  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 $(p_{4}, n_{5})$.
