Phi is hiding in there too ...

  succ(x)  = (x^2 + 2x) / (x^2 + 1). Newton's method [0]

  ~> Decimal expansion converges quadratically 

  0.618... = [0;1,1,1,1,...]. Continued fraction grows by (+1)/
   
  0.618..., 1.0, 1.618..., 2.618.... Powers of Phi.

  φ^2  = φ + 1

  e^iτ = 0 + 1. Euler's identity (τ version) [1]

  ~> The complex exponential of the circle constant is unity.
  
  Geometrically, multiplying by e^iθ corresponds to rotating a
  complex number by an angle θ in the complex plane, which
  suggests a second interpretation of Euler’s identity:

  ~> A rotation by one turn is 1.

  √φ ≈ 4/π. Maybe a coincidence, maybe not. 

  0.283... = 2 * (π - 3.00)

[0] phi https://en.wikipedia.org/wiki/Golden_ratio#Alternative_forms

[1] tau https://tauday.com/tau-manifesto#sec-euler_s_identity