conformal mapping of point P' on a sphere to point P on a horn torus, equivalent to  α → φ,
      when choosing α = 0 at the north pole  (later we change to version with α = 0 at south pole,
      which is symmetric, but more logical and consistent, because |z| is zero here simultaneous)

      φ = 2·arccot(-ln(|tan(α/2)|) - C)

      we observe a small circle around P' on sphere with radius dα   (α is latitude, 0 at north pole)
      the radius perpendicular to screen in P' with same length is dω = dα / sin(α) (ω is longitude)
      the projected small circle around P on horn torus has same radius dω  (as angle, not length)
      and for corresponding perpendicular radius dφ we respectively have:  dω = dφ / (1 - cos(φ))
      ω and dω, as angles, are identical on both figures, therefore:    dα / sin(α) = dφ / (1 - cos(φ))
       
      by integration and solving for φ we get above result
      values for α:  0 to π, resulting values for φ:  0 to 2·π
      approximations to both limits are highly asymptotic
      and absolute symmetrical from both sides of center *
      to get φ < 4 on horn torus, α has to be < 10^(-10),
      to get φ < 3 on horn torus, α has to be < 10^(-14),
      to get closer than 2 - my computer's capacity fails!  (α < 10^(-26))
      an extremly slow approach to the horn torus center
      reminds me of conditions at the edge of black holes
      and of the different views ahead and backwards, when riding a photon - with speed of light:   [] 

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       •  even slower 'approach' when mapping direct from the complex plane  (α = 2·arccot(|z|))
       •  latter inserted into above formula we eventually get:   φ = 2·arccot(-ln|tan(arccot(|z|))|)
       •  inverse mapping from horn torus to complex plane: |z| = cot(π/2 - arccot(e^(-cot(φ/2))))
       •  inverse mapping from horn torus back to Riemann sphere: α = π - 2·arccot(e^(-cot(φ/2)))
       •  most likely no stereographic projection possible, but mathematicians surely will look for!
       •  non-conformality creates pattern in the dynamic model, but conformality has a meaning
       
       *  we obtain much simpler formulas when we determine angle α to be zero at south pole 0
      my rough cross-checks with various angles α and differentials Δω indicate exact validity, but
      analytical evidence and search for a projective geometry  I gladly surrender to professionals  ...

       
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Riemann sphere:  0N = 1 , |z| = 0Z ,  ∠0ZN  =  ∠N0P'  =  α/2