At one point in our
sequence of pictures on **The
Projective Plane**, we have a loop which goes twice round the
Möbius Band, and then this loop is deformed off the band and over
to the disc, and then to a small loop, and finally to a point, which
represents a stationary rod.

This is related to the
famous **Dirac String Trick**. Take a square and tie the four
corners to another larger square by loose string as shown below
(alternatively, tie the initial square to the four corners of the
room).

Now rotate the small square by 360 degrees about a vertical axis, that is, in a horizontal plane. The strings will become somewhat tangled, and it is not possible to untangle them without rotating the square.

There is an old adage that there is nothing so complicated that it cannot, with sufficient trouble, be made more complicated. So we rotate through another 360 degrees, a total of 720 degrees. Contrary to the adage, it is now possible to untangle the string, without further rotation of the square, but simply by allowing enough space for the strings to be looped over the top of the square! You must check this for yourself. It is advisable for your experiments to use bulldog clips to attach the ribbons to the squares, so that it can be undone easily if it gets too tangled.

A similar idea works for a rotation through 720 degrees about any axis.

Another version of the
Dirac string trick has been called the Philippine wine glass trick. A
glass of water held in the hand can be rotated continuously through 720
degrees without spilling any water. **Try it!**

These geometrical demonstrations are related to the physical fact that an electron has spin 1/2! A particle with spin 1/2 is something like a ball attached to its surroundings with string. Its amplitude changes under a 360 degrees (2pi) rotation and is restored on rotation to 720 degrees (4pi).

The full and exact description of these phenomena needs some sophisticated mathematical ideas (algebra, groups, topology, quaternions, ..), all apparently `abstract' ideas. Indeed, the description of the structure of complex phenomena which are not part of our everyday experience, and so are not absorbed into ordinary language, needs abstract and general ideas, and the only way we have for expressing many of these ideas is that of mathematics.

These pictures were
taken from a 4 min silent video, **Pivoted lines and the Möbius
Band**, produced in 1992 with a story line by Ronald Brown and
programmed at IBMUK Research by Ramen Sen. This video was a part of
Ronald Brown's Royal Institution Friday Evening Discourse, 'Out of
Line', May, 1992. The video was copyright IBM/Mathematics and Knots.
Copies of the video are no longer available, but there is now a cheaper
and better way of viewing it. A QT movie made from the video can be
found on the CDRom **Raising
Public Awareness of Mathematics**.

**© Mathematics
and Knots/Edition Limitee 1996 - 2002**

This
material may be used freely for educational, artistic and scientific
purposes, but may not be used for commercial purposes, for profit or in
texts without the permission of the publishers.

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