Approximants of quasiperiodic structures generated by the inflation mapping. (English) Zbl 0719.52015

Duneau, Michel; Mosseri, Rémy; Oguey, Christophe

J. Phys. A, Math. Gen. 22, No. 21, 4549-4564 (1989).

Summary: The problem of deriving explicit coordinates for quasicrystal approximants is solved in all the cases where the quasicrystal has an inflation symmetry. In the higher-dimensional space \({\mathbb{R}}^ n\), 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.

MSC:

52C22	Tilings in \(n\) dimensions (aspects of discrete geometry)
82D25	Statistical mechanics of crystals

Keywords:

aperiodic tiling; quasicrystal approximants; inflation symmetry; quasiperiodic pattern; algorithm