Consider, for a polyomino $P$ made from $n$ unit squares joined at the edges, the arrangements of non-overlapping translations of $P$. Sometimes we can cover the infinite plane with such translations without any gaps:
Other times, we can't, such as with the $T$ pentomino:
When we can't do this, one can ask about the maximum density attainable, hereafter $\Delta(P)$; by a compactness argument, this will always be less than $1$ if $P$ does not tile the plane.
Many times, we can get the maximum packing density by adding $k$ squares to $P$ to produce an $n+k$-omino $Q$ which does tile the plane by translation, which gives us $\Delta(P) \ge \frac{n}{n+k}$. For instance, with the $T$ pentomino above and $k=1$:
By a 1984 result of Wijshoff and van Leeuwen, any $Q$ which tiles the plane via translation can do so in an isohedral manner, meaning that the symmetries of the plane are transitive on the tiles. This implies the same for the arrangements of $P\subseteq Q$ as well.
I am curious whether the above result generalizes to packing densities less than one, i.e. that maximal packings of $P$ are always possible to do by placing copies of $P$ in a lattice configuration. This is equivalent to the statement that the maximal density is equal to $\frac n{n+k}$, where $k$ is the smallest number of unit squares needed to add to $P$ to form a translationally-tiling polyomino.
I've verified manually that this is the case for all polyominoes with $6$ or fewer cells, but it's not clear to me how to proceed from here. Possibly the statement I'd like to show is equivalent to the $1984$ result, but if so I don't see the proof. It's also quite possible that there are straightforward counterexamples with some larger polyominoes - in particular, any non-tiling $n$-omino that packs with density larger than $n/(n+1)$ would work. Any suggestions or pointers to relevant literature would be welcome!