We present a modified version of the classical space filling curve of Hilbert, and we associate to this curve a geodesic lamination on the disk together with a transversal measure. The lamination helps us to understand how the points of the interval are mapped to the square. We study the geometry of this lamination. We generalize this construction to space filling curves from the interval to the regular $n$-gon and define an expanding dynamical system on this lamination.