"5차방정식과 정이십면체"의 두 판 사이의 차이

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*  Trott, M. "Solution of Quintics with Hypergeometric Functions." §3.13 in The Mathematica GuideBook for Symbolics. New York: Springer-Verlag, 2005.<br>http://books.google.com/books?id=3OtUpdFiXvkC&pg=PA1111&dq=icosahedron+and+quintic+mathematica&hl=ko&sa=X&ei=PMUIT4iUMqSLiAKtqaGNCQ&ved=0CDEQ6AEwAA#v=onepage&q=icosahedron%20and%20quintic%20mathematica&f=false[http://arxiv1.library.cornell.edu/abs/math/0412065v1 ]<br>
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*  Trott, M. "Solution of Quintics with Hypergeometric Functions." §3.13 in The Mathematica GuideBook for Symbolics. New York: Springer-Verlag, 2005.<br>http://books.google.com/books?id=3OtUpdFiXvkC&pg=PA1111&dq=icosahedron+and+quintic+mathematica&hl=ko&sa=X&ei=PMUIT4iUMqSLiAKtqaGNCQ&ved=0CDEQ6AEwAA#v=onepage&q=icosahedron%20and%20quintic%20mathematica&f=false<br>
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* [http://library.wolfram.com/examples/quintic/ Solving the Quintic with Mathematica]
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* http://books.google.com/books?id=txinPHIegGgC&pg=PA86&lpg=PA86&dq=icosahedral+equation+hypergeometric&source=bl&ots=moFmb96tvZ&sig=-_Ge7VpPR8mycWMJBZpcthe59cY&hl=en&sa=X&ei=gdMIT_nuB5LUiAKS4pGSCQ&ved=0CDEQ6AEwAg#v=onepage&q=icosahedral%20equation%20hypergeometric&f=false
  
 
 
 
 

2012년 1월 8일 (일) 09:34 판

이 항목의 스프링노트 원문주소

 

 

개요
  • 정이십면체의 대칭은 교대군 \(A_5\)
  • 생성원
    \(S=\begin{pmatrix} \zeta_{10} & 0 \\ 0 & \zeta_{10} \end{pmatrix} \), \(T={\begin{pmatrix} -1 & g \\ g & 1 \end{pmatrix}}\),  \(g=\frac{\sqrt{5}-1}{2}\)

 

 

invariants of the icosahedral group
  • vertex points
    \(V=F_1=z_1z_2(z_1^{10}+11z_1^5z_2^5-z_2^{10})\)
  • face points
    \(F=F_{2}=-(z_1^{20}+z_2^{20})+228(z_1^{15}z_2^{5}-z_1^{5}z_2^{15})-494z_1^{10}z_2^{10}\)
  • edge points
    \(E=F_3=(z_1^{30}+z_2^{30})+522(z_1^{25}z_2^{5}-z_1^{5}z_2^{25})-10005(z_1^{20}z_2^{10}+z_1^{10}z_2^{20})\)

 

 

syzygy relation
  • \(1728F_1^5-F_2^3-F_3^2=0\) 또는 \(1728V^5-E^2-F^3=0\)

 

 

Tschirnhaus transformation
  • Tschirnhaus transformation
  • principal quintic
    \(z^5+5az^2+5bz+c=0\)
  • 정이십면체 방정식(icosahedral equation)
    \(Z=\frac{F_1^{5}}{F_3^{2}}=\frac{z^{5}(z^{10}+11z^5-1)^{5}}{((z^{30}+1)+522(z^{25}-z^{5})-10005(z^{20}+z^{10}))^{2}}\)

 

 

초기하급수를 이용한 해
  • 초기하 미분방정식(Hypergeometric differential equations)
    \(z(1-z)\frac{d^2w}{dz^2}+(c-(a+b+1)z)\frac{dw}{dz}-abw = 0\)
  • 슈바르츠 삼각형 함수 (s-함수)
  • \(s(z)=\frac{z^{1-c}\,_2F_1(a',b';c';z)}{\,_2F_1(a,b;c;z)}=\frac{z^{1-c}\,_2F_1(a-c+1,b-c+1;2-c;z)}{\,_2F_1(a,b;c;z)}\)
  • \(\alpha=1-c,\beta=b-a,\gamma=c-a-b\) 로 두면, 상반평면을 \(\alpha\pi,\beta\pi,\gamma\pi\) 를 세 각으로 갖는 삼각형인 경우가 된다
  • \(\alpha=1/5, \beta=1/2, \gamma=1/3\) 로 두면, \(a=-1/60,b=29/60,c=4/5\) 를 얻는다
  • \(a=-1/60,b=29/60,c=4/5\) 를 이용하면,
    \(\frac{Z^{1/5}\,_2F_1(11/60,41/60;6/5;Z)}{\,_2F_1(-1/60,29/60;4/5;Z)}\)

 

 

역사
  • 1900 - 힐버트가 국제수학자대회 연설의 초반부에 클라인의 오차방정식과 정이십면체에 대한 연구를 언급
  • Mathematical Problems
    • Lecture delivered before the International Congress of Mathematicians at Paris in 1900 By Professor David Hilbert

But it often happens also that the same special problem finds application in the most unlike branches of mathematical knowledge. So, for example, the problem of the shortest line plays a chief and historically important part in the foundations of geometry, in the theory of curved lines and surfaces, in mechanics and in the calculus of variations. And how convincingly has F. Klein, in his work on the icosahedron, pictured the significance which attaches to the problem of the regular polyhedra in elementary geometry, in group theory, in the theory of equations and in that of linear differential equations.  

[/pages/2026224/attachments/2671447 icos1.jpg][/pages/2026224/attachments/2671449 icos2.jpg]

 

 

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