A Candle in the Dark

From xkcd,

So, how does one find the effective resistance?

The solution to this question, and many other network shapes, is present in Jozsef Cserti’s paper¹.

Additionally, this website² also clearly explains the technique to find the general equivalent resistance between any two nodes. The resistance between the two marked nodes in the xkcd question is,

which turns out to be . Check out Appendix A of Cserti’s paper for a technique to evaluate the above integral.

External Links
[1] – Application of the lattice Green’s function for calculating the resistance of an infinite network of resistors, Cserti József, American Journal of Physics, Volume 68, Issue 10, pp. 896-906 (2000). arXiv:cond-mat/9909120v4
[2] – Infinite 2D square grid of 1 ohm resistors