  We revisit the problem of self-similar properties of the Hofstadter butterfly
spectrum, focusing on spectral as well as topological characteristics. In our
studies involving any value of magnetic flux and arbitrary flux interval, we
single out the most dominant hierarchy in the spectrum, which is found to be
associated with an irrational number $\zeta=2+\sqrt{3}$ where nested set of
butterflies describe a kaleidoscope. Characterizing an intrinsic frustration at
smallest energy scale, this hidden quasicrystal encodes hierarchical set of
topological quantum numbers associated with Hall conductivity and their scaling
properties. This topological hierarchy maps to an {\it integral Apollonian
gasket} near-$D_3$ symmetry, revealing a hidden symmetry of the butterfly as
the energy and the magnetic flux intervals shrink to zero. With a periodic
drive that induces phase transitions in the system, the fine structure of the
butterfly is shown to be amplified making states with large topological
invariants accessible experimentally.
