  The approximation of a circle with the edges of a fine square grid distorts
the perimeter by a factor about $\tfrac{4}{\pi}$. We prove that this factor is
the same on average (in the ergodic sense) for approximations of any
rectifiable curve by the edges of any non-exotic Delaunay mosaic (known as
Voronoi path), and extend the results to all dimensions, generalizing Voronoi
paths to Voronoi scapes.
