Halos at Whistler B.C. Canada captured by David Hayes.
Why are halos so reproducible and geometrically exact?
Two reasons. Ice crystals have fixed and accurate geometries that echo their internal atomic order. The crystals are aligned very precisely by aerodynamic drag forces generated as they drift downwards.
Rather satisfyingly, as few as five main orientations are able to explain nearly all the halos ever observed. We can think of others and predict their halos - but they have not been observed in Nature - yet.
This display had four of the orientations (parhelia on other images confirm that plate crystals were present), only the elusive Lowitz orientation was missing.
Randomly oriented crystals point in any direction. Plates have the main crystal axis vertical. Singly oriented columns (SOCs) have the main axis horizontal and can rotate around it.
Parry oriented crystals also have the main axis horizontal but the crystal also has its top face horizontal.
Lowitz crystals can take rotational positions about a horizontal axis crossing opposite prism edges.