Great! Not sure if anyone noticed, but this creates an optical illusion of the whole thing floating upwards. That's rare, because I think most of the optical illusions are static, right? - and this one is produced by the dynamic part.

Very neat! Only a few minor things:

1. I would have computed verts like this:

   verts = Normalize /@ N[PolyhedronData[{"Dipyramid", 3}, "VertexCoordinates"]].RotationMatrix[-?/6, {0, 0, 1}]
   

(note the use of the inverse/transpose, since I'm post-multiplying instead of pre-multiplying)

1. As I pointed out in one of your previous posts, if your are already using inexact numbers, NullSpace[] will automatically return orthonormalized vectors (via SVD), so b = NullSpace[{#}] & /@ p suffices.

Otherwise, nicely done. :)