Date: Sun, 9 Jun 2002 00:13:25 -0400 (EDT)
From: "Keith F. Lynch" <kfl at keithlynch.net>
To: WSFAlist at keithlynch.net
Subject: [WSFA] Infinite resistor problems
Reply-To: WSFA members <WSFAlist at keithlynch.net>

At last night's meeting, Eric asked me what the "infinite resistor
problems" I mentioned having given to Marc Drexler and Carl Devore
late Friday night at Balticon.  (As mentioned in my Balticon report
in the latest WSFA Journal.)  I have also given at least one of them
to Ron Kean.  Here they are:

1. Given an infinite square grid (like a flyscreen) of one ohm
   resistors, what is the in-circuit resistance that would be
   measured across one of them.

2. The same, except with a 3-dimensional cubical grid, like a giant
   jungle gym filling all of space.

3. The same as 1, but a hexagonal grid, like the pattern on the floor
   of a Metro station.

4. The same as 1, but a triangular grid.

5. The same as 1, but what's the resistance across a "knight's move,"
   i.e. two units in one direction and one unit at right angles to that?

6. Instead of discrete resistors, fill an infinite flat plane with a
   uniformly resistive substance of uniform unit depth, such that
   between opposite faces of an isolated unit cube of the substance,
   resistance would be 1 ohm.  What's the in-circuit resistance when
   the cube is no longer isolated, but immersed in the infinite flat
   plane of identical cubes?

7. Generalize 6 to three dimensions.  Now all space is filled with the
   stuff.  What's the measured in-circuit resistance across one such
   cube.  (Ignore the problem of where the ohmmeter could possibly be,
   what with all of space already occupied.)

We also discussed the old non-infinite problem of the measured
in-circuit resistance between opposite corners of a cube whose 12
edges are all 1 ohm resistors.  Or across just one of the resistors.
or between opposite corners of one of the six faces.  Similarly with
the other four regular solids.

After Balticon, Ron Kean (who wasn't at that con) retaliated with a
simple triangle of resistors.  The trick is, you're given the three
unequal measured in-circuit resistances, and have to figure out what
the resistances of the three isolated resistors must be.  I beat it to
death with algebra, by hand, and got the correct answer.  I gave it to
Carl Devore, who had Maple (a computer algebra package) chew it up.
He came up with the same solution.

I hope that at least Eric and Adrienne will have fun with these,
though I hope to hear from others too.

If these are too easy, I can add inductors and capacitors, and ask
about reactances, phase angles, and impedences.
--
Keith F. Lynch - kfl at keithlynch.net - http://keithlynch.net/
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