straight line is said to have been cut in extreme and mean ratio when, as the whole line is to the greater segment, so is the greater to the less.''     

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(PD) Diagram: Joel Holdsworth
demonstrating a simple method to construct a Golden Ratio rectangle
PD Image
The old townhall in Leipzig. The tower is positioned between the left (a) and right (b) sections so that equals the golden ratio.

The golden ratio is a mathematical proportion that is important in the arts and interesting to mathematicians. In architecture and painting, some works have been proportioned to approximate the golden ratio ever since antiquity. It has been claimed that the golden ratio is found in the proportions of architectural structures ranging from the Great Pyramid of Giza[1], to the Parthenon[2], to Chartres Cathedral[3]. In the 20th century, the golden section was applied by the French architect Le Corbusier[4], who used the proportion in the design of virtually all his architectural works, and in his his painting, and who wrote a series of books on the subject called The Modulor. The ratio itself is also known as the mean and extreme ratio, the golden section, the golden mean, and the divine proportion.

The earliest existing description is found in the Elements of Euclid (Book 5, definition 3): A straight line is said to have been cut in extreme and mean ratio when, as the whole line is to the greater segment, so is the greater to the less.

According to Merriam-Webster's Collegiate Dictionary, Eleventh Edition, the proportion is derived from two segments in which "the ratio of the whole to the larger part is the same as the ratio of the larger part to the smaller."

To be more elaborate: if there is a longer line segment and and a shorter line segment , and if the ratio between and is equal to the ratio between the line segment and , this ratio is the golden ratio. The value of the golden ratio is

Properties

  • If   it follows that

With we could derive the infinite continued fraction of the golden ratio:

Thus


  • ,

where is the n-th term of the Fibonacci sequence.

References

  1. [1]The Golden Ratio & Squaring the Circle in the Great Pyramid, Paul Calter
  2. [2] The Golden Mean, William Harris
  3. [3] Chartres Cathedral, John James
  4. [4] Working With Corbusier, Jerzy Soltan