5차방정식과 정이십면체

 

 

 

 

invariants of the icosahedral group

• Stereographic projections
• vertex points
  
  • \(F_1=z_1z_2(z_1^{10}+11z_1^5z_2^5-z_2^{10})\)
• face points
  
  • \(F_2=-(z_1^{20}+z_2^{20})+228(z_1^{15}z_2^{5}-z_1^{5}z_2^{15})-494z_1^{10}z_2^{10}\)
• edge points
  
  • \(F_3=(z_1^{30}+z_2^{30})+522(z_1^{25}z_2^{5}-z_1^{5}z_2^{25})-10005(z_1^{20}z_2^{10}+z_1^{10}z_2^{20})\)

 

 

syzygy relation

• \(1728F_1^5-F_2^3-F_3^2=0\)
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역사

20세기 수학의 궤도를 제시한 힐버트의 역사적인 1900년 국제수학자대회 연설의 초반부에는 다음과 같은 언급이 있음. (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.

 

 

 

 

관련링크와 웹페이지

• http://library.wolfram.com/examples/quintic/main.html
• http://mathworld.wolfram.com/IcosahedralEquation.html

 

 

관련된 다른 주제들

• 일반적인 5차 이상의 방정식의 대수적 해가 존재하지 않음에 대한 아벨의 증명
• 뫼비우스 변환군과 기하학
• 구면기하학

 

 

표준적인 도서 및 추천도서

• Lectures on the Icosahedron and the Solution of Equations of the Fifth Degree
  
  • Felix Klein
• Geometry of the Quintic
  
  • Jerry Shurman
  • 위 클라인 책의 일부 내용이 학부생들도 충분히 접근할 수 있도록 잘 쓰여짐.
• Beyond the Quartic Equation
  
  • Bruce King

 

 

관련논문

• Solving the quintic by iteration
  
  • Peter Doyle and Curt McMullen
• Extensions icosaédriques (pdf)
  
  • J-P. Serre, Oeuvres III (no. 123 (1980)), Springer, 1986

 

 

블로그

• 펠릭스 클라인 : Lectures on the Icosahedron and the Solution of Equations of the Fifth Degree
  
  • 피타고라스의 창