Regular Wilson loops and MHV amplitudes at weak and strong coupling

Traditionally, the duality between Wilson loops and amplitudes beyond one loop in N = 4 SYM is characterised by the remainder function. Because of the perturbative origins of the BDS expression, the remainder function is more natural at weak than at strong coupling. We advocate instead a more direct approach, based on considering ratios of Wilson loops. This allows us to define a manifestly finite, regularisation independent, conformally invariant quantity. It does not make a direct reference to the BDS expression and the definition is regularisation independent. It is a natural object at weak and at strong coupling, and in the latter case is directly related to the free energy of an auxiliary integrable system. We then compute these ratios for continuous families of regular polygons for 6, 8 and 10 points at one and two-loops. These results are compared to expressions derived recently at strong coupling.