An infinite network (XKCD) - is it not clear?

[OJFord]

Hi, unless I'm missing something here, it seems to me that the answer is that it is infinite, and that that is pretty intuitive.

Is that not the case?

I would think it could be simplified to view as two parallel and infinite resistances, giving ∞²/2∞, which simplifies to half infinity, which is of course really (in as much as it is) still infinite.

 

[ZVdP]

I have no solution, but the resistance cannot be larger than 1.5 Ohm, since there are two independent paths of 3 resistors between the points. The other parallel paths can only lower the resistance. So it's definitely finite.

The forum page of this comic will no doubt contain the correct answer.

 

[Filip Larsen]

I think it is intuitive that the resistance can not be infinite.

With some hand-waving: whenever you have a parallel circuit the equivalent resistance is lower than any of the branch resistance, so the resistance between any to adjacent nodes must be less than 1 Ohm. This means you have a path between the two marked nodes as a series of three networks that can be replaced with a resistor less than 1 Ohm, totaling less than 3 Ohm.

The exact solution, I seem to remember from years back, is a bit harder to obtain. Found a derivation [1] that may be of some use (haven't read it thoroughly enough to say if it is correct or not).

[1] http://mathpages.com/home/kmath668/kmath668.htm

 

[OJFord]

I have no solution, but the resistance cannot be larger than 1.5 Ohm, since there are two independent paths of 3 resistors between the points. The other parallel paths can only lower the resistance. So it's definitely finite.

I think it is intuitive that the resistance can not be infinite.

With some hand-waving: whenever you have a parallel circuit the equivalent resistance is lower than any of the branch resistance, so the resistance between any to adjacent nodes must be less than 1 Ohm.
[1] http://mathpages.com/home/kmath668/kmath668.htm

Right, that was stupid of me. Thank you both - and for the link.

I see why it was comic-worthy now, the answer is certainly not trivial!

 

[alphysicist]

I would say that the concept of infinity is a complex and abstract one that can be difficult to fully comprehend. In the context of a network, an infinite network would mean that the number of nodes or connections is infinite, which is not something that we can easily visualize or understand. However, mathematically, it is possible to represent an infinite network using equations and models. In the example given, the approach of simplifying it to two parallel and infinite resistances is a valid way to conceptualize it, but it is important to note that this is just one way of looking at it and there may be other ways to model an infinite network. Ultimately, the answer to whether an infinite network is clear or not depends on the perspective and understanding of the individual.