I am working with arbitrary parallelopiped tilings given by projection from a higher dimensional space. The collection of tiles, and some properties of the higher dimensional space are specified by a finite collection of vectors forming the edges of the tiles. The tiles themselves are parallelopipeds given by all choices of three vectors from this set. A patch of such a tiling is shown below.
The tricky thing is trying to find sets of vectors where none of the parallelopipeds are nearly flat. This lead me to a more general question, a sort of dual problem to packing circles on a sphere.
Given $V$ a set of 3d vectors in general position, so that no two lie on the same line and no three lie on the same plane. Without loss of generality, we may assume they are unit vectors. We can consider $V_P$, the set of planes generated by all pairs of vectors in $V$. What arrangements of vectors $V$ will maximise the smallest angle between two planes in $V_P$?