# The Golden Ratio

Topics: Golden ratio, Fibonacci number, Kepler triangle Pages: 23 (7054 words) Published: July 7, 2011
The Golden Ratio
By : Kaavya.K
In mathematics and the arts, two quantities are in the golden ratio if the ratio of the sum of the quantities to the larger quantity is equal to the ratio of the larger quantity to the smaller one. The golden ratio is an irrational mathematical constant, approximately 1.6180339887. Other names frequently used for the golden ratio are the golden section and golden mean. Other terms encountered include extreme and mean ratio, medial section, divine proportion, divine section, golden proportion, golden cut, golden number, and mean of Phidias. The golden ratio is often denoted by the Greek letter phi, usually lower case (φ).

[pic]

The golden section is a line segment divided according to the golden ratio: The total length a + b is to the longer segment a as a is to the shorter segment b.

The figure on the right illustrates the geometric relationship that defines this constant. Expressed algebraically: [pic]
This equation has one positive solution in the algebraic irrational number [pic]
At least since the Renaissance, many artists and architects have proportioned their works to approximate the golden ratio—especially in the form of the golden rectangle, in which the ratio of the longer side to the shorter is the golden ratio—believing this proportion to be aesthetically pleasing. Mathematicians have studied the golden ratio because of its unique and interesting properties. [pic]

Construction of a golden rectangle:
1. Construct a unit square (red).
2. Draw a line from the midpoint of one side to an opposite corner. 3. Use that line as the radius to draw an arc that defines the long dimension of the rectangle. | |

Calculation

|List of numbers – Irrational and suspected irrational numbers | |γ – ζ(3) – √2 – √3 – √5 – φ – ρ – δS – α – e – π – δ | |Binary |1.1001111000110111011… | |Decimal |1.6180339887498948482… | |Hexadecimal |1.9E3779B97F4A7C15F39… | |Continued fraction |[pic] | |Algebraic form |[pic] | |Infinite series |[pic] |

Two quantities a and b are said to be in the golden ratio φ if: [pic]
This equation unambiguously defines φ.
The fraction on the left can be converted to
[pic]
Multiplying through by φ produces
[pic]
which can be rearranged to
[pic]
The only positive solution to this quadratic equation is
[pic]

History

[pic]

Mathematician Mark Barr proposed using the first letter in the name of Greek sculptor Phidias, phi, to symbolize the golden ratio. Usually, the lowercase form (φ) is used. Sometimes, the uppercase form (Φ) is used for the reciprocal of the golden ratio, 1/φ. The golden ratio has fascinated Western intellectuals of diverse interests for at least 2,400 years. According to Mario Livio: Some of the greatest mathematical minds of all ages, from Pythagoras and Euclid in ancient Greece, through the medieval Italian mathematician Leonardo of Pisa and the Renaissance astronomer Johannes Kepler, to present-day scientific figures such as Oxford physicist Roger 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 even mystics have pondered and debated the basis of its ubiquity and appeal. In fact, it is probably fair to say that the Golden Ratio has inspired thinkers of all disciplines like no other number in the...