We know that crystals (not including quasicrystals) have periodic structures. That is, they all have a translation invariant vector. My question is, do all crystals necessarily have a rotation axis? It seems that all the X-ray diffraction patterns of crystals have rotational symmetry, so I guess it is true, but how do I prove it?
The answer is no. Crystals in the triclinic crystal system do not possess any rotational symmetries at all. Crystals in the pinacoidal class only have an inversion center (which is not a rotational axis), whereas crystals in the pedial class have trivial symmetry. Crystals in the monoclinic domatic class also do not have any rotational axes. All other crystal structures do have rotational axes.