3.2.8 Mirror curves on different surfaces
The construction of mirror curves described above is independent from the metrical properties or the geometry of the surface, so the same principle of construction can be applied to any tiling (e.g., on a sphere) (Gerdes, 1996, 1999) or in the hyperbolic plane (Dunham, 2000; Sazdanovic and Sremcevic, 2002a,b). Let us consider any edge-to-edge tiling of a part of arbitrary surface be given. After connecting midpoints of adjacent edges, we obtain a 4-regular mid-edge graph. Using the rules for the introduction of mirrors, it can be converted in a single mirror curve. Open question: find a general formula for the number k of curves for any tiling, before mirrors are introduced.
In the same way, from these mirror curves we may obtain the corresponding Lunda designs, etc. All non-isomorphic Lunda designs on a regular octahedron are given in the corresponding figure. Open question: enumerate non-isomorphic Lunda designs obtained from regular polyhedra.