Pi hiding in prime regularities

Three mysterious Math Objects:

  • Pi, 
  • Complex Numbers, 
  • Prime Numbers

are hiding in circle.

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Circle in Different Representations

1-Dimensional Objects:
Affine Line: {\mathbb {A}^1}
Circle: {\mathbb {S}^1}

Six Representations of a Circle: {\mathbb {S}^1}
1) Euclidean Geometry (O-level Math)
Unit Circle : x^2 + y^2 = 1

2) Curve: (A-level Math)
Transcendental Parameterization :
\boxed { e(\theta) = (\cos \theta, \sin  \theta) \qquad  0 \leq \theta \leq 2\pi }

Rational Parameterisation :
\boxed { e(h) = \left(\frac {1-h^2} {1+h^2}  \: , \: \frac {2h} {1+h^2}\right) \quad \text { h any number or } \infty }

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3) Affine Plane (French Baccalaureate – equivalent A-level – Math) {\mathbb {A}^2}
1-Dim Sub-spaces = Projective Lines thru’ Origin

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4) Polygonal Representation (Undergraduate Math)

5) Identifying Intervals: (closed loop) (Undergraduate Math)
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6) \text {Translation } (\tau, {\tau}^{-1}) \text { on a Line } {\mathbb {A}^1}
(Honors Year Undergraduate / Graduate Math)

[Using Quotient Group Notation]:
\boxed {  {\mathbb {S}^1} \simeq {\mathbb {A}^1 } \Big/ { \langle \tau , {\tau}^{-1} \rangle} }
{\mathbb {S}^1} = \text { Space of all orbits}

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Question:
Are Circle and Line the same 1-dimensional object, i.e. are they Homeomorphic (同胚) in Topology ?

Answer: To be continued in the next blog “Homeomorphism