Variation of infinite grid of ideal one-ohm resistors: what would be the equivalent resistance between 2 points in a 3D lattice?

Question

I'm sure that many here are familiar with this famous problem that popped up on xkcd one day:

On this infinite grid of ideal one-ohm resistors, what's the equivalent resistance between the two marked nodes?

I am intensely curious, however: what if instead of an infinite grid of ideal one-ohm resistors, we had an infinite lattice? (interestingly, I could not find a single image on google illustrating this) This brings up two interesting questions:

• What is the equivalent resistance between two nodes that lie on the same plane of the lattice?
• What is the equivalent resistance between two nodes that do not lie on the same plane of the lattice?

I've searched the site a fair bit, and this question does not seem to have been asked. I don't know anywhere near enough about electrical engineering and circuit physics to even begin tackling this problem, but I would love to see something akin to a solution.

Answer 1

A good reference was given in an answer to a related question: Cserti 2000 (arXiv preprint, whose numbers I'll be referring to) solved a number of generalizations of the 2D lattice problem.

For a $d$-dimensional lattice, the resistance between the origin and the point $(l_1, \ldots, l_d)$ is given by eq. 18 in that paper: $$ R(l_1, \ldots, l_d) = R_0 \int_{-\pi}^\pi \frac{\mathrm{d}x_1}{2\pi} \cdots \int_{-\pi}^\pi \frac{\mathrm{d}x_d}{2\pi} \left(1 - \mathrm{e}^{\mathrm{i}(l_1x_1+\cdots+l_dx_d)}\right) \left(\sum_{i=1}^d (1 - \cos x_i)\right)^{-1}, $$ where $R_0$ is the resistance of a single resistor.

The author goes on to talk about the specific case of $d = 3$, where in eq. 35 it is shown that $$ R(l_1, l_2, l_3) = R_0 \int_{-\pi}^\pi \frac{\mathrm{d}x_1}{2\pi} \int_{-\pi}^\pi \frac{\mathrm{d}x_2}{2\pi} \int_{-\pi}^\pi \frac{\mathrm{d}x_3}{2\pi} \frac{1-\cos(l_1x_1+l_2x_2+l_3x_3)}{3-\cos x_1-\cos x_2-\cos x_3}. $$ Moreover, as one might expect with enough dimensions, the increase in the number of paths between points can grow fast enough to balance the increase in path length. Indeed, for $d = 3$ the resistance between two points asymptotes to a constant as the points diverge. The paper gives the constant in several forms, one of which is eq. 39: $$ R_\infty = \frac{\sqrt{3}-1}{96\pi^3} \Gamma^2\left(\frac{1}{24}\right) \Gamma^2\left(\frac{11}{24}\right) R_0 \approx 0.505 R_0. $$ Note that this means even if we stick to a single plane in a 3D lattice, the existence of the third dimension drastically alters the qualitative behavior as we move the two nodes apart, since in 2D $R_\infty$ diverges.