Constructing a dodecagonal structure by projection in two stages

Decagonal and octagonal tilings can be covered by a single cluster but two clusters are needed to guarantee a quasiperiodic dodecagonal structure. This might be due to the extra freedom when projecting from 6D to 2D. To elucidate this issue, we projected a 6D simple cubic lattice into 3D so that further projection to 2D might yield a dodecagonal structure. A 6D cube projects into a 3D (non-Keplerian) triacontahedron with symmetry 3̄m. The final result is a layer structure quasiperiodic in the basal plane and periodic in the perpendicular direction. The point symmetry of the structure is only 3̄m but it does show characteristic dodecagonal features. The plane pattern has symmetry 6m. It supports some well-known dodecagonal tilings, such as the square-triangle tiling. Some tilings appear to be periodic, the quasiperiodicity being hidden in the changing decoration. Physically the structure should be interpreted as a packing of dodecagonal prisms. Mathematically it is best to consider the point set as such. The structure has many surprising features that call for further study.