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Golden ratio - Mathematical uses | A Wisdom Archive on Golden ratio - Mathematical uses | | Golden ratio - Mathematical uses A selection of articles related to Golden ratio - Mathematical uses | |
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Golden ratio, Golden ratio - A startlingly quick proof of irrationality, Golden ratio - Aesthetic uses, Golden ratio - Alternate forms, Golden ratio - Decimal expansion, Golden ratio - Definition, Golden ratio - History, Golden ratio - Mathematical uses, Golden angle, Golden function, Golden rectangle, Golden section search, Golden section (page proportion), Logarithmic spiral, Fibonacci number, Modulor, Sacred geometry, The Roses of Heliogabalus, Plastic number, Penrose tiles, Dynamic symmetry, Golden ratio base, Vitruvian man
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ARTICLES RELATED TO Golden ratio - Mathematical uses | | | |  Golden ratio - Mathematical uses: Encyclopedia II - Golden ratio - Mathematical usesThe number φ turns up frequently in geometry, in particular in figures involving pentagonal symmetry. For instance the ratio of a regular pentagon's side and diagonal is equal to φ, and the vertices of a regular icosahedron are located on three orthogonal golden rectangles.
The explicit expression for the Fibonacci sequence involves the golden ratio:
The limit of ratios of successive terms of the Fibonacci sequence (or any Fibonacci-like sequence) equals the golden ratio; therefore, when a number in the F ...
See also:Golden ratio, Golden ratio - Definition, Golden ratio - History, Golden ratio - A startlingly quick proof of irrationality, Golden ratio - Alternate forms, Golden ratio - Mathematical uses, Golden ratio - Aesthetic uses, Golden ratio - Decimal expansion Read more here: » Golden ratio: Encyclopedia II - Golden ratio - Mathematical uses |
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| | | | Golden ratio - Mathematical uses: Encyclopedia II - Golden ratio - Definition Two quantities are said to be in the golden ratio, if "the whole (i.e., the sum of the two parts) is to the larger part as the larger part is to the smaller part", i.e. if
where a is the larger part and b is the smaller part.
Equivalently, they are in the golden ratio if the ratio of the larger one to the smaller one equals the ratio of the smaller one to their difference, i.e. if
After multiplying the first equation with a/b or the second equation with (a − b)/b, both of these equations are se ...
See also:Golden ratio, Golden ratio - Definition, Golden ratio - History, Golden ratio - A startlingly quick proof of irrationality, Golden ratio - Alternate forms, Golden ratio - Mathematical uses, Golden ratio - Aesthetic uses, Golden ratio - Decimal expansion Read more here: » Golden ratio: Encyclopedia II - Golden ratio - Definition |
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| | | | Golden ratio - Mathematical uses: Encyclopedia II - Golden ratio - A startlingly quick proof of irrationality Recall that we denoted the "larger part" by a and the "smaller part" by b, and concluded that
This gives a startlingly quick proof that the golden ratio is an irrational number. An irrational number is one that cannot be written as a/b where a and b are integers. If a/b is such a fraction, in lowest terms, then b/(a − b) is in even lower terms — a contradict ...
See also:Golden ratio, Golden ratio - Definition, Golden ratio - History, Golden ratio - A startlingly quick proof of irrationality, Golden ratio - Alternate forms, Golden ratio - Mathematical uses, Golden ratio - Aesthetic uses, Golden ratio - Decimal expansion Read more here: » Golden ratio: Encyclopedia II - Golden ratio - A startlingly quick proof of irrationality |
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