Pentagonal quasicrystals and their linear self-similarities

Abstract

Abstract We recall the construction of a cut-and-project scheme providing planar model sets with 5-fold symmetry. The cut-and-project construction consists in projecting points of a lattice ℒ ⊂ ℝ4 by two orthogonal projections π ||, π ⊥ to two 2-dimensional subspaces. We describe the set of all linear mappings preserving the ℤ-modules π ||(ℒ) and π ⊥(ℒ) and show that the set of all such mappings is in a correspondence with a 2-dimensional cut-and-project set. Finally, we describe linear self-similarities of a pentagonal cut-and-project set with circular acceptance window.