In Mathematics and the Art s, two quantities are in the Golden Ratio if the Ratio between the sum of those quantities and the larger one is the same as the ratio between the larger one and the smaller. The golden ratio is approximately 1.6180339887.
At least since the Renaissance , many Artist s and Architect s have proportioned their works to approximate the golden ratio—especially in the form of the , in which the ratio of the longer side to the shorter is the golden ratio—believing this proportion to be Aesthetically pleasing. Mathematician s have studied the golden ratio because of its unique and interesting properties.
The golden ratio can be expressed as a Mathematical Constant , usually denoted by the Greek letter ( Phi ). The figure of a illustrates the geometric relationship that defines this constant. Expressed algebraically:
:
This equation has as its unique positive solution the Algebraic Irrational Number
:
Other names frequently used for or closely related to the golden ratio are (Latin: ''sectio aurea''), '''golden mean''', '''golden number''', and the Greek letter '''phi''' (''''''.Jay Hambidge, ''Dynamic Symmetry: The Greek Vase'', New Haven CT: Yale University Press, 1920William Lidwell, Kritina Holden, Jill Butler, ''Universal Principles of Design: A Cross-Disciplinary Reference'', Gloucester MA: Rockport Publishers, 2003Pacioli, Luca. ''De divina proportione'', Luca Paganinem de Paganinus de Brescia (Antonio Capella) 1509, Venice.
:
1. Construct a unit square.
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.]]
proposed using the first letter in the name of Greek sculptor Phidias , ''phi'', to symbolize the golden ratio. Usually, the lowercase form () is used. Sometimes, the uppercase form () is used for the Reciprocal of the golden ratio, .]]
The golden ratio has fascinated intellectuals of diverse interests for at least 2,400 years:
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"http://wwwinformationdelightinfo/information/entry/Pythagoras" class="copylinks">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 history of mathematics
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and its reciprocal:
:
The
Convergent s of these continued fractions (1, 2, 3/2, 5/3, 8/5, 13/8, ..., or 1, 1/2, 2/3, 3/5, 5/8, 8/13, ...) are ratios of successive
Fibonacci Numbers .
The equation
likewise produces the continued
Square Root form:
:
Also:
:
:
:
These correspond to the fact that the length of the diagonal of a regular pentagon is φ times the length of its side, and similar relations in a
Pentagram .
If ''x'' agrees with
to ''n'' decimal places, then
agrees with it to 2''n'' decimal places.
An equation derived in 1994 connects the golden ratio to the
Number Of The Beast (666):x
:
Which can be combined into the expression:
:
.]]
The golden ratio can also be found by applying
Ptolemy's Theorem to the quadrilateral formed by removing one vertex from a regular pentagon. If the quadrilateral's long edge and diagonals are ''b'', and short edges are ''a'', then Ptolemy's theorem says ''b''
2 = ''a''
2 + ''ab'' which yields
The golden ratio is the
Limit of the ratios of successive terms of the Fibonacci sequence (or any Fibonacci-like sequence):
:
Therefore, if a Fibonacci number is divided by its immediate predecessor in the sequence, the quotient approximates φ; e.g., 987/610 ≈ 1.6180327868852. These approximations are alternately lower and higher than φ, and converge on φ as the Fibonacci numbers increase, and:
