"황금비"의 두 판 사이의 차이
| 64번째 줄: | 64번째 줄: | ||
| − | <h5>유리수 | + | <h5>유리수 근사와 황금비</h5> |
An interesting result, stemming from the fact that the continued fraction expansion for [http://en.wikipedia.org/wiki/Golden_ratio φ] doesn't use any integers greater than 1, is that φ is one of the most "difficult" real numbers to approximate with rational numbers. One theorem<ref></ref> | An interesting result, stemming from the fact that the continued fraction expansion for [http://en.wikipedia.org/wiki/Golden_ratio φ] doesn't use any integers greater than 1, is that φ is one of the most "difficult" real numbers to approximate with rational numbers. One theorem<ref></ref> | ||
| 76번째 줄: | 76번째 줄: | ||
While virtually all real numbers <em>k</em> will eventually have infinitely many convergents <em>m</em>/<em>n</em> whose distance from <em>k</em> is significantly smaller than this limit, the convergents for φ (i.e., the numbers 5/3, 8/5, 13/8, 21/13, etc.) consistently "toe the boundary", keeping a distance of almost exactly <math>{\scriptstyle{1 \over n^2 \sqrt 5}}</math> away from φ, thus never producing an approximation nearly as impressive as, for example, 355/113 for π. It can also be shown that every real number of the form (<em>a</em> + <em>b</em>φ)/(<em>c</em> + <em>d</em>φ) – where <em>a</em>, <em>b</em>, <em>c</em>, and <em>d</em> are integers such that <em>ad</em> − <em>bc</em> = ±1 – shares this property with the golden ratio φ. | While virtually all real numbers <em>k</em> will eventually have infinitely many convergents <em>m</em>/<em>n</em> whose distance from <em>k</em> is significantly smaller than this limit, the convergents for φ (i.e., the numbers 5/3, 8/5, 13/8, 21/13, etc.) consistently "toe the boundary", keeping a distance of almost exactly <math>{\scriptstyle{1 \over n^2 \sqrt 5}}</math> away from φ, thus never producing an approximation nearly as impressive as, for example, 355/113 for π. It can also be shown that every real number of the form (<em>a</em> + <em>b</em>φ)/(<em>c</em> + <em>d</em>φ) – where <em>a</em>, <em>b</em>, <em>c</em>, and <em>d</em> are integers such that <em>ad</em> − <em>bc</em> = ±1 – shares this property with the golden ratio φ. | ||
| + | |||
| + | * [[연분수와 유리수 근사|연분수]] 항목을 참조 | ||
2009년 5월 16일 (토) 15:08 판
목차----
- 정오각형과 황금비#
- 황금비와 피보나치 수열##
- 황금비와 정이십면체##
- 연분수##
- 로저스-라마누잔 연분수##
- Dilogarithm##
- 재미있는 사실##
- 관련된 단원##
- 많이 나오는 질문##
- 관련된 고교수학 또는 대학수학##
- 관련된 다른 주제들##
- 관련도서 및 추천도서##
- 참고할만한 자료##
- 동영상##
- 관련기사##
정오각형과 황금비##
- 정오각형의 한 변의 길이와 대각선의 길이의 비율은 황금비가 된다.
[/pages/3002548/attachments/1344232 180px-Ptolemy_Pentagon.svg.png]
\({b \over a}={{(1+\sqrt{5})}\over 2}\)
황금비와 피보나치 수열###
[/pages/2252978/attachments/1347082 goldenrectangle.jpg]
황금비와 정이십면체###
[[|Golden rectangles in an icosahedron]]
- 황금사각형 세 개가 이루는 꼭지점이 정이십면체의 꼭지점이 된다
연분수###
\(\frac{1+\sqrt5}{2}=1 + \cfrac{1}{1 + \cfrac{1}{1 + \cfrac{1}{1 + \ddots}}}\)
유리수 근사와 황금비
An interesting result, stemming from the fact that the continued fraction expansion for φ doesn't use any integers greater than 1, is that φ is one of the most "difficult" real numbers to approximate with rational numbers. One theorem인용 오류: <ref> 태그가 잘못되었습니다;
이름이 없는 ref 태그는 반드시 내용이 있어야 합니다
states that any real number k can be approximated by rational m/n with
\[\left| k - {m \over n}\right| < {1 \over n^2 \sqrt 5}.\]
While virtually all real numbers k will eventually have infinitely many convergents m/n whose distance from k is significantly smaller than this limit, the convergents for φ (i.e., the numbers 5/3, 8/5, 13/8, 21/13, etc.) consistently "toe the boundary", keeping a distance of almost exactly \({\scriptstyle{1 \over n^2 \sqrt 5}}\) away from φ, thus never producing an approximation nearly as impressive as, for example, 355/113 for π. It can also be shown that every real number of the form (a + bφ)/(c + dφ) – where a, b, c, and d are integers such that ad − bc = ±1 – shares this property with the golden ratio φ.
- 연분수 항목을 참조
로저스-라마누잔 연분수###
\(\cfrac{1}{1 + \cfrac{e^{-2\pi}}{1 + \cfrac{e^{-4\pi}}{1+\dots}}} = \left({\sqrt{5+\sqrt{5}\over 2}-{\sqrt{5}+1\over 2}}\right)e^{2\pi/5} = e^{2\pi/5}\left({\sqrt{\varphi\sqrt{5}}-\varphi}\right) = 0.9981360\dots\)
Dilogarithm###
\(\mbox{Li}_{2}(\frac{3-\sqrt{5}}{2})=\frac{\pi^2}{15}-\log^2(\frac{1+\sqrt{5}}{2})\)
\(\mbox{Li}_{2}(\frac{-1+\sqrt{5}}{2})=\frac{\pi^2}{10}-\log^2(\frac{1+\sqrt{5}}{2})\)
\(\mbox{Li}_{2}(\frac{1-\sqrt{5}}{2})=-\frac{\pi^2}{15}+\frac{1}{2}\log^2(\frac{1+\sqrt{5}}{2})\)
\(\mbox{Li}_{2}(\frac{-1-\sqrt{5}}{2})=-\frac{\pi^2}{10}+\frac{1}{2}\log^2(\frac{1+\sqrt{5}}{2})\)
재미있는 사실###
관련된 단원###
많이 나오는 질문###
관련된 고교수학 또는 대학수학###
관련된 다른 주제들###
관련도서 및 추천도서###
참고할만한 자료###
- This Week's Finds in Mathematical Physics (Week 203)
- John Baez
- Misconceptions about the Golden Ratio
- George Markowsky
- The College Mathematics Journal, Vol. 23, No. 1 (Jan., 1992), pp. 2-19
- http://ko.wikipedia.org/wiki/황금비
- http://en.wikipedia.org/wiki/golden_ratio
- 다음백과사전 http://enc.daum.net/dic100/search.do?q=
동영상###
관련기사###
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