The "butterfly fractal" describes a physical. Here we show that these abstract objects are related with a quantum fractal made up of integers, known as the Hofstadter Butterfly. In this Apollonian-Butterfly-Connection, dubbed as $\mathcal$) are iconic mathematical sets made up of integers that resonate with a wide spectrum of inquisitive minds. Integral Apollonian packing, the packing of circles with integer curvatures, where every circle is tangent to three other mutually tangent circles, is shown to encode the fractal structure of the energy spectrum of two-dimensional Bloch electrons in a magnetic field, known as the "Hofstadter butterfly". This paper also serves as a mini review of these fractals, emphasizing their hierarchical aspects in terms of Farey fractions. 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. In the Apollonian gaskets an infinite number of mutually tangent circles are nested inside each other, where each circle has integer curvature. In the Hofstadter butterfly, these integers encode the topological quantum numbers of quantum Hall conductivity. Both of these fractals are made up of integers. The geometric fractal is the integer Apollonian gasket characterized in terms of a 300 BC problem of mutually tangent circles. There are two circles that just touch all three: Since we have added new tangent circles, we can now take new combinations of circles three at a time that are mutually touching, and find the two circles that touch all three. 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. Each pair of circles touch each other at a single point, and the three points of contact are distinct. This paper unveils a mapping between a quantum fractal that describes a physical phenomena, and an abstract geometrical fractal.
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