Golden_ratio.pdf - Golden ratio In mathematics two quantities are in thegolden ratio if their ratio is the same as the ratio of their sum to the larger

Golden_ratio.pdf - Golden ratio In mathematics two...

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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, where the Greek letter phi ( or ) represents the golden ratio. [1] It is an irrational number with a value of: [2] The golden ratio is also called the golden mean or golden section (Latin: sectio aurea ). [3][4][5] Other names include extreme and mean ratio , [6] medial section , divine proportion , divine section (Latin: sectio divina ), golden proportion , golden cut , [7] and golden number . [8][9][10] 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. [11] The golden ratio appears in some patterns in nature, including the spiral arrangement of leaves and other plant parts. Some twentieth-century artists and architects, including Le Corbusier and Salvador 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. Calculation History Timeline Applications and observations Aesthetics Architecture Painting Book design Design Music Nature Optimization Line segments in the golden ratio A golden rectangle with longer side a 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 illustrates the relationship . Contents
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List of numbers · Irrational numbers ζ (3) · 2 · 3 · 5 · φ · e · π Binary 1.1001111000110111011... Decimal 1.6180339887498948482... [2] Hexadecimal 1.9E3779B97F4A7C15F39... Continued fraction Algebraic form Infinite series Perceptual studies Mathematics Irrationality Minimal polynomial Golden ratio conjugate Alternative forms Geometry Relationship to Fibonacci sequence Symmetries Other properties Decimal expansion Pyramids Mathematical pyramids and triangles Egyptian pyramids Disputed observations See also References and footnotes Further reading External links Two quantities a and b are said to be in the golden ratio φ if 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 , Therefore, Multiplying by φ gives which can be rearranged to Using the quadratic formula, two solutions are obtained: and Calculation
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Because φ is the ratio between positive quantities φ is necessarily positive: .
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