Construction of average unit cell for Penrose tiling

Abstract

Two appropriate reference lattices, oriented in different directions, have been used to construct the average unit cell for Penrose tiling. The obtained unit cell consists of four pentagons arranged in the physical plane. It has been shown that the pentagons are just the projections of the atomic surface from the 3D perpendicular space. For the two scattering vectors (corresponding to the reference lattices) an appropriate 2D plane embedded in 5D space has been constructed. The atomic surface is then projected perpendicularly to this plane. The trace of such projection on physical space gives the average unit cell, which can be used for calculation of peak intensities of the diffraction pattern for Penrose tiling.