So I put up that earlier post showing the Fourier transforms. For those of you who do know the answer, congratulations. You know the basics of symmetry! Crystals are said to be periodic. As such, they fill all space and have translational symmetry. 2, 3, 4 and 6 fold symmetry shows translation. If you look in the earlier post, then you’ll see that the pictures to the left have translational symmetry in the atoms, while the 5, 7, and 9 fold do not.
5, 7, and 9 fold symmetry does exist however, in the forms of quasi crystals. These are materials that are ordered, but nonperiodic in the short range, but have a long range order. They tend to produce X-ray diffraction patterns that are penrose tiles and all sorts of cool stuff. Quasicrystals can be solved by traditional X-ray crystallography, but for that you need to go into higher dimensional crystallography. That is another topic all together. And for now I’m tired so that’s where I’ll end. Huzzah!