Hybrid quasiperiodic-periodic structures constructed by projection in two stages

A two-stage variant of the cut-and-project method is presented, in which a periodic structure is cut and projected in a high-dimensional space onto three-dimensional physical space so that a second cut and projection onto a plane yields a quasiperiodic structure. The method is applied to the cases of octagonal, dodecagonal and pentagonal/decagonal symmetry. The focus is on the three-dimensional intermediate hybrid structures that are partly quasiperiodic and partly periodic. The method can be generalized to other symmetries as well as to include more intermediate steps.