In Graph Minors III, Robertson and Seymour write: "It seems that the
tree-width of a planar graph and the tree-width of its geometric dual are
approximately equal - indeed, we have convinced ourselves that they differ by
at most one". They never gave a proof of this. In this paper, we prove a
generalisation of this statement to embedding of hypergraphs on general
surfaces, and we prove that our bound is tight