Linear astral (n5)(n5) configurations with dihedral symmetry

A linear astral (nk)(nk) configuration is a collection of points and straight lines, so that each point lies on kk lines and each line passes through kk points, with ⌊k+12⌋ symmetry (transitivity) classes of points and lines under rotations and reflections mapping the configuration to itself. We discuss the possible structures of astral (n5)(n5) configurations with dihedral symmetry group DmDm in the Euclidean plane, and we provide methods to investigate the existence of such configurations. In doing so, we introduce a new class of astral (n3)(n3) configurations.