\nopagenumbers
\def\pt#1{\hbox{#1) }}
\input epsf
\headline={ESP Math 408D - AP\hfil Fall 1996}
\footline={\ifnum\pageno>1 \hfil Worksheet 18\hskip.3in Page \folio \fi}
\footnote{}{Mike Oehrtman}


\centerline{\bf WORKSHEET 18} 
\centerline{\bf Dirac's Belt Trick}

\bigskip 

\item{1.}  Carry out the following experiment:
\itemitem{}  Attach a belt to the back of a chair (or other upright
object). 
\itemitem{}  Twist the belt through 2 full turns ($720^\circ$).

\medskip
\hbox{\hskip.5in\vbox{
\epsfxsize=3.2truein\epsfysize=1.2truein
\epsfbox{/home/oehrtman/m210/chair.eps}}
\hskip.3in
\vbox{\hsize=2truein \baselineskip=18pt \noindent
The problem now is to untwist the belt without rotating the end of
the belt (the pencil) or moving the chair.\vskip.4in}}
\medskip

\item{2.}  Try the same experiment with more or less than 2 full turns.

\medskip
\hbox{\vbox{\hsize=4.5in
\item{3.}  The mathematical object that explains the above behavior is
called 
\smallskip
\centerline{\it the space of all rotations in 3-dimensions.}

\smallskip
\item{}  We now try to picture this object.
\smallskip

\item{}  A rotation in 3-dimensions is determined by two pieces of
information:  the axis about which to rotate and an angle through
which we rotate.  For example, rotate $30^\circ$ about the vertical
axis. (We need to pick a ``sense'' through which to rotate - here, we
choose a ``right handed'' rule.)}\hskip.5in
\vbox{\epsfysize=1.5truein\epsfbox{/home/oehrtman/m210/thirtydeg.eps}}}

\medskip
\itemitem{a)}  Consider a single die oriented in 3-space as shown
below: 

\centerline{
\epsfysize=1truein\epsfbox{/home/oehrtman/m210/die.eps}}

\smallskip
\itemitem{}  In the following four pictures the die has been rotated
about some axis by some angle.  For each picture, draw the axis of
rotation through the die and indicate the angle of rotation.

\vskip.1in
\centerline{
\epsfysize=.8truein\epsfbox{/home/oehrtman/m210/dice.eps}}
\vskip.1in

\itemitem{}What you have done is identify exactly which rotation
in ``the space of all rotations in 3-dimensions'' moved the die from
the orientation in the original picture to each of those above.

\medskip
\itemitem{b)} Draw a die which has been moved from its original
orientation as given in part a) by each of the following rotations:

\smallskip
\settabs\+\hskip.7in iii)&angle:$\pi/2$\hskip.5in&\cr
\+\hfill i) &angle: $\pi/2$ &axis: to the right, in the
plane of the page\cr

\smallskip
\+\hfill ii) &angle: $\pi/2$ &axis: to the left, in the
plane of the page\cr

\smallskip
\+\hfill iii) &angle: $\pi$ &axis: coming straight {\bf
out} of the plane of the page\cr

\smallskip
\+\hfill iv) &angle: $\pi$ &axis: going straight {\bf
into} the plane of the page\cr

\smallskip
\itemitem{}  Are rotations iii) and iv) really different?

\eject
\item{4.}  Now we construct an actual {\bf picture} of ``the space of
all rotations in 3-dimensions'':

\hbox{\vbox{\hsize=4.5in
\itemitem{}  Consider a solid ball of radius $\pi$.  Think of the
center point as ``zero rotation'' or the rotation of 3-space by {\bf
zero} angle (through any axis).  Now, any other rotation will also be
represented by a point in this solid ball.  Specifically, {\it a rotation
through an angle $\theta$ about an axis $L$ will be represented by a
point a distance $\theta$ out from the center along the axis $L$.}}
\hskip.3in
\epsfysize=1truein\epsfbox{/home/oehrtman/m210/rotation_rep.eps}}

\medskip
\itemitem{a)}  Locate a point in the solid ball that corresponds to
each of the eight rotations from problem 3.  

\medskip
\itemitem{b)}  Do some of these rotations correspond to more than one
point in the ball?

\medskip
\item{5.}  Now notice that a rotation through $\pi$ ($180^\circ$)
about $L$ is the same as a rotation through $\pi$ about the axis
pointing in the opposite direction to $L$:

\medskip
\hbox{\hskip.4in
\epsfysize=1truein\epsfbox{/home/oehrtman/m210/rotation_ident.eps}
\hskip.3in\vbox{\hsize=4.1in
\noindent \baselineskip=18pt Therefore in our picture,
we need to consider a point on the surface of the ball (at distance
$\pi$ from the center) along an axis $L$ as the {\bf same} rotation
which is represented by the point on the opposite side of the surface
(the antipodal point).}}

\medskip
\item{}  We are led to conclude that ``the space of all rotations in
3-dimensions'' can be pictured as a solid ball with antipodal points
on the boundary identified as the same rotation.  Every rotation is
represented by a point on this ball, and every point represents a
rotation.

\medskip
\item{}  Suppose you are a creature who lives inside of this
ball.  If you take a walk from the center (the zero rotation) to a
point on the boundary (a rotation by angle $\pi$) notice that you can
keep walking!  You will just reappear on the opposite side of the ball
coming inwards since we have identified these points!  (Just like a
3-d video game....)  The question is this: if you do this and then
return straight to the center, you will notice something strange.
What?

\medskip
\item{6.}  With the above picture in mind, we can now explain the
Dirac belt:

\medskip
\item{}  When we have rotated the belt through 2 turns, the belt
contains a graphical illustration of a path through our picture.  The
path makes two ``laps'' through the space: beginning at the center,
travelling out to point ``a,'' back to the center, out to point ``b''
then back to the center again.
\medskip
\hbox{\hskip.4in\vbox{
\epsfysize=1truein\epsfbox{/home/oehrtman/m210/twistedbelt.eps}
\vskip.4in}
\hskip.3in
\epsfysize=2truein\epsfbox{/home/oehrtman/m210/loop.eps}}

\medskip
\item{}  The sequence of moves which dissolves the rotations while
keeping $0$ and $4\pi$ fixed is shown below.  Here, we only show the
disk where all of the action is:

\smallskip
{\rightskip=0in
\hskip.04in\epsfysize=1truein\epsfbox{/home/oehrtman/m210/homotopy.eps}
\par}



\bye