Mathematician Emmy Noether changed the face of physics


Symmetry, Algebra and the “Monster”

Very good introduction of Modern Math concept “Group” to secondary school math students by an American high school teacher.


  • Symmetry of a Square
  • Isometry (*) or Rigid Motion (刚体运动) = no change in shape and size after a transformation
  • What is a Group (群 “CAN I” ) ? = Closure Associative Neutral Inverse
  • Monster Group = God ?
  • String Theory: Higgs boson (玻色子) aka “God Particles”

Note (*): “保距映射” (Isometry),是指在度量空间 (metric space) 之中保持距离不变的”同构“关系 (Isomorphism) 。几何学中的对应概念是 “全等变换”

Group is Symmetry

Landau’s book “Symmetry” explains it as follow:

Automorphism = Congruence= 叠合 has
1). Proper 真叠合 (symmetry: left= left, right = right)
2). Improper 非真叠合 (non-symmetry: reflection: left changed to right, vice-versa).
Congruence => preserve size / length
=> Movement 运动 (translation 平移, rotation about O )
= Proper congruence (Symmetry)

In Space S, the Automorphism that preserves the structure of S forms a Group Aut(G).
=> Group Aut(G) describes the Symmetry of Space S.

Hence Group is the language to describe Symmetry.


Noether Theorem: Symmetry

Symmetry (hence Group) explains :
1. Conservation of Energy;
2. Conservation of Angular Momentum;
3. Periodic Table;
4. Laws of Thermodynamic.

Emmy Noether Theorem (1918): Conservation Laws owes to Symmetry :
1. In Linear motion
=> Conservation of Momentum

2. In Angular movement
=> Conservation of Angular Momentum

3. In Time
=> Conservation of Energy