Geometric study of a 2D tiling related to the octagonal quasiperiodic tiling

A quasicrystal built with three types of tiles is related to the well-known octagonal tiling. The relationships between both tilings are investigated. More precisely, we show that the coordinates of the vertices can be obtained in two different but equivalent ways. The structure factor is calculated exactly. We emphazise the difficulty one can have to define the order of the symmetry of a quasicrystal, from a practical point of view, exhibiting a quasiperiodic tiling whose spectrum has a « quasi » eigth-fold symmetry. Finally, we show how to recover easily a class of octagonal-like quasicrystals.