Timeline for Why is there a $\sqrt{5}$ in Hurwitz's Theorem?
Current License: CC BY-SA 3.0
9 events
| when toggle format | what | by | license | comment | |
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| Jul 10, 2015 at 2:40 | comment | added | Halbort | Why was I downvoted? | |
| Jul 9, 2015 at 19:43 | history | edited | Halbort |
edited tags
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| Jul 7, 2015 at 22:45 | comment | added | Halbort | Here is a link to the article mentioned by Marty: cimat.mx/~gil/docencia/2008/elementales/circulos_ford.pdf | |
| Jul 7, 2015 at 22:41 | vote | accept | Halbort | ||
| Jul 7, 2015 at 21:40 | comment | added | Marty | My favorite proof is in Ford's aptly titled article "Fractions" (Amer. Math. Monthly, Vol 45, No 9 (Nov 1938)). He gives the "Ford circle" proof of Dirichlet's approximation theorem, and the $\sqrt{5}$ comes straight out of the geometry he uses. So, if "visual" suffices for "intuitive," this might suffice for your needs. | |
| Jul 7, 2015 at 21:28 | answer | added | Myshkin | timeline score: 27 | |
| Jul 7, 2015 at 19:22 | comment | added | Terry Tao | It relates to the Golden Ratio $\phi = \frac{1+\sqrt{5}}{2} = 1 + 1/ (1 + 1/ (1 + 1/ \dots))$, which is the "most badly approximable" irrational as its continued fraction has the lowest possible denominators. The rational approximants to $\phi$ are given by ratios $F_{n+1}/F_n$ of Fibonacci numbers $F_n = (\phi^n - (-\phi)^{-n})/\sqrt{5}$, which is basically where the $\sqrt{5}$ of Hurwitz's theorem arises from. | |
| Jul 7, 2015 at 19:20 | history | edited | Emil Jeřábek | CC BY-SA 3.0 |
fix spelling of a name
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| Jul 7, 2015 at 19:10 | history | asked | Halbort | CC BY-SA 3.0 |