On 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 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. 6 1 4 4 ... 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 v c these two zeroes merge into a real one, so that argument ϕ goes to 0. Thus, curve 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. [6], who constructed a sequence of logarithmic spiral segments that stay strictly away from the fire.) Second, we show that for any curve that visits the four coordinate half-axes in cyclic order, and in increasing distances from the origin the fire can not be contained if the speed v is less than 1.618..., the golden ratio. [ABSTRACT FROM AUTHOR]