Golden ratio について

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Golden ratio ：ウィキペディア英語版

Golden ratio

In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. The figure on the right illustrates the geometric relationship. Expressed algebraically, for quantities ''a'' and ''b'' with ''a'' > ''b'' > 0,
:

\frac = \frac \ \stackrel\ \varphi,

where the Greek letter phi (

\varphi

\phi

) represents the golden ratio. Its value is:
:

\varphi = \frac = 1.6180339887\ldots.

The golden ratio also is called the golden mean or golden section (Latin: ''sectio aurea'').〔〔Richard A Dunlap, ''The Golden Ratio and Fibonacci Numbers'', World Scientific Publishing, 1997〕 Other names include extreme and mean ratio,〔Euclid, ''(Elements )'', Book 6, Definition 3.〕 medial section, divine proportion, divine section (Latin: ''sectio divina''), golden proportion, golden cut,〔Summerson John, ''Heavenly Mansions: And Other Essays on Architecture'' (New York: W.W. Norton, 1963) p. 37. "And the same applies in architecture, to the rectangles representing these and other ratios (e.g. the 'golden cut'). The sole value of these ratios is that they are intellectually fruitful and suggest the rhythms of modular design."〕 and golden number.〔Jay Hambidge, ''Dynamic Symmetry: The Greek Vase'', New Haven CT: Yale University Press, 1920〕〔William Lidwell, Kritina Holden, Jill Butler, ''Universal Principles of Design: A Cross-Disciplinary Reference'', Gloucester MA: Rockport Publishers, 2003〕〔Pacioli, Luca. ''De divina proportione'', Luca Paganinem de Paganinus de Brescia (Antonio Capella) 1509, Venice.〕
Some twentieth-century artists and architects, including Le Corbusier and Dalí, 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. The golden ratio appears in some patterns in nature, including the spiral arrangement of leaves and other plant parts.
Mathematicians since Euclid have studied the properties of the golden ratio, including its appearance in the dimensions of a regular pentagon and in a golden rectangle, which may be cut into a square and a smaller rectangle with the same aspect ratio. The golden ratio has also been used to analyze the proportions of natural objects as well as man-made systems such as financial markets, in some cases based on dubious fits to data.
==Calculation==
}}}
|-
| Algebraic form
|

\frac

|-
| Infinite series
|

\frac+\sum_^\fracTwo quantities ''a'' and ''b'' are said to be in the ''golden ratio'' ''φ'' if: \frac = \frac = \varphi.

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,
:

\frac = 1 + \frac = 1 + \frac.

Therefore,
:

1 + \frac = \varphi.

Multiplying by ''φ'' gives
:

\varphi + 1 = \varphi^2

which can be rearranged to
:

^2 - \varphi - 1 = 0.

Using the quadratic formula, two solutions are obtained:
:

\varphi = \frac = 1.61803\,39887\dots

and
:

\varphi = \frac = -0.6180\,339887\dots

Because ''φ'' is the ratio between positive quantities ''φ'' is necessarily positive:
:

\varphi = \frac = 1.61803\,39887\dots

.

抄文引用元・出典: フリー百科事典『ウィキペディア（Wikipedia）』
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