Abelianization is a process that parameterizes nonabelian local systems on a singular flat surface by turning them into abelian ones. On a compact surface, the restriction maps of an abelianized local system are defined by random-looking infinite products, which are straightforward to evaluate numerically, but hard to describe analytically except in a few special cases. I'll demonstrate one of those special cases, which is related to the physics of electrons moving in a periodic material.