Quasiperiodic-periodic structures by projection in two stages: An illustrative example

Recently we have modified a two-stage variant of the cut-and-project method used to generate layered 8-fold (Ben-Abraham, 2004) and 12-fold (Ben-Abraham et al., 2004) quasiperiodic structures in three dimensions, and obtained 8-fold, 10-fold and 12-fold quasiperiodic structures via cut-and-project from various four dimensional lattices (Ben-Abraham and Quandt, 2006). However four dimensions are certainly not the most suitable playground to illustrate the basics of our general two-stage projection method. To this end we cut and project a periodic structure in three dimensions onto two-dimensional physical space, such that a second cut and projection onto a line yields a quasiperiodic structure. This simple example contains all of the basic elements of our two-stage projection method, but now the essential steps may directly be visualized.