We prove the convergence of a thresholding scheme to the motion by mean curvature of curves (filaments) in space ($R^3$). The scheme was first proposed by Ruuth-Merriman-Xin-Osher (2001). It generalizes to higher codimension (codimension$=2$) the classical thresholding scheme for hypersurfaces (codimension$=1$) which was initiated by Merriman-Bence-Osher in the 90's. The algorithm is essentially a time-splitting scheme, alternating two very simple steps: (1) linear diffusion and (2) projection to functions with norm one. Almost all the proofs of convergence in the hypersurface case make use of comparison principle. Such a technique is not applicable for higher codimensions. We will formulate the scheme using the gradient flow interpretation found by Esedoglu-Otto (2015) together with a computation by Lin for an energy estimate for Ginzburg-Landau functional. This is joint work with Tim Laux.