Generalized quasiperiodic patterns and superstructures in quasicrystals.

Patterns with octagonal and icosahedral symmetries obtained from periodic two-dimensional (2D) 8-grids and three-dimensional (3D) 12-grids by the dual method are shown to be superstructures of the Beenker pattern and the 3D Penrose pattern with the same edge length. The superstructures are described in the same spaces as those of the latter (4D for the Beenker and 6D for the 3D Penrose patterns) in the section method but their lattice constants are doubled. The occupation domains and the diffraction patterns for these cases are given.