  In crystalline systems, higher-order topology, characterized by topological
states of codimension greater than one, typically arises from the mismatch
between Wannier centers and atomic sites, leading to filling anomalies.
However, this phenomenon is less understood in aperiodic systems, such as
quasicrystals, where Wannier centers are not well defined. In this study, we
examine Fibonacci chains and squares, a quintessential type of quasicrystal, to
investigate their higher-order topological properties. We discover that
topological interfacial states, including corner states, can be inherited from
their higher-dimensional periodic counterparts, such as the two-dimensional
Su-Schrieffer-Heeger model. This finding is validated through numerical
simulations of both phononic and photonic Fibonacci quasicrystals by the finite
element method, revealing the emergence of topological edge and corner states
at interfaces between Fibonacci quasicrystals with differing topologies
inherited from their parent systems. Our results not only provide insight into
the higher-order topology of quasicrystals but also open avenues for exploring
novel topological phases in aperiodic structures.
