A lower bound on the distortion of embedding planar metrics into Euclidean space

(MATH) We exhibit a simple infinite family of series-parallel graphs that cannot be metrically embedded into Euclidean space with distortion smaller than (n\, ). This matches Rao's general upper bound for metric embedding of planar graphs into Euclidean space, 14, thus resolving the question of how well do planar metrics embed in Euclidean spaces.