The term “Affine” was coined by Euler (1748b), motivated by the idea that images related by affine transformation have an affinity with one another.
Note 1: The term “Affine Geometry” is never used in GCE A-level Math, but commonly taught in French Baccalaureate.
Note 2: “Affinity” 亲和力 => 模仿
Affine transformation => 仿射 变化
In geometry, an affine transformation, affine map or an affinity (from the Latin, affinis, “connected with”) is a function between affine spaces which preserves points, straight lines and planes. Also, sets of parallel lines remain parallel after an affine transformation. An affine transformation does not necessarily preserve angles between lines or distances between points, though it does preserve ratios of distances between points lying on a straight line.
Examples of affine transformations include translation, scaling, homothety, similarity transformation, reflection, rotation, shear mapping, and compositions of them in any combination and sequence.
The sum of all positive integers is divergent, but the result is ( -1/12).
Crazy but it is true in Quantum Physics (String Theory) !
Reimann’s Hypothesis : Zeta function explained :
Why -1/12 is a gold nugget:
Prove: (Euler Gamma Γ Function)
Integrate by parts:
Feynman trick: differentiating under integral => d/da left side of 
Differentiate the right side of :
Continue to differentiate with respect to ‘a’:
Set a = 1
Another Example using “Feynman Integration”:
; for b > -1
I(0) = 0= ln (1) + C = 0+C => C=0
Let b= 2
The Calculus of Friendship: What a Teacher and a Student Learned about Life while Corresponding about Math
Visualize by mapping Exp e (red and pink arrows):
Euler was invited by Peter I of Russia in 1727 to work in the
Petersburg Academy of Sciences. He introduced the fundamental math
disciplines in school math education:
these 4 are taught as separate and specific subjects, versus 19 duplicated disciplines in Europe.
Euler influenced not only in Russia schools, but in schools worldwide today.
Source: Russian Mathematics Education
Vol 1: History and world significance
Vol 2: Programs and practices
(Publisher: World Scientific)
1. Euler wrote to Goldbach @1750, “it astonishes me these properties have not been noticed by anyone else.”
Euler’s Polyhedron formula:
V- E + R = 2
Remember trick: “VERsion 2”
V= Vertices, E= Edges, R= Regions (or Faces)
Note: = 0 (instead of 2) for torus doughnut
2. Euler proved it 1 year later: intuition leads to discovery, then prove it by logic.
Football: Vertices=60, Edges E=90, Region R=32 faces (12 pentagons, 20 hexagons):
V- E + R = 60 – 90 + 32 = 2
A football (or soccer ball) icon. (Photo credit: Wikipedia)