Rotational invariance in resistor networks: two-point resistances around an n-fold corner
Résumé
Equivalent resistances between nodes in a resistor network are related when symmetries are present. In this paper, we establish, using van Steenwijk's method, that general relations can be derived around any n-corner node, which is a node with n branches holding an n-fold symmetry axis of the network. The expression of the equivalent resistance between an n-corner node and a neighbouring m-corner node can also be given. For networks with four- and five-fold rotational invariance, we illustrate the additional possibilities offered by rotational invariance and Kennelly's theorem (known as the star-triangle or Y-Δ transform). In particular, we derive general equations for the addition of an n-fold corner on top of any n-fold invariant network. The expressions of two-point resistances for 2 × 4 and 2 × 5 globe networks, for example, are hence obtained. Rotational symmetry, thus, allows exact analytical results to be obtained using elementary methods without heavy calculations, which can be useful for checking the results of advanced numerical methods in practical problems. These didactical examples with resistor networks illustrate the simplifications occurring in a physical system in the presence of rotational invariance.