A Fire Fighter's Problem

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 FFv 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 − sZ)−1, where w and s are real functions of v. For v > vc = 2.6144 . . . all zeroes are complex conjugate pairs. If φ denotes the complex argument of the conjugate pair nearest to the origin then, by residue calculus, the fire fighter needs Θ(1/φ) rounds before the fire is contained. As v decreases towards vc these two zeroes merge into a real one, so that argument φ goes to 0. Thus, curve FFv does not contain the fire if the fighter moves at speed v = vc. (That speed v > vc 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 . . ., the golden ratio, in order to contain the fire.