One of the examples of the problems of 5-fold symmetry is that pentagons tiled on a 2D plane do not completely fill that plane, leaving voids. This may be solved by "folding" it into 3D space, and forming a pyritohedron.
As far as I know, quasicrystals are the equivalent of this in 3D and higher dimensions. Does that mean that in the real 3D world, the inner structure of quasicrystals does not fill space? What real world implications are there for mathematically folding such a structure into a higher dimensional form?
This may be related to this question.