Twisted bilayer graphene (TBLG) subjected to a sequence of commensurate external periodic potentials [for a detailed review, see Ref.3] results in the formation of moir\'{e} fractals (MFs) [ref. 1] that can quantitatively describe the band structure of a large class of super-moir\'{e} systems. The self-similarity of the hierarchy of Brillouin zones (BZ) in such moir\'{e} fractals so produced results in a nested subband structure inside the bandwidth of the original moiré bands, and the number of such sub-bands can be directly computed with the help of fractal dimension. The fractal generators for such MFs are derived, and we examine their effect on the BZ edge hierarchy. We also find parallels between such MFs and those arising in economic geography related to transportation and correspondence within the framework of central place theory. We point out that the concept of MF is quite general and can be easily extended to non-graphene twisted heterostructures. The modification of band structure can strongly influence correlated phase and optical properties[ ref. 2], and various other transport and thermodynamic properties in such twisted heterostructures. Recent work shows lattice relaxations significantly alter the electronic band structure of twisted, layered structures. In further work, we show the robustness of MFs in the presence of lattice distortion-induced strain field and examine whether the number of in-gap states given by the fractal dimension remains the same under such strain [4].

 [1] D Aggarwal, Rohit Narula, and Sankalpa Ghosh, "Moiré fractals in twisted graphene layers," Phys. Rev. B 109, 125302 (2024) (Editor's Suggestion) [2] Disha Arora, Deepanshu Aggarwal, Sankalpa Ghosh, and Rohit Narula, "Interference effects in polarization-controlled Rayleigh scattering in twisted bilayer graphene,"Phys. Rev. B 107, 205423 (2023) [3] Deepanshu Aggarwal, Rohit Narula, and Sankalpa Ghosh "A primer on twistronics: a massless Dirac fermion's journey to moiré patterns and flat bands in twisted bilayer graphene," J. Phys.: Condens. Matter 35 143001 (2023) (Invited Topical Review) [4] Deepanshu Aggarwal, Rohit Narula and Sankalpa Ghosh., Moir\'{e} fractals under strain, (manuscript under preparation)