Platonic configurations of points and lines

We present some methods for constructing connected spatial geometric configurations $(p_q,n_k)$ of points and lines, preserved by the same rotations (and reflections) of Euclidean space $E^3$ as the chosen Platonic solid. In this paper we are primarily interested in balanced configurations $(n_3),(n_4)$ and $(n_5)$, but also in unbalanced configurations $(p_3,n_4),(p_3,n_5)$ and $(p_4,n_5)$.