Self similarity of dihedral tilings

We show that for any prime number n=2r+1⩾5n=2r+1⩾5 there exist r planar tilings with self-similar vertex set and the symmetry of a regular n  -gon (DnDn-symmetry). The tiles are the rhombi with angle πk/nπk/n for k=1,…,rk=1,…,r.