  In the paper "A Chaotic Search for $i$"~(\cite{strang1991chaotic}), Strang
completely explained the behaviour of Newton's method when using real initial
guesses on $f(x) = x^{2}+1$, which has only a pair of complex roots $\pm i$. He
explored an exact symbolic formula for the iteration, namely $x_{n}=\cot{
\left( 2^{n} \theta_{0} \right) }$, which is valid in exact arithmetic. In this
paper, we extend this to to $k^{th}$ order Householder methods, which include
Halley's method, and to the secant method. Two formulae, $x_{n}=\cot{ \left(
\theta_{n-1}+\theta_{n-2} \right) }$ with
$\theta_{n-1}=\mathrm{arccot}{\left(x_{n-1}\right)}$ and
$\theta_{n-2}=\mathrm{arccot}{\left(x_{n-2}\right)}$, and $x_{n}=\cot{ \left(
(k+1)^{n} \theta_{0} \right) }$ with $\theta_{0} = \mathrm{arccot}(x_{0})$, are
provided. The asymptotic behaviour and periodic character are illustrated by
experimental computation. We show that other methods (Schr\"{o}der iterations
of the first kind) are generally not so simple. We also explain an old method
that can be used to allow Maple's \textsl{Fractals[Newton]} package to
visualize general one-step iterations by disguising them as Newton iterations.
