Revealing the boundary Weyl physics of the four-dimensional Hall effect via phason engineering in metamaterials*

5. We fabricate a reconfigurable two-dimensional aperiodic acoustic crystal with a phason living on a 2-torus, giving us access to the four-dimensional quantum Hall physics. Here, we introduce the concept of phason engineering in metamaterials and use it as a vehicle to access and apply the quantum Hall physics in arbitrary dimensions. Also, we supply a direct experimental confirmation that the topological boundary spectrum assembles in a Weyl singularity when mapped as a function of the quasimomenta. We also demonstrate topological wave steering enabled by the Weyl physics of the three-dimensional boundaries.