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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?

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    $\begingroup$ Quasicrystalls do fill space. They just don't fill space with a single type of cell and they do not fill space regularly with multiple cells, either. If you do a Fourier transform, the peaks are on a (dense) lattice constructed from multiple base vectors that have irrational length ratios. This expresses the spatial ordering of cells, which is close to periodic, but not quite. Where the higher dimensionality enters is that one can construct these irrational latices by projecting a higher dimensional lattice under an irrational angle onto a subspace, but that's math, not physics. $\endgroup$
    – CuriousOne
    Sep 28, 2014 at 19:24

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