  We propose a new physical interpretation of the Diophantine equation of
$\sigma_{xy}$ for the Hofstadter problem. First, we divide the energy spectrum,
or Hofstadter's butterfly, into smaller self-similar areas called "subcells",
which were first introduced by Hofstadter to describe the recursive structure.
We find that in the energy gaps between subcells, there are two ways to account
for the quantization rule of $\sigma_{xy}$, that are consistent with the
Diophantine equation: Landau quantization of (i) massless Dirac fermions or
(ii) free fermions in Hofstadter's butterfly.
