Spira Mirabilis

Newton's method applied to f(z) = z²⁺ⁱ - z on the principal branch. The root condition z¹⁺ⁱ = 1 puts a whole family on the spiral r = exp(θ), but the principal branch keeps only z = 1 and z = -e^π, with the origin as a third attractor. The spiral survives anyway, since |z¹⁺ⁱ| = exp(ln r - θ) is constant along exactly that curve. The straight white horizon is the branch cut, where |f| jumps by e^(2π), and the cusped arcs are its preimages.