One of the rare examples of a tiling where the prototiles occur in infinitely many orientations. Apart from the pinwheel tiling and its generalizations [Sad98] there are only a few examples known which show infinite rotations.
The inflation factor of this one is a complex algebraic PV number of degree four. As the scaling and the rotations for the tiles are all given by algebraic units, every vertex of the tiling lies within a finitely generated Z-module.
Some generalizations of the pinwheel tiling
Discrete Comput. Geom., 1998, 20, 1, pp. 79--110, 1626703