Are chiral quasicrystals possible? Quasicrystals are materials that have long-range atomic order but lack the translational symmetry of conventional crystals. All quasicrystalline tilings and packings I have read about thus far, whilst lacking translational symmetry, have rotation axes and reflection planes. I'm wondering if anyone is aware of experimental evidence of quasicrystals which lack reflection symmetry (and thus possess implicit chirality), or can provide a theoretical rationale for why such structures can or cannot exist.

To clarify, a cursory google search turns up a paper interpreting a viral capsid as having chiral quasicrystalline order - Whilst I don't have access to the paper, I specifically exclude this sort of intepretation from consideration as I'm talking about structures with potentially infinite spatial extent, which viruses do not have.
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.