  Conduction through materials crucially depends on how ordered they are.
Periodically ordered systems exhibit extended Bloch waves that generate
metallic bands, whereas disorder is known to limit conduction and localize the
motion of particles in a medium. In this context, quasiperiodic systems, which
are neither periodic nor disordered, reveal exotic conduction properties,
self-similar wavefunctions, and critical phenomena. Here, we explore the
localization properties of waves in a novel family of quasiperiodic chains
obtained when continuously interpolating between two paradigmatic limits: the
Aubry-Andr\'e model, famous for its metal-to-insulator transition, and the
Fibonacci chain, known for its critical nature. Using both theoretical analysis
and experiments on cavity-polariton devices, we discover that the Aubry-Andr\'e
model evolves into criticality through a cascade of band-selective
localization/delocalization transitions that iteratively shape the self-similar
critical wavefunctions of the Fibonacci chain. Our findings offer (i) a unique
new insight into understanding the criticality of quasiperiodic chains, (ii) a
controllable knob by which to engineer band-selective pass filters, and (iii) a
versatile experimental platform with which to further study the interplay of
many-body interactions and dissipation in a wide range of quasiperiodic models.
