In many cases, the prototiles of a substitution tiling occur only in finitely many orientations. Here we list the tilings where this is not the case: the tiles occur in infinitely many different orientations. Thus these don’t show flc up to translation, even though they may show flc up to isometries.
In any primitive substitution tiling with tiles in infinitely many orientations the orientations are dense on the circle, and more: the orientations are equidistributed on the circle, see [fre08] .
Substitution tilings with statistical circular symmetry
European Journal of Combinatorics 2008, 29, pp. 1881-1893, arxiv 0704.2521