"5차방정식과 정이십면체"의 두 판 사이의 차이

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10번째 줄: 10번째 줄:
  
 
* [[정이십면체 뫼비우스 변환군]]
 
* [[정이십면체 뫼비우스 변환군]]
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*  vertex points<br>
+
*  vertex points
 
** <math>V=F_ 1=z_ 1z_ 2(z_ 1^{10}+11z_ 1^5z_ 2^5-z_ 2^{10})</math>
 
** <math>V=F_ 1=z_ 1z_ 2(z_ 1^{10}+11z_ 1^5z_ 2^5-z_ 2^{10})</math>
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*  face points<br>
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*  face points
 
** <math>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}</math>
 
** <math>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}</math>
−
*  edge points<br>
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*  edge points
 
** <math>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})</math>
 
** <math>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})</math>
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*  syzygy relation:<math>1728F_ 1^5-F_ 2^3-F_ 3^2=0</math> 또는 <math>1728V^5-E^2-F^3=0</math><br>
+
*  syzygy relation:<math>1728F_ 1^5-F_ 2^3-F_ 3^2=0</math> 또는 <math>1728V^5-E^2-F^3=0</math>
  
 
   
 
   
32번째 줄: 32번째 줄:
 
==정이십면체 방정식과 초기하급수 해==
 
==정이십면체 방정식과 초기하급수 해==
  
−
*  정이십면체 방정식(icosahedral equation):<math>w=\frac{V (z)^{5}}{E (z)^{2}}=\frac{z^{5}(z^{10}+11z^5-1)^{5}}{((z^{30}+1)+522(z^{25}-z^{5})-10005(z^{20}+z^{10}))^{2}}</math><br> 다시 쓰면, :<math>z^5 \left(z^{10}+11 z^5-1\right)^5-w \left(z^{30}+522 \left(z^{25}-z^5\right)-10005 \left(z^{20}+z^{10}\right)+1\right)^2=0</math><br> 또는 :<math>w z^{60}+1044 w z^{55}+252474 w z^{50}+\cdots =0</math><br>
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*  정이십면체 방정식(icosahedral equation):<math>w=\frac{V (z)^{5}}{E (z)^{2}}=\frac{z^{5}(z^{10}+11z^5-1)^{5}}{((z^{30}+1)+522(z^{25}-z^{5})-10005(z^{20}+z^{10}))^{2}}</math> 다시 쓰면, :<math>z^5 \left(z^{10}+11 z^5-1\right)^5-w \left(z^{30}+522 \left(z^{25}-z^5\right)-10005 \left(z^{20}+z^{10}\right)+1\right)^2=0</math> 또는 :<math>w z^{60}+1044 w z^{55}+252474 w z^{50}+\cdots =0</math>
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*  이 60차방정식의 해는 초기하급수를 사용하여 표현할 수 있다:<math>z=\frac{\, _ 2F_ 1\left(-\frac{1}{60},\frac{29}{60};\frac{4}{5};1728 w \right)}{w^{1/5} \, _ 2F_ 1\left(\frac{11}{60},\frac{41}{60};\frac{6}{5};1728 w \right)}</math><br>
+
*  이 60차방정식의 해는 초기하급수를 사용하여 표현할 수 있다:<math>z=\frac{\, _ 2F_ 1\left(-\frac{1}{60},\frac{29}{60};\frac{4}{5};1728 w \right)}{w^{1/5} \, _ 2F_ 1\left(\frac{11}{60},\frac{41}{60};\frac{6}{5};1728 w \right)}</math>
  
 
   
 
   
41번째 줄: 41번째 줄:
 
==슈바르츠 삼각형 함수==
 
==슈바르츠 삼각형 함수==
  
−
* [[초기하 미분방정식(Hypergeometric differential equations)]]:<math>z (1-z)\frac{d^2w}{dz^2}+(c-(a+b+1)z)\frac{dw}{dz}-abw = 0</math><br>
+
* [[초기하 미분방정식(Hypergeometric differential equations)]]:<math>z (1-z)\frac{d^2w}{dz^2}+(c-(a+b+1)z)\frac{dw}{dz}-abw = 0</math>
−
* [[슈바르츠 삼각형 함수|슈바르츠 삼각형 함수 (s-함수)]]:<math>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)}</math><br>
+
* [[슈바르츠 삼각형 함수|슈바르츠 삼각형 함수 (s-함수)]]:<math>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)}</math>
−
* <math>\alpha=1-c,\beta=b-a,\gamma=c-a-b</math> 로 두면, 상반평면을 <math>\alpha\pi,\beta\pi,\gamma\pi</math> 를 세 각으로 갖는 삼각형인 경우가 된다<br>
+
* <math>\alpha=1-c,\beta=b-a,\gamma=c-a-b</math> 로 두면, 상반평면을 <math>\alpha\pi,\beta\pi,\gamma\pi</math> 를 세 각으로 갖는 삼각형인 경우가 된다
−
* <math>\alpha=1/5, \beta=1/2, \gamma=1/3</math> 로 두면, <math>a=-1/60,b=29/60,c=4/5</math> 를 얻는다<br>
+
* <math>\alpha=1/5, \beta=1/2, \gamma=1/3</math> 로 두면, <math>a=-1/60,b=29/60,c=4/5</math> 를 얻는다
−
* <math>a=-1/60,b=29/60,c=4/5</math> 를 이용하면,:<math>\frac{Z^{1/5}\,_ 2F_ 1(11/60,41/60;6/5;Z)}{\,_ 2F_ 1(-1/60,29/60;4/5;Z)}</math><br>
+
* <math>a=-1/60,b=29/60,c=4/5</math> 를 이용하면,:<math>\frac{Z^{1/5}\,_ 2F_ 1(11/60,41/60;6/5;Z)}{\,_ 2F_ 1(-1/60,29/60;4/5;Z)}</math>
  
 
   
 
   
59번째 줄: 59번째 줄:
  
 
* 1900 - 힐버트가 국제수학자대회 연설의 초반부에 클라인의 오차방정식과 정이십면체에 대한 연구를 언급
 
* 1900 - 힐버트가 국제수학자대회 연설의 초반부에 클라인의 오차방정식과 정이십면체에 대한 연구를 언급
−
* [http://aleph0.clarku.edu/%7Edjoyce/hilbert/problems.html Mathematical Problems]<br>
+
* [http://aleph0.clarku.edu/%7Edjoyce/hilbert/problems.html Mathematical Problems]
 
** Lecture delivered before the International Congress of Mathematicians at Paris in 1900 By Professor David Hilbert
 
** 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.'''
 
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.'''
74번째 줄: 74번째 줄:
 
* Nagano, Atsuhira. “Icosahedral Invariants, CM Points and Class Fields.” arXiv:1504.07500 [math], April 28, 2015. http://arxiv.org/abs/1504.07500.
 
* Nagano, Atsuhira. “Icosahedral Invariants, CM Points and Class Fields.” arXiv:1504.07500 [math], April 28, 2015. http://arxiv.org/abs/1504.07500.
  
−
*  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 %20 and %20 quintic %20 mathematica&f=false<br>
+
*  Trott, M. "Solution of Quintics with Hypergeometric Functions." §3 .13 in The Mathematica GuideBook for Symbolics. New York: Springer-Verlag, 2005.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 %20 and %20 quintic %20 mathematica&f=false
 
* [http://library.wolfram.com/examples/quintic/ Solving the Quintic with Mathematica]
 
* [http://library.wolfram.com/examples/quintic/ Solving the Quintic with Mathematica]
 
* 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 %20 equation %20 hypergeometric&f=false
 
* 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 %20 equation %20 hypergeometric&f=false
143번째 줄: 143번째 줄:
 
==블로그==
 
==블로그==
  
−
* [http://bomber0.byus.net/index.php/2008/10/07/813 펠릭스 클라인 : Lectures on the Icosahedron and the Solution of Equations of the Fifth Degree]<br>
+
* [http://bomber0.byus.net/index.php/2008/10/07/813 펠릭스 클라인 : Lectures on the Icosahedron and the Solution of Equations of the Fifth Degree]
 
** 피타고라스의 창
 
** 피타고라스의 창
 
[[분류:방정식과 근의 공식]]
 
[[분류:방정식과 근의 공식]]

2020년 11월 12일 (목) 23:16 판

개요



정이십면체 뫼비우스 변환군의 불변량

  • 정이십면체 뫼비우스 변환군
  • 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 을 이용하여 일반적인 5차방정식 \(x^5+Ax^4+Bx^3+Cx^2+Dx+E=0\) 을 principal quintic 즉, \(z^5+5az^2+5bz+c=0\) 형태로 바꿀 수 있다



정이십면체 방정식과 초기하급수 해

  • 정이십면체 방정식(icosahedral equation)\[w=\frac{V (z)^{5}}{E (z)^{2}}=\frac{z^{5}(z^{10}+11z^5-1)^{5}}{((z^{30}+1)+522(z^{25}-z^{5})-10005(z^{20}+z^{10}))^{2}}\] 다시 쓰면, \[z^5 \left(z^{10}+11 z^5-1\right)^5-w \left(z^{30}+522 \left(z^{25}-z^5\right)-10005 \left(z^{20}+z^{10}\right)+1\right)^2=0\] 또는 \[w z^{60}+1044 w z^{55}+252474 w z^{50}+\cdots =0\]
  • 이 60차방정식의 해는 초기하급수를 사용하여 표현할 수 있다\[z=\frac{\, _ 2F_ 1\left(-\frac{1}{60},\frac{29}{60};\frac{4}{5};1728 w \right)}{w^{1/5} \, _ 2F_ 1\left(\frac{11}{60},\frac{41}{60};\frac{6}{5};1728 w \right)}\]



슈바르츠 삼각형 함수

  • 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.


2026224-icos1.jpg 2026224-icos2.jpg


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관련논문

  • Crass, Scott. 2014. “Dynamics of a Soccer Ball.” arXiv:1404.3170 [math], April. http://arxiv.org/abs/1404.3170.
  • Yang, Lei. 2004. “Hessian Polyhedra, Invariant Theory and Appell Hypergeometric Partial Differential Equations.” arXiv:math/0412065,
  • Crass, Scott. 1999. “Solving the Quintic by Iteration in Three Dimensions.” arXiv:math/9903054, March. http://arxiv.org/abs/math/9903054.
  • Doyle, Peter, and Curt McMullen. 1989. “Solving the Quintic by Iteration.” Acta Mathematica 163 (1): 151–80. doi:10.1007/BF02392735.
  • J-P. Serre, Extensions icosaédriques (pdf), Oeuvres III, p .550-554 (no. 123 (1980)), Springer, 1986



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