  This paper unveils a mapping between a quantum fractal that describes a
physical phenomena, and an abstract geometrical fractal. The quantum fractal is
the Hofstadter butterfly discovered in 1976 in an iconic condensed matter
problem of electrons moving in a two-dimensional lattice in a transverse
magnetic field. The geometric fractal is the integer Apollonian gasket
characterized in terms of a 300 BC problem of mutually tangent circles. Both of
these fractals are made up of integers. In the Hofstadter butterfly, these
integers encode the topological quantum numbers of quantum Hall conductivity.
In the Apollonian gaskets an infinite number of mutually tangent circles are
nested inside each other, where each circle has integer curvature. The mapping
between these two fractals reveals a hidden threefold symmetry embedded in the
kaleidoscopic images that describe the asymptotic scaling properties of the
butterfly. This paper also serves as a mini review of these fractals,
emphasizing their hierarchical aspects in terms of Farey fractions.
