Infinite Grid of Resistors
66 comments
·June 14, 2025neepi
Taniwha
Eventually you need to pullin a physicist too who will point out that at an appropriate distance quantum effects will dominate - because eventually at a far enough distance the number of electrons moving per second (ie current flow) will be either 0 or 1 at some nodes
__MatrixMan__
I've not studied QED directly, so by all means correct me if I'm wrong, but it seems to me that we'd get a double-slit like scenario where it's as if a partial electron went through either path. We might want to say that surely a whole electron took one path and not the other but we couldn't say which and if we tried to instrument to and find out we'd affect the resistance.
But that's fine because knowing which path the electron took is not part of the problem. Both paths contributed to the resistance even if one was not taken.
We only have to worry about quantum effects if the probabilities are not a decent proxy for the partial-particles that we suspect don't exist. In this case, the Physicist can probably proceed directly to the bar and have a drink with the Mathematician.
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mjevans
Intuitively I knew this class of problem was theoretical only BS when it came up in college...
I hadn't considered that sort of strange effect though! Makes me feel not so bad for 'never really getting it' because I just couldn't wrap my mind around the problem description's obvious inanity and the infinite edges.
aydyn
The question is not pretending to be realistic. No one is thinking it is possible to build an infinite grid of resistors.
It's simply an evaluation of your mathematical ability to manipulate the equations and overall understanding of them, wrapped up in a cute little thought experiment. This evaluation IS relevant to more realistic scenarios and therefore your grade and engineering ability.
divbzero
An infinite grid of resistors is clearly a toy scenario, but the infinite universe is a reality that astrophysicists try to reason about. I wonder if there are blindspots in astrophysics because we lack intuition about the universe at that scale and are forced to approach it from theory.
red75prime
> The electrical engineer suggests it's not measurable unless you apply current and also asks "when"
Just wait infinite time for all the transient responses to die down. The grid to enter steady state and to became true to the schematic.
repiret
I think there are two interpretations of schematics.
One is where the components on the schematic represent physical things, where the resistors have some inductance and some non-linearity, and some capacitance to the ground plane and so on. This is what we mean by schematics when we’re using OrCad or whatever.
There is another interpretation where resistors are ideal ohms law devices, the traces have no inductance or propagation delay or resistance. Where connecting a trace between both ends of a voltage source is akin to division by zero.
Sometimes you translate from the first interpretation to the second, adding explicit resistors and inductors and so on to model the real world behavior of traces etc. if you don’t, then maybe SPICE does for you.
Infinite resistor lattices exist only in the second interpretation.
bravesoul2
Given an infinite grid of resistors... would you expect planets to form?
corysama
They say hydrogen is an odorless colorless gas which, in sufficient quantities, given enough time, turns into people. I’m sure the same could be true of resistors.
bravesoul2
Resistors are made of heavier elements though. And I remember something like everything wants to become iron (fuse if lighter, decay if heavier)
That said there might be enough energy (infinity!) for anything to be possible.
inopinatus
People are resistors too.
QuadmasterXLII
I think it collapses into a black hole. black hole mass scales with radius, grid mass scales with radius squared
jfengel
It becomes a black hole, but it doesn't necessarily collapse, at least not at first. A supermassive black hole has very low density and a very gentle gravitational gradient.
All of the mass does end up in the singularity, in finite time (at least for any finite subset of the black hole), but it doesn't automatically become super dense just because it's a black hole. It can remain quite ordinary for a very long time.
pixl97
Thats what I'm not sure about here...
If we assume this is an infinite grid in its own universe then nothing can actually move. The gravitational pull should be the same from every direction. If we assume the grid is perfect then there is no nucleation sites to start a collapse. The grid would be in perfect balance.
The same is thought about our universe. If there hadn't been small quantum fluctuations during the inflationary stage it would have taken much longer for what we see in the modern universe to form.
rzzzt
Assume perfectly spherical through-hole resistors soldered on an infinite PCB.
sandworm101
And the electrician knows he can get a 99% answer out of a 10x10 grid on a workbench. The engineer is free to then add more resisters to the periphery until either the grant money runs out or the physicist's publishing deadline approaches.
A really difficult question: At each distance, what percentage of soldering errors in the grid can be tolerated before the fluke meter across the center square detects the fault? (That might actually be a thing as I've heard people talk about using changes of local resistance to detect remote cracks in conductive structures ... like maybe in a carbon fiber submarine hull.)
nerdsniper
For measuring corrosion in conductive surfaces, “eddy current” testing is often used. It uses AC current of some frequency, so it’s technically measuring inductance rather than resistance.
kayson
A much more useful (in the educational sense) question to ask, in my opinion, is the resistance between opposite corners of a cube of 1ohm resistors. There are some neat intuitions it can help build (circuit symmetry, KCL, etc). The infinite grid is too much an obscure math problem that seems like it might be solvable in an introductory circuits class.
mmastrac
This was the question I hated in my EE degree. The thought exercise was a favourite of the profs.
praptak
What I never got about the simple symmetry-based solution is "if we accept the idea that we can treat the current fields for the positive and negative nodes separately".
Why are the currents in the two node solution (not symmetric) a simple sum of the currents of two single node solutions (symmetric)?
Obviously the 2 node solution still has some symmetries but not the original ones that let us infer same current in every direction.
IronyMan100
the Maxwell Equations are linear in the electric and magnetic fields, then you can add Up and subtract fields and Potentials from each other. It's the same Argument for why interfernce works or optical gratings
pyman
Re: the infinite resistor grid
If you take an endless grid made of identical resistors and try to measure the resistance between two neighbouring points, the answer turns out to be about one-third of a resistor
quibono
There's one thing I don't get about the symmetric+superposition explanation. Why are there alpha - beta - alpha on the adjacent nodes, and not alpha-alpha-alpha? I.e. why is one of the directions distinct while the other two are considered the same?
magicalhippo
Start by assuming they could potentially be all different, so denote the currents i_1 to i_12.
However note the problem is symmetrical about the vertical axis, so flip the figure. The current passing through the flipped paths should be the same as before the flip, so note down which i's equate to each other due to this.
Note that the problem is symmetrical about the horizontal axis, and do the same there. Note that the problem is symmetric when rotated 90 degrees, so do that. And so on.
In the end you'll have a bunch of i's that are equal, and you can group those into two distinct groups. Call those groups alpha and beta.
edit: Another way to look at it is that you can't use the available symmetry operations to take you from any of the alphas to a beta. This is unlike alpha to alpha, or beta to beta.
clbrmbr
The finite grid of resistors (or arbitrary impedances) is actually of great practical usefulness.
Kirr
This may be as good time as any to plug my calculator for finite resistor networks (including grids) [1]. It works by eliminating non-terminal nodes one by one with the Star-Mesh transform, while keeping the exact rational resistances at each point.
[1] https://kirill-kryukov.com/electronics/resistor-network-solv...
sriku
This is cool and I have my own take on it after being nerd sniped by XKCD - https://sriku.org/posts/nerdsniped/ - I link to this article at the end but that post specifically solves the xkcd puzzle.
nimish
In the integral, the h_m(s) are chebyshev polynomials of the first kind
petschge
See also https://xkcd.com/356/
ordu
Why mathematicians are three points? I think it is easier to disable a mathematician. Look at this discussion, for example. EE engineers and physicists are dismissing the problem outright, while mathematicians have no issues thinking about it.
quinndexter
-Why mathematicians are three points?
Possibly based on this ranking. Everything sub-mathematician is 2 points? Maybe there's subdivision of points.
Mawr
See alsoer https://youtu.be/zJOS0sV2a24?t=932
causality0
My math isn't strong enough to follow the whole article, but my intuition as someone who works in electronics is that when a quantized system interacts with an infinity, the infinity is restricted based on the magnitude of the quantized factor. Electric charge is quantized. Less than one electron cannot pass through a node, therefore an infinite grid of resistors is effectively a finite grid of resistors whose size changes based on how much charge is dumped into the system.
morepedantic
That was my initial thought, but on further reflection it feels wrong. The electron is also a wave, and that wave can spread across the entire grid.
Another interesting aspect is that in an infinite grid, a spontaneous high voltage is going to exist somewhere at all times. It is probably very far away from you, but it's still weird.
yusina
Funny to put "intuition" and "infinity" into the same sentence.
The only type of person for whom intuition about infinity to form is not entirely unlikely are mathematicians.
shove
Word on the street was that my Physics professor at NCSSM (Dr Britton) worked on this problem during his doctorate
I'm a bit mathematician and a bit electrical engineer.
The electrical engineer suggests it's not measurable unless you apply current and also asks "when" after the current is applied referring to the distributed inductive and capacitive element and the speed of field propagation. The mathematician goes to a bar and has a stiff drink after hearing that.