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

Abstract

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» eight-fold symmetry. Finally, we show how to recover easily a class of octagonal-like quasicrystals Au moyen de trois tuiles, nous construisons un pavage quasiperiodique du plan, que nous relions au quasicristal octogonal. Ainsi, nous montrons que les coordonnees des nœuds peuvent etre obtenues de deux manieres differentes. Le facteur de structure est calcule exactement. Ce pavage qui possede «presque» une symetrie d'ordre huit, souleve la difficulte de la determination pratique de la symetrie d'un quasicristal. Finalement, nous montrons comment construire une large classe de pavage du type de l'octogonal, a partir de ce nouveau pavage