Indexing approximants of icosahedral quasicrystals

It is well known that the crystallography of approxi­mants is directly related to that of the parent quasicrystal, once its unit-cell vectors are identified as parallel projections of certain N-dimensional lattice nodes . Derived here are explicit simple relations for calculating the shear matrices and the related crystallographic properties of the corresponding approximants, including diffraction indexing and the determination of the lattice in perpendicular space. Applied to low-dimensional approximants, the derivation shows that the systematic `accidental' extinction rules observed in the pentagonal phases are generic extinctions that are due to the geometrical properties of the projected 1D lattice and are independent of the actual model of the quasicrystal.