Revision #1269 → #1834 · back to history
modifiedGolden ratio (lead definition)15899c69e484
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| anchors | [{"section":"(Lead)","snippet":"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"},{"type":"math_alttext","value":"{\\displaystyle {\\frac {a+b}{a}}={\\frac {a}{b}}=\\varphi ,}"}] | — |
modifiedMinimal polynomial of the golden ratiodf1ac2fba62e
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| anchors | [{"section":"Minimal polynomial","snippet":"the polynomial of lowest degree with integer coefficients that has the golden ratio as a root"},{"type":"math_alttext","value":"{\\displaystyle x^{2}-x-1.}"}] | — |
modifiedBoth roots are algebraic integers3e8e1be6b894
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| anchors | [{"section":"Minimal polynomial","snippet":"both roots are algebraic integers"},{"type":"math_alttext","value":"{\\displaystyle x^{2}-x-1.}"}] | — |
modifiedUnique property among positive numbers72baee029703
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| anchors | [{"section":"Golden ratio conjugate and powers","snippet":"This illustrates the unique property of the golden ratio among positive numbers"},{"type":"math_alttext","value":"{\\displaystyle {\\frac {1}{\\varphi }}=\\varphi -1,}"}] | — |
modifiedPowers satisfy Fibonacci-like recurrence6eb9627be70b
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| anchors | [{"section":"Golden ratio conjugate and powers","snippet":"is equal to the sum of the two immediately preceding powers"},{"type":"math_alttext","value":"{\\displaystyle \\varphi ^{n}=\\varphi ^{n-1}+\\varphi ^{n-2}=\\varphi \\cdot \\operatorname {F} _{n}+\\operatorname {F} _{n-1}.}"}] | — |
modifiedSimple continued fraction expansion6672564b3157
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| anchors | [{"section":"Continued fraction and square root","snippet":"can be expanded recursively to obtain a simple continued fraction for the golden ratio"},{"type":"math_alttext","value":"{\\displaystyle \\varphi =[1;1,1,1,\\dots ]=1+{\\cfrac {1}{1+{\\cfrac {1}{1+{\\cfrac {1}{1+{{\\vphantom {1}} \\atop \\ddots }}}}}}}}"}] | — |
modifiedHurwitz inequality for Diophantine approximationse066d2c4f7f3
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| anchors | [{"section":"Continued fraction and square root","snippet":"Hurwitz inequality for Diophantine approximations"},{"type":"math_alttext","value":"{\\displaystyle \\left|\\xi -{\\frac {p}{q}}\\right|<{\\frac {1}{{\\sqrt {5}}q^{2}}}.}"}] | — |
modifiedContinued square root forma02089c66c52
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| anchors | [{"section":"Continued fraction and square root","snippet":"A continued square root form"},{"type":"math_alttext","value":"{\\displaystyle \\varphi ={\\sqrt {1+{\\sqrt {\\textstyle 1+{\\sqrt {1+\\cdots {\\vphantom {)}}}}}}}}.}"}] | — |
modifiedFibonacci sequence recurrence5bac6c5d106a
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| anchor.snippet | each term | In the Fibonacci sequence, each term |
| provenance | ai | ai-moderated |
modifiedGolden ratio as limit of Fibonacci/Lucas ratiosd51bb962849f
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| anchors | [{"section":"Relationship to Fibonacci and Lucas numbers","snippet":"the golden ratio is equal to the limit of the ratios of successive terms in the Fibonacci sequence and sequence of Lucas numbers"},{"type":"math_alttext","value":"{\\displaystyle \\lim _{n\\to \\infty }{\\frac {F_{n+1}}{F_{n}}}=\\lim _{n\\to \\infty }{\\frac {L_{n+1}}{L_{n}}}=\\varphi .}"}] | — |
modifiedLimit of Lucas/Fibonacci quotient equals sqrt(5)9b5a71a66ede
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| anchors | [{"section":"Relationship to Fibonacci and Lucas numbers","snippet":"the limit of the quotient of Lucas numbers by Fibonacci numbers as equal to the square root of five"},{"type":"math_alttext","value":"{\\displaystyle \\lim _{n\\to \\infty }{\\frac {L_{n}}{F_{n}}}={\\sqrt {5}}.}"}] | — |
modifiedDiagonal-to-side ratio of regular pentagon7adccd307d95
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| anchors | [{"section":"Pentagonal symmetry system","snippet":"In a regular pentagon the ratio of a diagonal to a side is the golden ratio"},{"type":"math_alttext","value":"{\\displaystyle {\\frac {a}{b}}={\\frac {a+b}{a}}=\\varphi .}"}] | — |
modifiedKepler triangle26eca3acb2a9
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| anchors | [{"section":"In triangles and quadrilaterals","snippet":"is the unique right triangle with sides in geometric progression"},{"type":"math_alttext","value":"{\\displaystyle 1\\mathbin {:} {\\sqrt {\\varphi {\\vphantom {+}}}}\\mathbin {:} \\varphi .}"}] | — |
modifiedGolden spiralcf0466f615a2
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| anchors | [{"section":"Golden spiral","snippet":"A logarithmic spiral whose radius increases by a factor of the golden ratio for each quarter-turn is called the golden spiral"},{"type":"math_alttext","value":"{\\displaystyle r=\\varphi ^{2\\theta /\\pi }.}"}] | — |
modifiedGolden ratio in modular functions38110a6d8752
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| anchors | [{"section":"Other properties","snippet":"The golden ratio appears in the theory of modular functions"},{"type":"math_alttext","value":"{\\displaystyle R(q)={\\cfrac {q^{1/5}}{1+{\\cfrac {q}{1+{\\cfrac {q^{2}}{1+{\\cfrac {q^{3}}{1+{{\\vphantom {1}} \\atop \\ddots }}}}}}}}}.}"},{"type":"math_alttext","value":"{\\displaystyle R(e^{-2\\pi })={\\sqrt {\\varphi {\\sqrt {5}}}}-\\varphi ,\\quad R(-e^{-\\pi })=\\varphi ^{-1}-{\\sqrt {2-\\varphi ^{-1}}}}"},{"type":"math_alttext","value":"{\\displaystyle R(e^{-2\\pi i/\\tau })={\\frac {1-\\varphi R(e^{2\\pi i\\tau })}{\\varphi +R(e^{2\\pi i\\tau })}}}"}] | — |
addedQuadratic equation satisfied by φ961f88155f10
addedPowers of φ round to Lucas numbersc8fbdbb82a07
addedPolynomial in φ reduces to linear expression9f34a08f3b68
addedConvergents alternate above and below φ0098f320b2a5