“Bulletin Board”

 School of Physics - September 13, 2001

Unravelling the myth

It is possible to tie a knot that cannot be untied. Keith Devlin explains Thursday September 13, 2001

 A Polish physicist and a Swiss biologist have used computer simulation to recreate what might have been the Gordian knot. Piotr Pieranski of the Poznan University of Technology in Poland and Andrzej Stasiak of the University of Lausanne in Switzerland acknowledge that the story of the Gordian knot is just myth. But entering into the spirit of the myth, they show how it would be possible to tie a knot that cannot be untied. We now use the phrase "Gordian knot" to refer to any problem that seems too complicated to resolve. According to ancient Greek legend, however, it was a real knot.
The story goes like this. One day, a poor peasant called Gordius arrived with his wife in a public square of Phrygia in an ox cart. Unbeknown to Gordus, an oracle had informed the populace that their future king would come into town riding in a wagon. Seeing Gordius, therefore, the people made him king. In gratitude, Gordius dedicated his ox cart to Zeus, tying it up with a highly intricate knot - the Gordian knot. Another oracle - or maybe the same one, the legend is not specific - foretold that the person who untied the knot would rule Asia. The problem of untying the Gordian knot resisted all solutions until the year 333 BC, when Alexander the Great cut through it with a sword. Although it would have been unwise to have pointed it out in his presence, Alexander's method did seem to go against the spirit of the challenge, which was supposed to be solved by manipulating the knot. That is the traditional story. Here is how Pieranski and Stasiak approached it. Alexander was no dummy, they observed. He was, after all, a former student of Aristotle, and thus would have been no stranger to logical puzzles. Perhaps he resorted to the sword because he could see that the knot could not be untied simply by manipulating the rope. If so, then the knot surely could not have had any free ends. The two ends of the rope must have been spliced together. This means that the Gordian knot was a knot in the sense that present-day knot theorists use the term, to mean a closed loop of string that winds around itself. However, before Alexander came along, many fine minds had been stumped by the Gordian knot problem, and yet no one had claimed the puzzle was unsolvable. Surely, then, it must have appeared that, in principle, the knot could be untied. This means that the loop of rope cannot have been tied into a knot before the ends were spliced together. The knot must have been constructed by first splicing the two ends of a length of rope to form a circular loop, which was then "tied up" (wrapped around itself in some way) to disguise the fact that it was not really knotted. Now, when modern mathematicians study knots, they assume the knots are constructed out of flexible, stretchable, infinitely thin string. Under those assumptions, any unknotted loop can always be untied, no matter how complicatedly it is wrapped around itself. More precisely, you can always manipulate it to be in the form of a simple loop that does not cross itself.
Thus, the only thing that could make it absolutely necessary to resort to a sword to untie the Gordian knot would be that the physical thickness of the actual rope prevented the necessary manipulations being carried out. For example, the rope could have been soaked before tying, then dried rapidly in the sun after tying to make it shrink. But was there a way of wrapping the rope up that would make this method work? Pieranski has developed a computer program called Sono (Shrink On No Overlaps) to simulate the manipulation of such knots. Using this program, he has showed that most ways of trying to construct a Gordian knot will fail - Sono eventually found a way to unravel them. But recently he discovered a knot that worked. Sono, which had not been programmed to make use of an algorithmic sword, was unable to unravel it. Maybe, just maybe, Pieranski and Stasiak suggest, this was the actual structure of the Gordian knot.
Incidentally, there is a seri ous side to all of this. Physicists, such as Pieranski, are interested in knots because the latest theories of matter postulate that everything is made up of tightly coiled (and maybe knotted) loops of space-time.
Biologists like Stasiak are interested in knots because the long, string-like molecules of DNA coil themselves up tightly to fit inside the cell. These two scientists have been collaborating in a study of knots that can be constructed from real, physical material that has, in particular, a fixed diameter. This restriction makes the subject very different from the knot theory traditionally studied by mathematicians, and could lead to advances in both physics and biology.
back to top
scroll left or right