At one point in our sequence of pictures, 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).
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.