황금비

목차----

1. 정오각형과 황금비#
2. 황금비와 피보나치 수열##
3. 황금비와 정이십면체##
4. 연분수##
5. 로저스-라마누잔 연분수##
6. Dilogarithm##
7. 재미있는 사실##
8. 관련된 단원##
9. 많이 나오는 질문##
10. 관련된 고교수학 또는 대학수학##
11. 관련된 다른 주제들##
12. 관련도서 및 추천도서##
13. 참고할만한 자료##
14. 동영상##
15. 관련기사##

 

 

정오각형과 황금비^##

• 정오각형의 한 변의 길이와 대각선의 길이의 비율은 황금비가 된다.

[/pages/3002548/attachments/1344232 180px-Ptolemy_Pentagon.svg.png]

 

\({b \over a}={{(1+\sqrt{5})}\over 2}\)

• 정오각형

 

 

황금비와 피보나치 수열^###

[/pages/2252978/attachments/1347082 goldenrectangle.jpg]

• 피보나치 수열의 여러가지 성질

 

 

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 근사과 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 theorem

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.

 

 

로저스-라마누잔 연분수^###

\(\cfrac{1}{1 + \cfrac{e^{-2\pi}}{1 + \cfrac{e^{-4\pi}}{1+\dots}}} = \left({\sqrt{5+\sqrt{5}\over 2}-{\sqrt{5}+1\over 2}}\right)e^{2\pi/5} = e^{2\pi/5}\left({\sqrt{\varphi\sqrt{5}}-\varphi}\right) = 0.9981360\dots\)

• 라마누잔의 연분수

 

Dilogarithm^###

\(\mbox{Li}_{2}(\frac{3-\sqrt{5}}{2})=\frac{\pi^2}{15}-\log^2(\frac{1+\sqrt{5}}{2})\)

\(\mbox{Li}_{2}(\frac{-1+\sqrt{5}}{2})=\frac{\pi^2}{10}-\log^2(\frac{1+\sqrt{5}}{2})\)

\(\mbox{Li}_{2}(\frac{1-\sqrt{5}}{2})=-\frac{\pi^2}{15}+\frac{1}{2}\log^2(\frac{1+\sqrt{5}}{2})\)

\(\mbox{Li}_{2}(\frac{-1-\sqrt{5}}{2})=-\frac{\pi^2}{10}+\frac{1}{2}\log^2(\frac{1+\sqrt{5}}{2})\)

• Dilogarithm

 

 

 

 

 

 

많이 나오는 질문^###

• 네이버 지식인
  
  • http://kin.search.naver.com/search.naver?where=kin_qna&query=황금비

 

 

 

관련된 다른 주제들^###

• 정오각형
• 정다면체
• 피보나치 수열의 여러가지 성질
• Dilogarithm
• 라마누잔의 연분수

 

 

 

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^###

• http://www.youtube.com/results?search_type=&search_query=

 

관련기사^###

네이버 뉴스 검색 (키워드 수정)

• http://news.search.naver.com/search.naver?where=news&x=0&y=0&sm=tab_hty&query=
• http://news.search.naver.com/search.naver?where=news&x=0&y=0&sm=tab_hty&query=