  We take a deeper dive into the geometry and the number theory that underlay
the butterfly graphs of the Harper and the generalized Harper models of Bloch
electrons in a magnetic field. Root of the number theoretical characteristics
of the fractal spectrum is traced to a close relationship between the Farey
tree -- the hierarchical tree that generates all rationals and the Wannier
diagram -- a graph that labels all the gaps of the butterfly graph. The
resulting Farey-Wannier hierarchical lattice of trapezoids provides geometrical
representation of the nested pattern of butterflies in the butterfly graph.
Some features of the energy spectrum such as absence of some of the Wannier
trajectories in the butterfly graph fall outside the number theoretical
framework, can be stated as a simple rule of "minimal violation of mirror
symmetry". In a generalized Harper model, Farey-Wannier representation prevails
as the lattice regroups to form some hexagonal unit cells creating new {\it
species} of butterflies
