Abstract

Recent experimental observations have been argued to demonstrate the equivalence of one-dimensional quasicrystals to two-dimensional quantum Hall systems. If correct this would lend strong evidence to the claim that three-dimensional quasicrystals such as certain Ho-Mg-Zn alloys are, in a very real sense, higher dimensional objects living in our world. The interpretation is controversial, but the objections mathematically subtle, based on the topological classification of free fermion systems. In recent work we showed that the mathematics of both quasicrystals and the quantum Hall effect can be considered as different limits of a third problem: incommensurate charge order. We use the equivalence, and a couple of properties of fractal patterns, to resolve the debate by means of a simple analogy.