Simple Question

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No. Not all infinite sums are infinite, so that adding an infinite number of conductivities in parallel need not yield infinite conductivity (or zero resistance) if the successive values progress in an appropriate way.

Yes. That's true. What do you think is the solution?

I don't know offhand. It'll probably involve a double integral.

There's a paper on just this topic on the American Journal of Physics site, but I don't have a subscription to be able to download it. Pity. It's at:

Another paper which is freely available online is at:

The short and sweet of it after wasting so much time is as I see it: If the grid is infinite, it does not matter what resistors you use (1 ohm or

I miss added the resistors as you did also. for a 100 x 100 grid you should get by adding it as rows of 100 x 100 = 10,000 + 100 x 100 colombs = 20,000 resistors. The 10 x 10 I mentioned is 200 resistors not 121 I said (late nite = dumb thinking I guess). Just wanted to correct myself :)..

Nope; as was already mentioned, the number resulting from the sum of an infinite series doesn't have to itself be infinite (or zero). A more practical example, though, was already given - imagine an "infinite" sheet of copper (or better, some poorer conductor). Do you really expect to read exactly zero if you touch probes to its surface, some finite distance apart? Of course not. As you already noted in:

Of course there IS such a thing as "infinite." How many points are there along even a finite line? How many points are there on a circle?

Because each additional bit of area is NOT merely another parallel-connected resistance - there's also an effective series component to be considered.

Bob M.

This is an old puzzle, and the answer is trivial to determine if the two points on the grid are adjacent. Imagine that we inject 1 amp into point A; by symmetry the current divides into 4 equal amounts in the 4 resistors leading away from point A, so that the resistor between points A and B carries .25 amp. Now inject -1 amp (withdraw 1 amp, in other words) at point B. Again the current divides equally in the 4 resistors connected to point B, with .25 amp in the resistor connecting point A and B. Thus the current in the resistor connecting point A and B is .5 amp; .25 from the first current source, and .25 from the second. When you have .5 amps in a

If the points are diagonally positioned on the grid, the problem becomes more complicated. There have been some references given in this thread, and another one is at:

Notice that when there is a diagonal offset, the result involves Pi.

To those who think the resistance would be zero between a particular pair of points, consider this: If your reasoning correctly concludes that the resistance is zero between a particular pair of points, then it would be zero between *any* pair of points by the same reasoning (the reasoning being that all those resistors in parallel extending out to infinity would reduce the resistance between a pair of points to zero). But, it is plainly not zero between two adjacent points as can be seen by the symmetry argument I gave above.

Theory says differently. And so does practice. Someone else already has pointed out the thinking process that should qualitatively tell you that you are wrong about this, without the need for quantitative theory. But quantitative theory clearly says the actual value is non-zero and provides an actual number, in fact.

This way of thinking says more about you, than nature itself.

Nope. I solved it correctly. You could learn from that.

Jon

I wanted to add one more thing to aid your ability to imagine. Take a look at one of the web sites I pointed Chris at:

Read carefully the much more easily solved case and easily imagined case of an infinite ladder of resistors. If there is some aspect you don't follow, just ask here. I can help, I think. But see if you can follow the argument and see if you can work through ALL of the algebraic manipulations to arrive at the same answer they show (not just the steps they illustrate, but all the intermediary steps you may feel you need, in between.)

Your argument would seem to disagree with their results. Yet I can assure you that they are right about it.

Think in detail, James. Not just handwave. It makes a difference.

Jon

Right. If it were always the case that an infinite sum resulted in the extreme conclusion of either infinity or zero, there is a lot of the world we'd still not understand well and calculus would be pretty much rendered useless on the whole.

The very suggestion that Leo makes tells me that he hasn't been usefully exposed to calculus.

Jon

Nope.

The resistance across the diagonal of one square in an infinite grid of

From:

I'm guessing but it looks like the knight's move resistance is maybe 4/PI - you do the math!

Bibliography

As long as you're writing the program, how about a sort of chart, or graph, of the node-to-node resistance for a whole family of matrix sizes - 4 x 4, 9 x 9, 10 x 10, 100 x 100, and so on, and see if it approaches a limit - it might be interesting to see the difference between, say, 80 x 80 vs. say, 1000 x 1000.

Thanks, Rich

Then you use ellipses: '...'

Cheers! Rich

Trivial to do, actually. It's just that the run time right now using interpreted basic is painful. I may get around to it. But I've posted my code, too, so anyone else can try a hack at it if they want to.

Jon

(8-pi)/(2*pi), actually. I did the math.

Jon

this is incorrect.

all current leaves point A by 4 resistors and enters point B by 4 resistors

if the rest of the grid were to be replaced by superconductor you'd have a resistance of R/2

(unless A and B are neighbours 2R/5)

Bye. Jasen

"Jonathan Kirwan" schrieb im Newsbeitrag news: snipped-for-privacy@4ax.com...

Yes. That's correct :-) It's not something I particularly regret, though ;-)

Leo

"Bob Myers" schrieb im Newsbeitrag news:ZMxSg.409$ snipped-for-privacy@news.cpqcorp.net...

Really? Show me a thing that is denoted by an adjective ;-)

I don't know. I haven't counted them. Nor has anybody. How do you know they are infinite in number? How can you be sure that they aren't more than infinite in number? How can you be sure that you haven't simply defined it to be as you would like to have it?

This is becoming an interesting discussion, but getting slightly OT.

Let me conclude: You have convinced me - I was wrong on the 0 ohms. But let me say this:

Infinity is just a word; it is a concept in our minds. It is not a "thing", nor is it an attribute that anything in the real world has. It's not even a "real" number (pun intended), because you can't calculate properly with it. It is a useful crutch used by mathematicians. It is a recursively defined concept. What is the main characteristic of "infinity"? That it's infinite, nothing else. There is no such "thing".

For further discussions we should move to math.philosophy if there is such a newsgroup ;-)

Regards, Leo

A simple thought experiment suffices. Pick a point anywhere on the line or circle. Pick another point. Unless that second point is PRECISELY in the same location as the first (in which case it is the SAME point), it logically must be located some small - perhaps VERY small - spatial distance from the first. As any distance separating the points could always be divided, you can now pick a new point between the initial two. This process can in theory continue on forever, hence the number of points on the original line segment or circle must be infinite.

You might argue that you will eventually reach a point where it is beyond our physical ability to measure the distances separating the points - but that is irrelevant, since we are talking about th existance of such additional points solely from the perspective of mathematical theory.

While there ARE higher "degrees" of infinity, all such may still be said to be "infinite." Thus "more than infinite" is a meaningless distinction in this context.

Clearly, you CAN "calculate properly with it" - the fact that you yourself are not familiar with the techniques for mathmetically dealing with infinite quantities, series, etc., does not mean that such do not exist.

The statement that "infinitely is just a word, it is a concept in our minds" has more to do with philosophy (semantics and metaphysics, specifically) than mathematics. If you want to go in that direction, then the next logical challenge would be for you to demonstrate conclusively that there is ANYTHING which "actually" exists outside of your own mind. That quickly becomes a rather interesting, although perhaps pointless, exercise.

Bob M.