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 $\Z$, leading to a phase transition. Based on joint work with Chris Wells.