Re: Don Kelly's pentagon problem



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"Benj" <bjacoby@xxxxxxxxxxx> wrote in message news:dee7b4ce-150f-4422-a7fe-1a8c87c3c2ff@xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
On Oct 1, 12:29 am, Benj <bjac...@xxxxxxxxxxx> wrote:
On Sep 30, 7:41 pm, "Don Kelly" <d...@xxxxxxxxxxxx> wrote:

> Any more tries?

I say it ends up with two one ohm resistors and one two ohm resistor
all in parallel. Which of course gives .4 Ohms. But then I'm not very
good at puzzles.

OK. I'll reply to myself. Next I looked at the arrangement for
resistance between vertices with one in between rather than adjacent.
Again I basically came up with two one ohm resistors and one two ohm
resistors. Which gives .4 Ohms.

Comments.

Of course the essence of the puzzle is that if you have a balanced
bridge arrangement, any resistance across the balanced part doesn't
count because there is no current there. The surprise to Salmon Egg
and to me was that the resistance comes out the same no matter which
two vertexes you measure between.

Of course symmetry is a very powerful argument in physics and the
extremely symmetrical arrangement of the pentagon with the 5 pointed
"star" on top of it, strongly suggests that such an outcome should not
be surprising.

But I'm not much for puzzles and hey, this is a PHYSICS group, not
some mathematical fairyland. So... I built the damn thing to "do the
experiment". First trial was a failure. I had a large drawer full of
precision 1% one Ohm resistors so I built the pentagon. Bad news. The
low final resistance plus the lead resistance and lack of meter range
made the readings too inaccurate to be reliable. OK. Well, hey
experimental physics is about trying things (and thinking about them)
until you get the data, right? So I built the thing out of less
accurate 10K resistors. YES! Sure enough, within experimental
(resistor) error, you can see that the measurement between ANY two
vertexes was 4 K. Amazing!

So there you have it. A mathematical model verified by experiment!
What more can you ask for? Well, I suppose instead of 5% resistors
where I got 4.21 k as an answer one could refine this with 1%
resistors to see if the theoretical 4k ohms is approached! And so on.
Of course this physics was a lot cheaper on the taxpayers (me in this
case) with ten 1 cent resistors than to try things by building a super-
collider!

And finally shall we take note of the fact that absolutely NONE of the
usual clowns who spend their time on the net calling other people
stooopid and "idiot" and suggesting that they learn about science, had
enough hairs on their collective butts to even TRY to guess an answer
let alone do the experiment. Empty barrels make the most noise.

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Bonus for the measurements. I note that neither you nor Salmon Armtried circuit analysis methods but applied critical thinking instead. Good for you.
However, this approach may run into problems when the number of vertices increases-or one considers a cube where every vertex is tied to the other 7 vertices, or some other n-vertex polygon. At what point does visualiztion reach a limit?
How do you model the equivalent resistance between any pair of vertices? It can be done.

Yes, it is a mathematical model and the physics involved is no more than Kirchoff's and Ohm's laws-mathematical expressions of what is observed. Does the math fit the observations? In this case, it does. So,score 1 for the "mathematical" model which does what it should do -reflect reality. If it doesn't do so- then pitch it- or (horrors) pitch the "physics" upon which the mathematic is based.

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Don Kelly
dhky@xxxxxxxxxxxx
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