Most sphere/circle/square-packing problems are interested in the highest possible packing density, or the average density of a random packing. I am interested in what is the worst that one can do, for circles, squares, etc. For nontriviality, I am assuming the packing is maximal, in the sense that no one shape can be added to the packing anywhere without overlapping some shape that already exists.
Given a plane shape $S$, what is the (limiting) smallest possible packing density of a maximal packing of the plane with the shape $S$?
For squares, I thought at first that the answer might be $1/2$, as the following picture indicates.
But after thinking about it a little longer, I realized you could get as low as $1/4$, as the following image shows. I am not sure if this can be improved upon.
And what about circles, and other shapes?
Edit: I just realized that the square can't be as far apart as in the above picture, or else you could fit a rotated square in the white space between four squares. If the radius of the squares is $1$, then the distance between them needs to be less than $\sqrt{2}$ to prevent this, which gives a density of $4/(2+\sqrt{2})^2 \approx 0.34314\ldots$
Here is a partial answer for the case of circle packings, dependent on the fact that circles look the same no matter how you rotate them.
Let $B_1$ be the open unit ball of radius $1$ and suppose $C \subset \mathbb{R}^2$ is a set of points such that $\{B_1 + c : c \in C\}$ is a maximal circle packing. That means that for every $x \in \mathbb{R}^2$, if $x \notin C$ then there exists a $c \in C$ such that $(B_1 + x)\cap (B_1 + c) \neq \varnothing$, which implies $d(x,c) < 2$.
That means that $\{B_2 + c : c \in C\}$ is a collection which actually covers the whole plane. In a hypothetical extreme case where $\{B_2 + c : c \in C\}$ is actually a partition of $\mathbb{R}^2$ (impossible for actual circles), the density of the packing would be precisely $\pi(1)^2 / \pi(2)^2 = 1/4$.
This shows that $1/4$ is a lower bound for the density of any maximal plane packing by circles, but is there actually a maximal circle packing (resp. family of circle packings) which achieves (resp. approaches) this bound?