Very good introduction of Modern Math concept “Group” to secondary school math students by an American high school teacher.
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”。
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.
Positron is the anti-matter of Electron, born by symmetry of universe.
It disappears the moment it is created, to neutralise to nothingness.
Application in medicine: ‘PET‘ scanning (P= Positron).
Positron is smaller than electron, so can scan smaller tumor than CT Scan machine.
Maxwell boldy derived from Faraday experimental results by symmetry to get the Maxwell Equation for Electro-Magnetic Fields:
1) rot E = -1/c ∂H/∂t
div H = 0
By symmetry (swap E <-> H )
2) rot H = +1/c ∂E/∂t
div E = 0
E = Electric Field
H= Magnetic Field
c = Speed of light
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