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.
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
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.)
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.
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.
Going on something of a tangent: in engineering, it seems unusual to talk about “applying current,” it’s usually voltage (say, across a resistor) or some sort of an “electromotive force.”
People think this is not relevant to real world problems but it actually is, albeit all the calculations aren't that relevant. Silicon substrate's resistance is basically an infinitely large grid of unut resistances at the distances relevant for a local point of an IC. Note that silicon substrate is often heavily doped (p-type) and all info you get from the fab is it's resistivity (often somewhere between 1 to 100 ohm per cm). For the most advanced tech nodes its often 10 ohm/cm. If you need to develop some intuition about noise coupling via the substrate you have to think that it's a grid instead of just calculating the resustance between point A and B. We need to distribute a grid of substrate contacts to collect the noisy currents too. So the grid shows up again!
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.
I saw this question only once, as the first of 4 problems on the final exam for my very first EE introductory course. The course had covered an infinite ladder of resistors, but at the time it seemed like quite a leap to apply that knowledge to this problem.
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.
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
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?
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.
Odd. As an undergrad in physics, we had a project for our team which involved percolation theory and "testing" it. So, we had to make differing grids of conductive ink, with a certain number of "links" (resistors, edges in the graph) as missing. Getting even-flowing conductive ink was hard. I wrote all of the software for the XY plotter, pushing out instructions to make rectangular and triangular grids. Then we would measure the resistance from one side to another.
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.
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.
That only matters if you're measuring in the time domain and seeing the noise due to individual carriers. Often you just care about averages over some time and space (e.g. the macroscopic flow of water behaves quite different from the speeds of the individual molecules).
This is a discrete case of "sheet resistance."[1] The resistance between any two points, nodes in this case, is the same. We covered this in the EE uni curriculum back in the day, but I don't remember the solution derivation anymore.
Aside: Veritasium had a great video similar to this on the paths that light takes. I'll link to the part where they do the best physics demo I have ever seen:
sadly the demo is not so impressive as they make it out to be: The extra-path light is 'caused' by exit diffraction of the light source.
Now, the same underlying theory also explains why there will always be diffraction from any finite boundary resulting in a reality which is indistinguishable from one where light actually takes all possible paths. But to argue that it actually does is arguably more meta-physics than physics. The demonstration is further compromised by the fact that the laser's diffraction performance is also presumably far from the physical limit.
So a cynic seeing that demo would say "Isn't that just due to some of the light from the laser being off-axis" -- and it absolutely is. The physics means that some of the light will always be off-axis, but the demonstration does nothing to establish that.
At infinite scale this reduces to the bulk equation R = rl/A for a rectangular block where r is resistivity, l is length, and A is the area of the block.
Both l and A are infinite. So you get infinity/infinity, which is undefined, proving it's a silly problem and you should go do something useful with your time instead.
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
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.
Because it is effectively a superconductor! An electron (or proton) with some velocity will keep going in a straight line forever. This is neglecting the influence of stray background magnetic fields and gravitational fields, but the general notion applies.
Resistance is V/I. You literally cannot have current flowing in a resistor without a voltage across it (either the voltage causes the current to flow, or the resistor in the path of a flowing current has a voltage appear across it).
A voltage drop with a flowing current is power (P=VI).
There is literally nothing you can do to avoid resistors dissipating that power as heat, it's just what they are. If they didn't do it, they wouldn't be resistors.
What you can do is use larger resistances which need less current to see the same voltage (e.g. change a pull up from 10k to 100k or higher, but that's more sensitive to noise), or smaller resistances that drop less power from a given current (e.g. a miiliohm-range current shunt, and then you need a more sensitive input circuit) or find another way to do what you want (e.g. a switched-mode power supply is far more efficient than a voltage divider at stepping down voltage). This is usually much more complex and often requires fiddly active control, but is worth it in power-constrained applications, and with modern integrated technology, there's often a chip that does what you need "magically" for not much money.
You're describing thin film resistors, and they exist.
They also just convert current to heat though. Some amount of current moving through a material with some amount of resistance always produces a fixed amount of current in accordance with Ohm's law. You can't really get away from that.
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.
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.
Now I'm wondering if anyone has built a very large grid of resistors to try to approximate / curve fit this. Surely there's a youtube video in this ...
I imagine that a 100x100 grid of resistors all terminated in a massive resistor would be the same as long as you kept to the middle of the grid for testing.
I don't get why EE education emphasizes problems of this sort. The infinite grid is an extreme example, but solving weirdly complicated problems involving Kirchoff's laws and Thevenin's theorem was a common way to torture students back in my day...
Here, I don't think it's even useful to look at this problem in electronic terms. It's a pure math puzzle centered around an "infinite grid of linear A=B/C equations". Not the puzzle I ever felt the need to know the answer to, but I certainly don't judge others for geeking out about it.
I was about to say "they still torture students this way" but stopped myself when I remembered I took Circuits 1 and 2 back in 2007. So maybe my knowledge is dated too...
It's a weird butterfly effect moment in my career though. I had an awesome professor for circuits 1, and ended up switching majors to EE after that. Then got two more degrees on top of the bachelor's
> I don't get why EE education emphasizes problems of this sort
Last I checked, they don't. I certainly never hit an "infinite grid of resistors" in general circuits and systems except as some weird "bonus" problem in the textbook.
Occasionally, I would hit something like this when we would be talking about "transmission lines" to make life easier, not harder. ("Why can we approximate an infinite grid of inductors and capacitors to look like a resistor?")
It's possible that infinite grid/infinite cube might have some pedagogical context when talking about fields and antennas, but I don't remember any.
If all you mind is the EE curriculum then ok. Or else there is an interesting work of Gerald Westendorp on the web [1] on how allowing other classical passive components (Ls & Cs) you can get discretizations (and hence alternative views) of a very wide class of iconic Physics partial differential equations (to the point that the question is more what cannot be fit to this technique). G. W. is alive and kicking in mathstodon.
> Here, I don't think it's even useful to look at this problem in electronic terms
I always thought this problem was a funny choice for the comic, because it’s not that esoteric! It’s equivalent to asking about a 2d simple random walk on a lattice, which is fairly common. And in general the electrical network <-> random walk correspondence is a useful perspective too
One of my core grips with STEM education, mainly the math heavy part of it, which frankly is most of it, is that it is taught primarily by people who love math.
The people who loved application and practical solutions went to industry, the people who got off spending a weekend grinding a theoretical infinite resistor grid solution went into academia.
neepi|8 months ago
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.
Taniwha|8 months ago
sandworm101|8 months ago
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.)
repiret|8 months ago
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.
red75prime|8 months ago
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.
bravesoul2|8 months ago
divbzero|8 months ago
nofunsir|8 months ago
Koshkin|8 months ago
Going on something of a tangent: in engineering, it seems unusual to talk about “applying current,” it’s usually voltage (say, across a resistor) or some sort of an “electromotive force.”
bgnn|8 months ago
ChoGGi|8 months ago
I'll see myself out.
eternauta3k|8 months ago
fraserphysics|8 months ago
kayson|8 months ago
mmastrac|8 months ago
dcassett|8 months ago
praptak|8 months ago
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|8 months ago
quibono|8 months ago
magicalhippo|8 months ago
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.
at_a_remove|8 months ago
causality0|8 months ago
morepedantic|8 months ago
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.
eternauta3k|8 months ago
yusina|8 months ago
The only type of person for whom intuition about infinity to form is not entirely unlikely are mathematicians.
dogman1050|8 months ago
[1] https://en.m.wikipedia.org/wiki/Sheet_resistance
Balgair|8 months ago
https://www.youtube.com/watch?v=qJZ1Ez28C-A&t=1500
nullc|8 months ago
Now, the same underlying theory also explains why there will always be diffraction from any finite boundary resulting in a reality which is indistinguishable from one where light actually takes all possible paths. But to argue that it actually does is arguably more meta-physics than physics. The demonstration is further compromised by the fact that the laser's diffraction performance is also presumably far from the physical limit.
So a cynic seeing that demo would say "Isn't that just due to some of the light from the laser being off-axis" -- and it absolutely is. The physics means that some of the light will always be off-axis, but the demonstration does nothing to establish that.
TheOtherHobbes|8 months ago
Both l and A are infinite. So you get infinity/infinity, which is undefined, proving it's a silly problem and you should go do something useful with your time instead.
1970-01-01|8 months ago
pyman|8 months ago
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
Koshkin|8 months ago
shove|8 months ago
clbrmbr|8 months ago
Kirr|8 months ago
[1] https://kirill-kryukov.com/electronics/resistor-network-solv...
bilsbie|8 months ago
jiggawatts|8 months ago
nimish|8 months ago
bilsbie|8 months ago
Wouldn’t that be more efficient than converting current to heat?
grues-dinner|8 months ago
A voltage drop with a flowing current is power (P=VI).
There is literally nothing you can do to avoid resistors dissipating that power as heat, it's just what they are. If they didn't do it, they wouldn't be resistors.
What you can do is use larger resistances which need less current to see the same voltage (e.g. change a pull up from 10k to 100k or higher, but that's more sensitive to noise), or smaller resistances that drop less power from a given current (e.g. a miiliohm-range current shunt, and then you need a more sensitive input circuit) or find another way to do what you want (e.g. a switched-mode power supply is far more efficient than a voltage divider at stepping down voltage). This is usually much more complex and often requires fiddly active control, but is worth it in power-constrained applications, and with modern integrated technology, there's often a chip that does what you need "magically" for not much money.
mort96|8 months ago
They also just convert current to heat though. Some amount of current moving through a material with some amount of resistance always produces a fixed amount of current in accordance with Ohm's law. You can't really get away from that.
unknown|8 months ago
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petschge|8 months ago
ordu|8 months ago
Mawr|8 months ago
sriku|8 months ago
kevinmhickey|8 months ago
personjerry|8 months ago
rwmj|8 months ago
Kirr|8 months ago
BizarroLand|8 months ago
unknown|8 months ago
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steamrolled|8 months ago
Here, I don't think it's even useful to look at this problem in electronic terms. It's a pure math puzzle centered around an "infinite grid of linear A=B/C equations". Not the puzzle I ever felt the need to know the answer to, but I certainly don't judge others for geeking out about it.
goochphd|8 months ago
It's a weird butterfly effect moment in my career though. I had an awesome professor for circuits 1, and ended up switching majors to EE after that. Then got two more degrees on top of the bachelor's
bsder|8 months ago
Last I checked, they don't. I certainly never hit an "infinite grid of resistors" in general circuits and systems except as some weird "bonus" problem in the textbook.
Occasionally, I would hit something like this when we would be talking about "transmission lines" to make life easier, not harder. ("Why can we approximate an infinite grid of inductors and capacitors to look like a resistor?")
It's possible that infinite grid/infinite cube might have some pedagogical context when talking about fields and antennas, but I don't remember any.
jesuslop|8 months ago
[1] https://westy31.nl/Electric.html
choonway|8 months ago
ohxh|8 months ago
I always thought this problem was a funny choice for the comic, because it’s not that esoteric! It’s equivalent to asking about a 2d simple random walk on a lattice, which is fairly common. And in general the electrical network <-> random walk correspondence is a useful perspective too
bobmcnamara|8 months ago
Hey now, those actually come up sometimes.
Workaccount2|8 months ago
The people who loved application and practical solutions went to industry, the people who got off spending a weekend grinding a theoretical infinite resistor grid solution went into academia.
esafak|8 months ago
unknown|8 months ago
[deleted]
tekla|8 months ago
unknown|8 months ago
[deleted]