An example of the relevance of symmetry in physics: corner theorems in grids and cubic resistor networks

Equivalent resistances between various points of a resistor network are related. In this paper, we establish general relations for the corner of a planar grid and for the corner of a three-dimensional network with three-fold symmetry. In three dimensions, two demonstrations are given using Kennelly's theorem or alternatively using van Steenwijk's method. When three-fold symmetry is not satisfied, but when a plane of symmetry exists, then two relations can be proven relating the four corner resistances. These exact relations are useful to check detailed analytical or numerical solutions, and, when corner resistances are only partially known, to derive the values of the desired missing resistances. Examples of applications are also given in the case of regular polytopes or repeating networks such as ladders and scaffolding.