On a Fire Fighter’s Problem

Abstract

Suppose that a circular fire spreads in the plane at unit speed. A single fire fighter can build a barrier at speed [Formula: see text]. How large must [Formula: see text] 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 [Formula: see text] that develops when the fighter keeps building, at speed [Formula: see text], a barrier along the boundary of the expanding fire. We prove that the behavior of this spiralling curve is governed by a complex function [Formula: see text], where [Formula: see text] and [Formula: see text] are real functions of [Formula: see text]. For [Formula: see text] all zeroes are complex conjugate pairs. If [Formula: see text] denotes the complex argument of the conjugate pair nearest to the origin then, by residue calculus, the fire fighter needs [Formula: see text] rounds before the fire is contained. As [Formula: see text] decreases towards [Formula: see text] these two zeroes merge into a real one, so that argument [Formula: see text] goes to 0. Thus, curve [Formula: see text] does not contain the fire if the fighter moves at speed [Formula: see text]. (That speed [Formula: see text] 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 [Formula: see text] is less than 1.618…, the golden ratio.