Re: qualitative reasoning

[John Denker <jsd@MONMOUTH.COM>]

Regarding the skewered cube, at 06:13 AM 6/26/00 -0400, David Bowman wrote:
> This example is even better than the previous one.

I'm glad you like it.

> John, did you dream this up yourself, or did you see it elsewhere?

I dreamed it up when I was in grad school.  The Wigner-Eckart theorem had
just been covered in class.  A bunch of my friends didn't understand it,
and they started pestering me for an explanation.  So I skewered a cube and
let them twirl it.

I also skewered a disk through its center at a cockeyed angle, so people
could twirl it and feel the wobble that results when the angular momentum
is not aligned with the angular velocity.

> I'm impressed.  Do you have any more examples where these came from?  A
> book on the wonders of various symmetries perhaps?

I have a few more, at various levels of sophistication.  Some original,
some not.  I'll post them when I get the chance.

Actually I know of distressingly few resources on the topic.  I like to
call the topic "qualitative reasoning".  It includes things like
 -- symmetries
 -- conservation laws
 -- scaling laws
 -- dimensional analysis
 -- expanding to lowest order
 -- mnemonics for remembering the sizes of key quantities

AltaVista generates thousands of hits on "qualitative reasoning" but most
of them are from the field of fuzzy logic, where the term just means "close
match", not the sort of high-power high-accuracy qualitative reasoning that
we're talking about here.

 *) Feynman had quite a collection of examples.
 *) I once heard Weisskopf give a very nice lecture on the subject.
 *) Wigner was a skillful practitioner.

... but beyond the Wigner-Eckart theorem, I'm not sure how much of this got
written up.  I once saw a book entitled _The Buckingham Pi Theorem_ or some
such;  an entire book seemed like more than the subject deserved.  Of
course any book on fluid dynamics will talk about dimensional analysis, and
any book on critical phenomena will talk about scaling laws -- but these
are hardly suitable introductions to qualitative reasoning.

If anybody knows of good resources on the topic, please let us know.