From: dan@orion.math.uiuc.edu (Dan Grayson)
Newsgroups: sci.math.research
Subject: Re: Visualizing Pi_1(SO(3))
Date: 24 Jul 1997 21:43:45 GMT
landsbur@laforge.cc.rochester.edu (Steven E. Landsburg) writes:
>It is not hard to see that SO(3) is double covered by the three-sphere
>S^3, and that therefore the fundamental group of SO(3) is Z/2Z. It is
>also not hard to trace through this construction and discover that the
>nontrivial element of Pi_1(SO(3) is represented by a 360 degree rotation
>around (say) the z-axis.
>
>Every few years, somebody tells me that you can visualize some aspect of
>this situation in the following way: Extend your arm with palm up. Now
>rotate your hand 360 degrees under your shoulder, always with palm up. Your
>arm will be twisted. Now rotate your hand another 360 degrees in the same
>direction---the constraints of anatomy will force you to go over the
>shoulder rather than under---still keeping your palm up---and the arm
>will untwist. (I'm sorry not to be able to provide a good ASCII diagram
>of all this.)
>
>It is alleged that this experiment "demonstrates" that the generator
>of Pi_1(SO(3))---that is, the 360 degree rotation around a fixed axis---
>is either nontrivial, or of order at most 2, or both. But although there
>seems to be a vague connection, I cannot figure out exactly how to make
>that connection precise.
Put your arm out straight in front of you, and imagine that you attach the
standard frame to each point of your arm and shoulder; then perform some
motion of your arm that keeps your shoulder fixed. A frame is an orthonormal
basis, so giving a frame is equivalent to giving an element of O(3). A
snapshot of your arm provides a path in O(3) which starts out at the identity
element (at your shoulder). These paths form the universal covering space of
O(3) if you equate two paths that are connected by a homotopy fixing both
endpoints. So after rotating the palm of your hand 360 degrees, the arm
represents a loop, and the evident impossibility of deforming the rest of the
arm (leaving the palm fixed) to restore it to its original state shows you
that you have a nontrivial element of pi_1, appearing as the fiber of the
universal covering space over the base point. Doing it twice amounts to
lifting the same path to the universal covering space twice, hence to
composition in the fundamental group, and shows you have an element of order
two.
==============================================================================
From: landsbur@troi.cc.rochester.edu (Steven E. Landsburg)
Newsgroups: sci.math.research
Subject: Re: Visualizing Pi_1(SO(3))
Date: Fri, 25 Jul 97 16:07:39 GMT
Dan Grayson answered my question about how a 360 degree rotation of
your hand around your shoulder, resulting in a twisted arm, illustrates
the nontriviality of the generator of Pi_1(O(3)).
Because I've recently learned that a *lot* of people are as mystified
by this demonstration as I was, I think it's worth highlighting the
key point, which was somewhat buried in Dan's posting.
I---and apparently many others---was mistaken about how the arm motion
represents a loop in O(3). I had always thought that the loop was
parameterized by *time* and traced out by my fingertips. But now---
thanks to a second reading of Dan's posting---I understand that the
loop is instead parameterized by *distance from the shoulder* after
the motion is completed.
If there were a homotopy demonstrating the triviality of that loop, then
one could parameterize that homotopy by time and use it to describe a
way of untwisting one's arm without moving one's fingertips.
So the mistake is to think that the loop is parameterized by time and
the (purported) homotopy by distance, whereas in fact exactly the
opposite is the case.
I think this all makes sense now.
Steve Landsburg
landsbur@troi.cc.rochester.edu