  In this paper, we investigate $q$-Varchenko matrices for some hyperplane
arrangements with symmetry in two and three dimensions, and prove that they
have a Smith normal form over $\mathbb Z[q]$. In particular, we examine the
hyperplane arrangement for the regular $n$-gon in the plane and the dihedral
model in the space and Platonic polyhedra. In each case, we prove that the
$q$-Varchenko matrix associated with the hyperplane arrangement has a Smith
normal form over $\mathbb Z[q]$ and realize their congruent transformation
matrices over $\mathbb Z[q]$ as well.
