5차방정식과 정이십면체

이 항목의 스프링노트 원문주소

• 오차방정식과 정이십면체

 

 

개요

• 정이십면체의 대칭은 교대군 \(A_5\)

 

 

invariants of the icosahedral group

• Stereographic projections
• vertex points
  
  • \(V=F_1=z_1z_2(z_1^{10}+11z_1^5z_2^5-z_2^{10})\)
• 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
  \(F=F_3=(z_1^{30}+z_2^{30})+522(z_1^{25}z_2^{5}-z_1^{5}z_2^{25})-10005(z_1^{20}z_2^{10}+z_1^{10}z_2^{20})\)

 

syzygy relation

• \(1728F_1^5-F_2^3-F_3^2=0\)

 

 

Tschirnhaus transformation

• principal quintic
  \(z^5+5az^2+5bz+c=0\)
  

• \(Z=\frac{F_1^{5}}{F_3^{2}}=\frac{z^{5}(z^{10}+11z^5-1)^{5}}{((z^{30}+1)+522(z^{25}-z^{5})-10005(z^{20}+z^{10}))^{2}}\)

 

 

초기하급수를 이용한 해

• 초기하 미분방정식(Hypergeometric differential equations)
  \(z(1-z)\frac{d^2w}{dz^2}+(c-(a+b+1)z)\frac{dw}{dz}-abw = 0\)
  
• 슈워츠 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

• 1824 - 아벨이 일반적인 5차 이상의 방정식의 근의 공식이 없음을 증명함. 5차방정식의 근의 공식과 아벨의 증명 참조
• In 1858, Hermite and Kronecker solved the equation of the fifth degree by elliptic functions
• In 1877, Klein published Lectures on the Icosahedron and the Solution of Equations of the Fifth Degree.

• 힐버트의 1900년 국제수학자대회 연설의 초반부에 클라인의 오차방정식과 정이십면체에 대한 연구가 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]

• [1]http://www.google.com/search?hl=en&tbs=tl:1&q=quintic+equation
• http://www.google.com/search?q=quintic+equation+klein
• 수학사연표
•  

 

 

메모

• Hessian polyhedra, invariant theory and Appell hypergeometric partial differential equations

 

 

관련된 항목들

• 일반적인 5차 이상의 방정식의 대수적 해가 존재하지 않음에 대한 아벨의 증명
• 뫼비우스 변환군과 기하학
• 구면기하학
• 로저스-라마누잔 연분수와 항등식

 

 

사전형태의 자료

• http://en.wikipedia.org/wiki/Bring_radical

 

 

관련도서

• Lectures on the Icosahedron and the Solution of Equations of the Fifth Degree
  
  • Felix Klein, Part II. chapter III.
• Geometry of the Quintic
  
  • Jerry Shurman
  • 위 클라인 책의 일부 내용이 학부생들도 충분히 접근할 수 있도록 잘 쓰여짐.
• Beyond the Quartic Equation
  
  • Bruce King
• Finite Möbius groups, minimal immersions of spheres, and moduli
  
  • Gabor Toth, 66p

• Glimpses of Algebra and Geometry (Undergraduate Texts in Mathematics)
  
  • Gabor Toth

 

 

관련논문

• Solving the quintic by iteration
  
  • Peter Doyle and Curt McMullen
• Extensions icosaédriques (pdf)
  
  • J-P. Serre, Oeuvres III, p.550-554 (no. 123 (1980)), Springer, 1986

 

 

관련링크와 웹페이지

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

 

 

블로그

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