Interpenetrating Incommensurately Modulated Lattices with Icosahedral Symmetry

This paper derives an infinite class of quasiperiodic lattices, composed of incommensurately modulated sublattices with displacive modulations, whose diffraction patterns have icosahedral symmetry. The sublattice reference lattices, as well as the sublattice displacement fields, belong to the space group $R\overline{3}m$ and are equivalent under the icosahedral point group. These quasiperiodic lattices are fundamentally different from those constructed by the projection and grid methods. Nevertheless, a special displacement field reproduces a three-dimensional generalization of the Penrose tiling.