  Fractal lattices, featuring the self-similarity symmetry, are often geometric
descents of parent crystals, possessing all their discrete symmetries (such as
rotations and reflections) except the translational ones. Here, we formulate
three different general approaches to construct real space Hamiltonian on a
fractal lattice starting from the Bloch Hamiltonian on the parent crystal,
fostering for example strong and crystalline topological insulators resulting
from the interplay between the nontrivial geometry of the underlying electronic
wavefunctions and the crystal symmetries. As a demonstrative example, we
consider a generalized square lattice Chern insulator model, and within the
framework of all three methods we successfully showcase incarnations of strong
and crystalline Chern insulators on the Sierpi\'nski carpet fractal lattices.
The proposed theoretical framework thus lays a generic foundation to build a
tower of topological phases on the landscape of fractal lattices.
