숫자 12와 24

간단한 소개

• 수학에서 숫자 12와 24는 매우 흥미로운 수.
• 모듈라 군(modular group)과 깊게 관련되어 있음.
  
  • 12 = cusp form이 가질수 있는 가장 작은 weight
    \(\Delta(\tau)= q\prod_{n>0}(1-q^n)^{24}= q-24q+252q^2+\cdots\)
    는 weight 12 cusp form
    
  • \(SL(2,Z)_{ab}=C_{12}\)
  • \(\chi(SL(2,Z))=-\frac{1}{12}\)
• 24
  
  • The Eisenstein series
    \(E_2=1+24\sum_{n=1}^{\infty}\sigma_1(n)q^n\)
     
    
  • Leech 격자의 차원
  • Sporadic group M24
  • If we take a double cover Mp2(Z) of SL2(Z), we have (Mp2(Z))ab = Z/24.
• 26=24+2 is the critical dimension in bosonic string theory

 

하위주제들

• Square pyramid puzzles
• 리만제타함수의 -1에서의 값
  
  • \(\zeta(-1)= -\frac{1}{12}\)
• 정수계수 2x2 행렬군의 분류
• Leech lattice
• Bosonic string theory
• Lattice polygons

 

 

 

 

 

관련된 다른 주제들

• 라마누잔의 수학
• j-invariant

 

 

 

위키링크

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

 

참고할만한 자료

• Lattice Polygons and the Number 12
  
  • Bjorn Poonen and Fernando Rodriguez-Villegas
  • The American Mathematical Monthly, Vol. 107, No. 3 (Mar., 2000), pp. 238-250
• My Favorite Numbers : 5, 8, and 24
  
  • John Baez
  • The Rankin Lectures, University of Glasgow, September 15-19, 2008
• A short proof of the twelve-point theorem
  
  • Repovscaron D.; Skopenkov M.; Cencelj M.
  • Mathematical Notes, Volume 77, Number 1, January 2005 , pp. 108-111(4)
• The Square Pyramid Puzzle
  
  • W. S. Anglin
  • The American Mathematical Monthly, Vol. 97, No. 2 (Feb., 1990), pp. 120-124
• This Week's Finds in Mathematical Physics (Week 125)
  
  • John Baez
  • November 3, 1998
• Picard Groups of Moduli Problems
  
  • David Mumford
  • Arithmetical Algebraic Geometry, Proceedings of a Conference Held at Purdue University