Are chiral quasicrystals possible?
Some recent developments shown in article Transversely quasicrystallographic 3D photonic chiral lattices: Polarization-controllable complex photonic band gap structures, where autors claimed experimantal results on produsing 3d chiral photonic quasicrystals. 




  Figure. Computer simulation of the light intensity distribution of
  the interference pattern for transversely quasicrystallographic 3D
  chiral photonic lattices respectively with rotational symmetries
  (a)5-fold (d) 8-fold (g) 12-fold. (b), (e) and (h) are their respective
  x-y planes. (c), (f) and (i) are their respective x-z planes
  
  "So the fabricated class of 3D PQC chiral lattices integrate both these wonder features on asingle platform in view of their possible applications in new generation integrated optical photonic devices with tunable polarization band gaps as well as frequency band gaps"


I also found fascinating that some researchers, use combination of structures like simple lattice and chiral lattice. This presentation Photonic Crystals and Metamaterials
has a lot of pictures, which could be good "food for thoughts".



Hope this was helpful, I think your area of research is very interesting.