This site is based on the following article from
The Mathematical Intelligencer Vol.19, No.2, 1997, 50-52.

Curriculum Vitae



WMY 2000


Dirac's String Problem

Vagn Lundsgaard Hansen

In 1982 the University Library in Copenhagen celebrated its 500th anniversary. One of the chief librarians for the natural and medical sciences suggested that to celebrate the occasion, the various subjects in the library be presented on shopping bags from IRMA, a food store chain in the Copenhagen area. (IRMA is just an old Danish girl's name.) The University Library arranged to decorate a series of 15 of IRMA's brown paper shopping bags.

I was asked to design a shopping bag with a mathematical theme. This gave me a lot to think about. I wanted to show that mathematics is not only theoretical but has something to do with grasping the world in which we live, and that a theory can result from the combined efforts of mathematicians in many countries over a comparatively long period.

This was not an easy task, but I ended up choosing Dirac's String Problem. The solution of this problem was the result of mathematicians in several countries gradually developing a full understanding. The final step was made by Ed Fadell, whose work I had come to admire in connection with my research on braids, configuration spaces, and their relations to polynomial covering spaces.

Briefly, Dirac's String Problem is to explain mathematically why you can untwist a double twisting of loose (or elastic) strings with fixed ends without cutting it up, and why, on the other hand, you cannot do it with strings having only a single twisting.

The problem originated in Dirac's notion of spin which he introduced in the 1920's in connection with relativistic quantum mechanical models for the elementary particles. There is also an application to the problem of transferring electrical current to a rotating plate without the wires getting tangled and breaking. The application does not really need an understanding of the mathematics. But never mind, showing people that you can untwist a number of strings with a double twisting always surprises them; as a matter of fact, many people don't believe it until they untwist the strings themselves. I think that the problem illustrates how mathematics is like a sixth sense in human beings, by which we "sense" (understand) counter-intuitive phenomena.

When the bag was produced, it was the first time four colors were used on a shopping bag in Denmark. I was fortunate that some of my colleagues at the Institute of Drawing made excellent figures out of my sketches. The bag was produced in a quantity of 200,000. I received 25 "reprints" and a small number of unfolded bags, which will serve as posters. From one of these posters, the present copies, in reduced size, have been made. When I am in high spirits, I say that I must be the mathematician who has had the most copies published, except for Euclid. When I am in low spirits, I say that I must be the mathematician who's work has been placed in the most wastebaskets. Presenting the shopping bag to the Rector of the Technical University of Denmark, where I am a professor of mathematics, I told him, "Here you see a piece of applied mathematics which can actually be carried out."

It was quite entertaining to do the work. And it did generate some interest - I have had quite a few comments on it over the years.

Since we mathematicians are so modest (true!), I was afraid that Dirac's String Problem would not be enough to completely decorate one shopping bag. In my proposal, I therefore added a small "mathematical gem" on how to move a ring from one end of a string passing through a hole, obviously too small, to the other end of the string. This gem was used, but on a second mathematical shopping bag.


  • E. Fadell, Homotopy groups of configuration spaces and the string problem of Dirac, Duke Math.J. 29 (1962), 231-242.
  • V.L. Hansen, "Braids and Coverings - Selected topics", London Math. Soc. Student Texts 18, Cambridge University Press, 1989.
  • V.L. Hansen, The Magic World of Geometry - III. The Dirac String Problem, Elemente der Mathematik 49 (1994), 149-154.