Topological edge modes by smart patterning

We study identical coupled mechanical resonators whose collective dynamics are fully determined by the patterns in which they are arranged. In this work, we call a system topological if (1) boundary resonant modes fully fill all existing spectral gaps whenever the system is halved, and (2) if the boundary spectrum cannot be removed or gapped by any boundary condition. We demonstrate that such topological characteristics can be induced solely through patterning, in a manner entirely independent of the structure of the resonators and the details of the couplings. The existence of such patterns is proven using K theory and exemplified using an experimental platform based on magnetically coupled spinners. Topological metamaterials built on these principles can be easily engineered at any scale, providing a practical platform for applications and devices.