# DIRAC'S STRING TRICK

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).

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.

## Video

These pictures are 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 is copyright IBM/Mathematics and Knots. Copies are available price 15 pounds from Mathematics and Knots

## THE MATHEMATICAL THEMES

• Borromean Rings - what they are and why they don't exist!
• The Möbius Band - what one looks like, experiments to try, amd a beautiful rotating golden one enabling you to really see what one looks like in 3D (this is optional as 90Kb).
• Bernard Morin and the Brehm Model - how Bernard Morin showed John Robinson the Brehm Model of the Möbius Band and how to make one!
• Fibre Bundles - what they are, how to make them, and examples of them in John Robinson's Work.
• Knots and Links - Introduction to the subject of Knot Theory, includes history of the subject, and a rotating mathematically constructed Immortality (the trefoil being one of the most basic knots).
• Torus Knots - two pages explaining the basics about torus knots with the help of excellent colourful graphics. There are also 3D moving images of John Robinson's sculptures of the Gordian Knot and the Rhythm of Life.
• Fractals 3 pages, introducing Fractals, considering iteration, The Sierpinski Gasket and the applications of the subject.

• Exhibition "Mathematics and Knots"
• Ronnie Brown's Homepage
• John Robinson's Symbolic Sculpture
• Brochure giving a guided Tour around the Symbolic Sculptures at the University of Wales, Bangor.
• About the Centre for the Popularisation of Mathematics