Hmm, interesting. If we added a point at infinity to R^n as a topological space, there is a natural way to view this as S^n, the n-sphere. But what is the rigorous mathematical object corresponding to the edges of the integer grid in R^n (n >= 2) after a "point at infinity" has been added. Just curious. --Dan Veit wrote: << Connect the boundary of a large hypercubic resistor grid to "ground". Ground acts as a current source at zero potential. Now apply a voltage V to the node at the origin such that 1 amp is extracted there. By symmetry, the current flowing in each of the resistors connected to the origin node is 1/(2D) amp, where D is the dimension. The potential across these resistors is 1/(2D) volt since their resistance is 1 ohm. Now consider what happens when instead, voltage -V is applied to a neighboring node, say [1,0]. We have the same symmetrical distribution of currents, only the center is shifted and the current directions are reversed. On the other hand, the voltage across the resistor R connecting [0,0] to [1,0] is the same (in magnitude and sign) as it was before. Now superimpose these two current/voltage distributions. The voltage across R will be 2/(2D) and exactly one amp will flow out of [0,0] and into [1,0]. The equivalent resistance is therefore 1/D.

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