Covering the Plane by a Sequence of Circular Disks with a Constraint

Abstract

We are interested in the following problem of covering the plane by a sequence of congruent circular disks with a constraint on the distance between consecutive disks. Let $(\mathcal{D}_n)_{n \in \mathbb N}$ be a sequence of closed unit circular disks such that $\cup_{n \in \mathbb{N}} \mathcal{D}_n = \mathbb {R}^2$ with the condition that for $n \ge 2$, the center of the disk $\mathcal{D}_n$ lies in $\mathcal{D}_{n-1}$. What is a "most economical" or an optimal way of placing $\mathcal{D}_n$ for all $n \in \mathbb{N}$? We answer this question in the case where no "sharp" turn is allowed, i.e. if $C_n$ is the center of the disk $\mathcal{D}_n$, then for all $n \ge 2$, % $\angle C_{n-1}C_nC_{n+1}$ is not very small. We also consider a related problem. We wish to find out an optimal way to cover the plane with unit circular disks with the constraint that each disk contains the centers of at least two other disks. We find out the answer in the case when the centers of the disks form a two-dimensional lattice.