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Golden Ratio

Golden Ratio



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Published by Nassouh Mahfouz

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Published by: Nassouh Mahfouz on Dec 04, 2012
Copyright:Attribution Non-commercial


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This article is about the number. For the pop music album, see The Golden Ratio (album). For calendardates, see Golden number (time).Line segments in the golden ratioA golden rectangle with longer side and shorter side b, when placed adjacent to a square with sides of length a, will produce a similar golden rectangle with longer side a + b and shorter side a. This illustratesthe relationship .The golden ratio () is also called the golden section (Latin: sectio aurea) or golden mean. Other namesinclude extreme and mean ratio, medial section, divine proportion, divine section (Latin: sectiodivina), golden proportion, golden cut, golden number, and mean of Phidias.In mathematics and the arts, two quantities are in the golden ratio if the ratio of the sum of thequantities to the larger quantity is equal to the ratio of the larger quantity to the smaller one. The figureon the right illustrates the geometric relationship. Expressed algebraically:
 where the Greek letter phi () represents the golden ratio. Its value is:At least since the 20th century, many artists and architects have proportioned their works toapproximate the golden ratio
especially in the form of the golden rectangle, in which the ratio of thelonger side to the shorter is the golden ratio
believing this proportion to be aesthetically pleasing(see Applications and observations below). Mathematicians since Euclid have studied the properties of the golden ratio, including its appearance in the dimensions of a regular pentagon and in a goldenrectangle, which can be cut into a square and a smaller rectangle with the same aspect ratio. The goldenratio has also been used to analyze the proportions of natural objects as well as man-made systems suchas financial markets, in some cases based on dubious fits to data.Two quantities a and b are said to be in the golden ratio
One method for finding the value of φ is to start
with the left fraction. Through simplifying the fraction
and substituting in b/a = 1/φ,
 it is shown thatMultiplying by
giveswhich can be rearranged toUsing the quadratic formula, the only positive solution is
 MathematicianMark Barrproposed using the first letter in the name of Greek sculptorPhidias,phi, to symbolize the golden ratio. Usually, the lowerc
ase form (φ) is used. Sometimes, the uppercase form (Φ)
is used for thereciprocal
of the golden ratio, 1/φ.
The golden ratio has fascinated Western intellectuals of diverse interests for at least 2,400 years.According toMario Livio: Some of the greatest mathematical minds of all ages, fromPythagorasandEuclidinancient Greece,  through the medieval Italian mathematicianLeonardo of Pisaand the Renaissance astronomerJohannes Kepler,to present-day scientific figures such as Oxford physicistRoger Penrose,have spent endless hours over this simple ratio and its properties. But the fascination with the Golden Ratio is not confined just to mathematicians. Biologists, artists, musicians, historians, architects, psychologists, and evenmystics have pondered and debated the basis of its ubiquity and appeal. In fact, it is probably fair to saythat the Golden Ratio has inspired thinkers of all disciplines like no other number in the history of mathematics.Ancient Greekmathematicians first studied what we now call the golden ratio because of its frequentappearance ingeometry.The division of a line into "extreme and mean ratio" (the golden section) isimportant in the geometry of regularpentagramsandpentagons.The Greeks usually attributed discovery of this concept toPythagorasor hisfollowers.The regular pentagram, which has a regular pentagon inscribed within it, was the Pythagoreans' symbol.Euclid'sElements(Greek: 
Στοιχεῖα) provides the first known written def 
inition of what is now called thegolden ratio: "A straight line is said to have been cut in extreme and mean ratio when, as the whole lineis to the greater segment, so is the greater to the less." Euclid explains a construction for cutting(sectioning) a line "in extreme and mean ratio", i.e. the golden ratio. Throughout the Elements, severalpropositions(theoremsin modern terminology) and their proofs employ the golden ratio. Some of thesepropositions show that the golden ratio is anirrational number. The name "extreme and mean ratio" was the principal term used from the 3rd century BC until aboutthe 18th century.The modern history of the golden ratio starts withLuca Pacioli'sDe divina proportioneof 1509, which captured the imagination[citation needed]of artists, architects, scientists, and mystics with theproperties, mathematical and otherwise, of the golden ratio.

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