Newton Cosh

Newton's usual root-finding step changes character when f(z) = cosh z − 1. Its roots are 2πik for every integer k, and each is a double root. A polynomial gives an orbit only finitely many possible destinations; these roots keep marching along the imaginary axis at equal intervals. The fractal therefore arranges itself into bands running upward and downward, rather than settling into a few basins.

The correction (cosh z − 1) / sinh z simplifies, away from zeros, to tanh(z/2). The denominator sinh z vanishes at every πik: the even multiples are the roots and are removable in the simplified quotient, while the genuine poles and sensitive singularities are the odd multiples (2k+1)πi. Near those odd multiples, a tiny change in the starting point can greatly change the step.

You can see the periodicity along the imaginary axis at a glance — no math background needed. The horizontal bands repeat above and below one another, one for each period of the roots. At their joins, fine vertical threads record where one basin hands an orbit to the next.

Start with the horizontal bands along the imaginary axis. A period of roots makes one band, and the convergence pattern returns in the next. The borders do not run straight: they bend and fold, then pull out fine vertical filaments like taut threads.

Within each band, the basin geometry resembles its neighbours, with small distortions. Think of a ribbon folded repeatedly. The warm and cool zones come from the colouring; the sharp breaks gather near the odd multiples of πi, where the Newton correction has genuine poles.