  The translation symmetry of a lattice is greatly modified when subjected to a
perpendicular magnetic field [Zak, Phys. Rev. \textbf{134}, A1602 (1964)]. This
change in symmetry can lead to magnetic unit cells that are substantially
larger than the original ones. Similarly, the translation properties of a
double-layered lattice alters drastically while two monolayers are relatively
twisted by a small angle, resulting in large-scale moir\'{e} unit cells.
Intrigued by the resemblance, we calculate the complete band structures of a
twisted bilayer optical lattice and show that the geometric moir\'{e} effect
can induce fractal band structures. The fractals are controlled by the twist
angle between two monolayers and are closely connected to the celebrated
butterfly spectrum of two-dimensional Bloch electrons in a magnetic field
[Hofstadter, Phys. Rev. B \textbf{14}, 2239 (1976)]. We demonstrate this by
proving that the twisted bilayer optical lattice can be mapped to a generalized
Hofstadter's model with long-range hopping. Furthermore, we provide numerical
evidence on the infinite recursive structures of the spectrum and give an
algorithm for computing these structures.
