The discovery of Quasicrystals — materials which are neither ordered nor disordered — changed the definition of crystals. Recently, the unrelated discovery of Topological Insulators defined a new type of materials classified by their topology. Here we show a
connection between quasicrystals and topological matter, namely that quasicrystals exhibit non-trivial topological phases attributed to dimensions higher than their own. Specifically, we show theoretically and experimentally using photonic lattices, that one-dimensional quasicrystals exhibit topologically-protected boundary states equivalent to the edge states of a two-dimensional topological system. We harness this property to adiabatically pump light across the quasicrystal, and generalize our results to higher dimensional systems. Hence, quasicrystals offer a new platform for the study of topological phases while their topology may better explain their surface properties.
