  Suppose that a circular fire spreads in the plane at unit speed. A single
fire fighter can build a barrier at speed $v>1$. How large must $v$ be to
ensure that the fire can be contained, and how should the fire fighter proceed?
We contribute two results.
  First, we analyze the natural curve $\mbox{FF}_v$ that develops when the
fighter keeps building, at speed $v$, a barrier along the boundary of the
expanding fire. We prove that the behavior of this spiralling curve is governed
by a complex function $(e^{w Z} - s \, Z)^{-1}$, where $w$ and $s$ are real
functions of $v$. For $v>v_c=2.6144 \ldots$ all zeroes are complex conjugate
pairs. If $\phi$ denotes the complex argument of the conjugate pair nearest to
the origin then, by residue calculus, the fire fighter needs $\Theta( 1/\phi)$
rounds before the fire is contained. As $v$ decreases towards $v_c$ these two
zeroes merge into a real one, so that argument $\phi$ goes to~0. Thus, curve
$\mbox{FF}_v$ does not contain the fire if the fighter moves at speed $v=v_c$.
(That speed $v>v_c$ is sufficient for containing the fire has been proposed
before by Bressan et al. [7], who constructed a sequence of logarithmic spiral
segments that stay strictly away from the fire.)
  Second, we show that any curve that visits the four coordinate half-axes in
cyclic order, and in inreasing distances from the origin, needs speed
$v>1.618\ldots$, the golden ratio, in order to contain the fire.
  Keywords: Motion Planning, Dynamic Environments, Spiralling strategies, Lower
and upper bounds
