Constructing a dodecagonal structure by projection in two stages
Shelomo Izhaq Ben-Abraham, Y. Lerer, Y. Snapir · Acta Crystallographica Section A Foundations of Crystallography · 2002
The covering cluster approach offers a plausible mechanism for the formation of aperiodic (as well as periodic) crystals.Thus, decagonal and octagonal tilings can be constructed by covering with a single cluster (or patch).The approach fails in the dodecagonal case where two clusters are needed to guarantee a quasiperiodic structure.Considering the cut-and-project method one may guess that this is due to the extra freedom when projecting from 6D to 2D.To explore this line of thought, we projected a 6D simple cubic lattice into 3D so that a further projection to 2D yields a dodecagonal structure.A 6D cube projects into a 3D triacontahedron different from the Keplerian; its symmetry is -3m.The final result is a dodecagonal layer structure quasiperiodic in the basal plane and periodic in the perpendicular direction.We also explore this approach in the octagonal case where it yields a structure quasiperiodic in one direction and periodic in the two perpendiculars.