Calculus 3 Labs

Lab 1: Dirac's belt trick


Carry out the following experiment:

Attach a belt to the back of a chair (or other upright object). Twist the belt through 2 full turns.

chair

The problem now is to untwist the belt without rotating the end of the belt (the pencil) or moving the chair.

 Try the same experiment with more or less than 2 full turns.

 

The mathematical object that captures the above behavior is called

the space of all rotations in 3-dimensions.

 

We now try to picture this object.

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-hand" rule.)

 

Consider a single die oriented in 3-space as shown below:

Lab 1   Rotation axis

Lab 1   Original die

  1. 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.
    Lab 1   Rotated dice
    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.

  2. Draw a die which has been moved from its original orientation as given at the beinning of Question 1 through each of the following rotations:
    1. angle: \(\pi/2\)             axis: coming straight out of the face with 2 dots
    2. angle: \(\pi/2\)              axis: coming straight out of the face with 3 dots
    3. angle: \(\pi\)                 axis: coming straight out of the edge between the 2 and 3
    4. angle: \(\pi\)                 axis: going straight into the the edge between the 2 and 3

      Are rotations c and d really different?

  3. rotation repNow we construct an actual picture of "the space of all rotations in 3-dimensions":

    Consider a solid ball of radius \(\pi\). Think of the center point as "zero rotation" or the rotation of 3-space by zero angle (through any axis).

    Any other rotation will also be represented by a point in this solid ball. Specifically, 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 \(L\).

    Locate a point in the solid ball that corresponds to each of the eight rotations from Questions 1 and 2. 
    Do some of these rotations correspond to more than one point in the ball?

Notice that a rotation through \(\pi\) (\(180^\circ\)) about \(L\) has the same outcome as a rotation through \(\pi\) about the axis pointing in the opposite direction to \(L\):

rotation identTherefore 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 same rotation which is represented by the point on the opposite side of the surface (the antipodal point).

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.

 

With the above picture in mind, we can now explain the Dirac belt:

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.

twistedbeltloop
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:

homotopy