Approximants of quasiperiodic structures generated by the inflation mapping

Abstract

The problem of deriving explicit coordinates for quasicrystal approximants is solved in all cases where the quasicrystal has an inflation symmetry. In the higher-dimensional space Rn, from which the quasiperiodic pattern is obtained by the cut method, the inflation symmetry is represented by a hyperbolic modular matrix (with integer entries) leaving the 'physical' space invariant. But this matrix also generates, by iteration, a sequence of (rational) approximant spaces which converges to the irrational space. A simple algorithm is described, providing the approximant periodic lattice and the set of vertices within a unit cell.