  The unrelated discoveries of quasicrystals and topological insulators have in
turn challenged prevailing paradigms in condensed-matter physics. We find a
surprising connection between quasicrystals and topological phases of matter:
(i) quasicrystals exhibit nontrivial topological properties and (ii) these
properties are attributed to dimensions higher than that of the quasicrystal.
Specifically, we show, both theoretically and experimentally, that
one-dimensional quasicrystals are assigned two-dimensional Chern numbers and,
respectively, exhibit topologically protected boundary states equivalent to the
edge states of a two-dimensional quantum Hall system.We harness the topological
nature of these states to adiabatically pump light across the quasicrystal. We
generalize our results to higher-dimensional systems and other topological
indices. Hence, quasicrystals offer a new platform for the study of topological
phases while their topology may better explain their surface properties.
