Re: CUBE RESISTANCE

["John S. Denker" <jsd@MONMOUTH.COM>]

Hi --

Let's analyze what Bernard Cleyet wrote:

> http://216.239.53.100/search?q=cache:FR1M90QRFqwC:www.cs.york.ac.uk/~wf/tuts/WF-3.ex.ps+cube+resistance+ohms+vertices&hl=en&ie=UTF-8
>
> another:
> http://www.mathematik.uni-bielefeld.de/~sillke/PROBLEMS/resistance
>
> best one so far: http://rec-puzzles.org/new/sol.pl/physics/resistors

OK, he exhibits three fish.

> hint search: cube resistance ohms vertices

He shows how to find fish in the world-wide pond.

> bc who simplifies by shorting vertices that have the same potential

He explains how to create fish dinner from scratch;
no pond required.

That byline/postscript is worth a hundred times
more than the rest of the note.

Let's take a step back and get some perspective
on this problem.  Why assign such a problem?  They
don't assign it in electrical engineering class,
because cubical resistor networks have no engineering
significance.  So why assign it in physics class?

Is physics just the study of things that are too
complicated and too useless to be of interest to
engineers?

I don't think so!!!!

This is a fine puzzle, an excellent puzzle.  But
you have to present it in the right way or the
customers will miss the point.

At the highest level, the point is that there are
multiple ways to solve any problem.
 -- There's usually a brute-force approach.  I
envision hundreds of medieval villagers carrying
a giant log and trying to break down the front
door of the monster's castle.
 -- There's usually a sophisticated approach.  I
envision the village wizard getting on his bicycle,
riding around to the back, and picking the lock on
the back door of the castle.

There's a style point here, the notion that with a
little sophistication you can get any given task
done with a whole lot less work (or with any given
amount of work you can do N times as many tasks
and get paid N times as much).

At the next level of detail, the point of the cubical
resistor network is that there are certain tricks that
physicists use to simplify problems.  High on the
list is _appeal to symmetry_.

This is the third time in four days that I've used
a symmetry argument to answer a phys-l question.

You can solve the cubical resistor network in your
head, in less time than it takes to say it, by
appealing to symmetry.  Actually I frown upon the
three references cited above, because they make it
sound like a hard problem.

Physics is like one of those theme parks where you
have to pay an admission fee at the gate, but once
you get in, all the rides are free.  That is,
you need a certain amount of sophistication before
the resistor-cube turns into an easy problem.  In
this case, part of the "price of admission" to
PhysicsLand is knowing the symmetries of a cube.
It doesn't do any good to tell the kids APPEAL TO
SYMMETRY if they don't know the symmetries of a
cube.  Most of them think that cubes have fourfold
symmetry (if they think about it at all) ... most
of them don't realize that a cube has threefold
symmetry also.

To repeat:  this problem has got nothing to do
with electronics.  It has everything to do with
symmetry.  The point of this problem is to motivate
a discussion of the symmetry of a cube, and the
role of symmetry in problem-solving.

Most kids can't visualize things in three dimensions.
So they're not even going to follow you if you just
_tell_ them that a cube has a threefold symmetry.
So hand out some cubes for them to play with.

Also if at all possible hand out some octahedrons.
(Why is jsd talking about octahedrons?  Heretofore
the subject was cubes, wasn't it?)

So here is how I would answer the original problem:
 -- Electrodes at the corners are "special" points.
 -- Threefold symmetry about the special points.
 -- Do the sub-problem:  R + two Rs in parallel + R
 -- Three copies of the subproblem makes 2.5 thirds.

The goal is not to teach 'em the answer.  The goal
is to teach 'em enough about symmetry so that they
can figure out the answer in their heads in less
time than it takes to state the problem.