숫자 12와 24

간단한 소개

• 수학에서 숫자 12와 24는 매우 흥미로운 수.
• 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
    
• 26= Critical dimension in bosonic string theory
  25−1 = 24 is related to:
  1. D(t) = h(t)24, where D is the discriminant cusp form of weight 12, and h is the
  Dedekind function.
  2. The constant term of j(t)−720 = q−1 +24+196884q+. . . , which is the “right
  normalization” of this modular function.
  3. The Eisenstein series E2 = 1−24ås1(n)qn.
  4. Dimension of the Leech Lattice.
  5. Sporadic group M24
  6. If we take a double cover Mp2(Z) of SL2(Z), we have (Mp2(Z))ab = Z/24.

 

 

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