Abstract:
We study $(n,k)$-scatters, which are regular $n$-gon tiles arranged so that each tile shares edges with at least $k$ others. To measure how much freedom there is in arranging scatters, we ask which shapes are unavoidable. It turns out that for a few choices of $(n,k)$ there are infinite unavoidable shapes, but otherwise they are finite. We discuss the infinite case as an analogue of crystallization.
The main result here is that besides the trivial situations when there’s a unique scatter, there are only four instances of this. Scatters crystallize nontrivially just when $(n, k) = (5, 3)$, $(7, 3)$, $(10, 4)$, or $(14, 4)$.
