Abstract
AbstractThe approximation of a circle with the edges of a fine square grid distorts the perimeter by a factor about $$\tfrac{4}{\pi }$$ 4 π . 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.
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