On the structure of axial quasicrystals

A unified treatment. valid for all the axial quasicrystals exhibiting 5-, 8-, 10-, and 12-fold symmetries. is presented for the linear distribution of atomic sites. The starting point is a cyclotomic integer basis that employs non-crystallographic roots involving Pisot-Vijayaraghavan algebraic integers. The general solution is expressed in the form of a theorem. An explicit method is given for determining the basis vectors that are involved. Both two- and three-dimensional quasicrystals with these axial symmetries are considered.