Barber and Erde asked the following question: if $B$ generates ${\mathbb{Z}}^n$ as an additive group, then must the extremal sets for the isoperimetric inequality on the Cayley graph $({\mathbb{Z}}^n,B)$ form a nested family? We answer this question negatively for both the vertex- and edge-isoperimetric inequalities, already when $n=1$. The key is to show that the structure of the cylinder $\mathbb{Z}\times (\mathbb{Z}/k\mathbb{Z})$ can be mimicked in certain Cayley graphs on $\text{Z}$, leading to a phase transition. Based on joint work with Chris Wells.