Construction of average lattices for quasiperiodic structures by the section method

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Abstract
The construction of an average lattice with bounded modulation, for one dimensional quasiperiodic tilings, is considered from the viewpoint of the higher dimensional space R2. The 1D quasiperiodic tilings are : a) the canonical 1D tiling obtained e.g., by the cut and project method, b) tilings generated by a circle map algorithm, for particular values of the parameters defining the model. In this last case, the construction bridges a gap between the cut and project, or section, methods, and the circle map model, and provides an alternative proof of the quasiperiodic ordering : we build suitable 2D periodic tilings yielding the quasiperiodic ones by section. This geometrical approach gives also an intuitive image of the mechanism of the disappearance of the average latice, and of the quasiperiodic ordering, for generic values of the parameters of the model. The considerations given here may serve as a basis for the construction of average lattices with bounded modulation, if they exist, of higher dimensional tilings