4 ms·

How does that work? ... Explanations invited. Hold the 1 fixed. Notice in both sequences the remaining numbers are 3 interleaved ordered sets, in multiples of

• espeed 7y ago
  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 https://en.wikipedia.org/wiki/Golden_ratio#Alternative_forms [1] tau https://tauday.com/tau-manifesto#sec-euler_s_identity https://tauday.com/tau-manifesto#sec-euler_s_identity