  This paper further studies the matroid realization space of a specific
deformation of the regular $n$-gon with its lines of symmetry. Recently, we
obtained that these particular realization spaces are birational to the
elliptic modular surfaces $\Xi_{1}(n)$ over the modular curve $X_1(n)$. Here,
we focus on the peculiar cases when $n=7,8$ in more detail. We obtain concrete
quartic surfaces in $\mathbb{P}^3$ equipped with a dominant rational self-map
stemming from an operator on line arrangements, which yields K3 surfaces with a
dynamical system that is semi-conjugated to the plane.
