A series of coverings of the regular n-gon

We define an infinite series of translation coverings of Veech’s double-n-gon for odd n ≥ 5 which share the same Veech group. Additionally we give an infinite series of translation coverings with constant Veech group of a regular n-gon for even n ≥ 8. These families give rise to explicit examples of infinite translation surfaces with lattice Veech group.