Pinning of scroll waves to flat and highly branched unexcitable heterogeneities

System heterogeneities such as organelles, cells, and anatomical features strongly affect nonlinear wave patterns in biological systems. These effects are more readily studied in otherwise homogeneous chemical reactions that allow the introduction of tailored structures. Following this approach, we investigate the dynamics of three-dimensional excitation vortices pinned to inert sheets with circular holes arranged on a hexagonal lattice. Experiments with the Belousov-Zhabotinsky reaction and numerical simulations of an excitable reaction-diffusion model reveal vortex pinning that circumvents the rapid collapse of free vortex rings. The pinned scroll waves are affected by the topological mismatch between their looplike rotation backbone and the branched pinning structure. Depending on the initial condition, a multitude of stable vortex states exist, all of which obey topological constraints, suggesting spinlike states for the involved obstacle holes.