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Golden ratio - Alternate forms | | Golden ratio - Alternate forms: Encyclopedia II - Golden ratio - Alternate forms | | The formula can be expanded recursively to obtain a continued fraction for the golden ratio:
and its reciprocal:
Note that the successive convergents of these continued fractions are ratios of 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 ...
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 | | | 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 | | |
| | | Golden ratio: Encyclopedia II - Golden ratio - Alternate forms
Golden ratio - Alternate forms
The formula can be expanded recursively to obtain a continued fraction for the golden ratio:
and its reciprocal:
Note that the successive convergents of these continued fractions are ratios of 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.
Other related archivesA4, Acropolis, Architect, Architecture, Béla Bartók, Dynamic symmetry, Egyptians, Erik Satie, Euclid, Fibonacci number, Fibonacci numbers, Fibonacci sequence, For Ann (rising), Golden angle, Golden function, Golden mean base, Golden ratio base, Golden rectangle, Golden section (page proportion), Golden section search, Greek letter, Hippasus of Metapontum, James Tenney, Lagrange, Le Corbusier, Leonardo Da Vinci, Logarithmic spiral, Luca Pacioli, Modulor, OEIS, Parthenon, Penrose tiles, Phidias, Pi, Pisot-Vijayaraghavan number, Plastic number, Pyramids of Giza, Pythagoras, Pythagoreans, Sacred geometry, Shepard tone, The Roses of Heliogabalus, Theodorus, Venus, Vitruvian man, acoustic scale, aesthetically, algebraic number field, ancient Egyptians, ancient Greeks, approximation theorem, beauty, continued fraction, equal tempered, fifth, geometry, glissandoing, golden rectangles, icosahedron, integers, irrational number, justly tuned, limit, lowest terms, major sixth, minor, numeral system, octave, paper sizes, pentagon, pentagram, phi, phyllotaxis, quadratic equation, ratio, reciprocal, recurrence, rule of thirds, rule of thumb, sixths, square root, symmetry, tau, woodwork
 Adapted from the Wikipedia article "Alternate forms", under the G.N U Free Docmentation License. Please also see http://en.wikipedia.org/wiki |
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