[math-fun] Winding line puzzle

As is well know, for any irrational number c, there exists an injective immersion of the reals into the square torus, I_c: R -> T = R^2/Z^2 given by I_c(x) = q(x, cx) where q: R^2 -> R^2/Z^2 is the quotient mapping. (Such an image I_c(R) is called a winding line in the torus.) The image I_c(R) inherits a the subspace topology from T. PUZZLE: Given any irrational numbers c <> d, are the spaces I_c(R) and I_d(R) (with the subspact topologies) necessarily homeomorphic? --Dan