Subject: SciAm reference re Stasheff
Date: Mon, 19 Jan 2004 09:34:59 -0500 (EST)
From: Jack Morava <jack@math.jhu.edu>

You may be tired of this topic by now, but here are some
remarks by Herb Bernstein about the Scientific American
article which Jim asked about. I'm embarassed that I'd
forgotten that another close friend, Tony Phillips at
Stonybrook, was co-author of the article.

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Date: Sun, 18 Jan 2004 14:05:36 -0500
From: Herbert J Bernstein <hjbNS@hampshire.edu>
Subject: Re: SciAm

         Fiber Bundles and Quantum Theory

the article was in July '81 Sci Am 245, Number 1, p. 122; Almost every
figure caption is incorrect or the figure, or its graphed-axis
scales etc are incorrect.

The illustration of the Indian temple dancer is really a direct
example of the non-trivial topology of the O(3) group itself due to the
identification in a simple parametrization by axis of rotation and
angle of rotation (mapped in the obvious way to our actual
3-space as a ball of radius R) has an important identification of
each point at +R in a given direction with the one at -R. I
believe they never published our errata; it's not SciAM style to do
so. I have a copy of all the errors I found somewhere in my office.

Incidentally I think there is nothing except unintelligibility wrong
with the illustration of that dancer's move; we wanted them to use a
multiple flash time-sequence exposure and comissioned a famous
photographer to do the photo our way but despite his boasts it didn't
come out great that way either.

We is Tony Phillips & me, we co-authored the piece.
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Subject: Re: 6 different on pi_1(SO(3))
Date: Mon, 19 Jan 2004 10:54:25 -0500 (EST)
From: Robert Bruner <rrb@math.wayne.edu>

I don't know if you want more on pi_1(SO(3)), but these comments
don't seem to be in what I have read so far.

I agree with David Pengelley that the demonstration using the
rotation of a water glass has always seemed a bit unsatisfactory
because you don't repeat the same motion twice.  Instead the
first pass goes under and the second goes over.  Convincing
someone who doesn't know any math that you haven't `cheated' can
be a little hard.  I don't know how to do this, but it does make
me want to convince myself that I haven't cheated.

The solution to this i have always used is to attach a framing
of R^3 to each point of the arm, from shoulder to hand, starting
with the constant framing (using parallel transport in R^3).
Then each pass, under or over, composes with the same element of
pi_1(SO(3)).

The most satisfactory demonstration I have seen is similar to
but different from Pengelley's `moves' using string and was
shown to me by either Rob Thompson, Tom Hunter, Hal Sadofsky or
Mark Hovey. (I apologize for not remembering exactly who showed
me this.)  It was called the marionette trick when i was shown
it, but any asymmetrical object can be substituted for the
marionette.

Tie three strings from the marionette to a  fixed object.  I
usually use three legs of an upended chair.  Rotate the object
through 720 degrees and the strings will now appear to be wound
around one another.  But with a little effort, you can
untangle them without moving the object.  This is nontrivial
to do the first time, but is quite compelling, I find.