I meant "a given orientation of the tile will be present many times (or infinitely many)". It's very probable (I did not read the paper, only their website pag

I meant "a given orientation of the tile will be present many times (or infinitely many)". It's very probable (I did not read the paper, only their website page) that the tile only occupies a finite amount of orientations in the tiling and therefore at least (and probably more if not all) one orientation will also be present an infinite amount of times. However this does not imply periodicity of the tiling.