Fractal hierarchy enables exponential scaling of topological boundary states

Exponential growth describes an extremely rapid process ubiquitous in mathematics and across diverse physical, biological, and technological systems. Here, we introduce a class of fractal-inspired lattices that combine long-range periodic order with self-similar hierarchy, establishing a structural motif that enables exponential scaling of topological boundary states. We demonstrate this phenomenon in (i) a quasi-one-dimensional lattice chain constructed from Koch-curve unit cells and (ii) a two-dimensional periodic tiling lattice composed of Sierpiński-gasket unit cells. We show that, for suitable coupling parameters, both the number of topological boundary states $N_{\ell }$ and the number of topological minigaps $M_{\ell }$ grow exponentially with the fractal generation index $\ell$ . We find that $N_{\ell }$ is an integer multiple of $M_{\ell }$ , with the integer determined by the underlying symmetry. This hierarchical scaling law is captured by  the multi-topological-phase theory and confirmed experimentally in laser-written photonic lattices. Our results identify fractal hierarchy as a design principle for controlling boundary-state multiplicity, revealing a fundamental interplay between topology, self-similar geometry, and periodic order. More broadly, this work suggests a route toward synthetic materials and integrated photonic platforms in which large numbers of robust boundary modes can be engineered within hierarchically structured architectures.