Interesting Physics Problem (Resistors)?

I recall this sort of problem, which in fact will give you a resistance value between 0 and 1, depending on which 2 nodes are picked. Let me check the web to see if I can find something for you. It's far from a "classroom" problem. Whole papers have been devoted to this subject, as the first link indicates.

I know, for example, for a infinite square network of 1 Ω resistors, the effective resistance between two diagonally opposite nodes in a square is 2/π, which is an interesting example of how π has a way of popping up in unexpected places. What's stranger is that this is the same value in Buffon's Needle Problem where the spacing of parallel lines equals the length of the needle, 2/π.

This is not an "idle" poser, since it's related to finding the effective resistance between two points where electrodes touch an infinite sheet of a semi-conductor.

Edit: Okay, I have found your answer. The effective resistance between the 2 nodes as in the cartoon is:

-(1/2) + 4/π = 0.77324...

You can check this and other values in the PDF paper in the 2nd link. See 2nd page.

The key word here is "infinite" - an infinite number of resistances in parallel will absolutely result in zero resistance - or a number infinitely close to zero.

What is the EMF? The voltage given by the battery is it shown on the diagram??