Candle in the woods

摘要

We consider the problem of determining the minimum number Nd of unit disks that is required to block all rays emanating from a point P in the two-dimensional space, where each disk has at least a distance d to point P and to any other disk. We study the asymptotic behavior of Nd, as d tends to infinity. By deriving upper bounds and lower bounds, we prove that pi2/16 = lim_{d - infinity} N_d/d2 = 18/pi2, where the upper bound is based on establishing an interesting link between unit disks positioned on a regular triangular grid and Farey sequences from number theory. By positioning point P as well as the centers of the disks on the grid points of such a triangular grid, we create hexagonal rings of disks around P. We prove that we need exactly d-1 of these hexagons to block all rays emanating from P.