Articles for kids about math in everyday life, written by Ivars Peterson for Muse magazine.

March 3, 2008

Knotty Cords

You're done with your iPod, so you carefully coil the headphone cord around the player and stuff it in your pocket. The next time you take it out, however, you find that you have to unravel the cord and undo a knot before you can go back to listening to your music. In fact, it's pretty amazing how easily knots can form by themselves—not only in headphone cords, but also in necklaces, coiled ropes, strings of holiday lights, hanks of yarn, or garden hoses.

A coiled headphone cord, once unraveled, is likely to contain a knot.

Two physicists at the University of California, San Diego recently decided to do some experiments to try to find out why knots form so easily in coiled strings. They looked at what happens when a long string is coiled into a box and the box is then tumbled.

The researchers, Dorian Raymer and Douglas E. Smith, found that complex knots form within seconds—if the string is long and flexible enough. So, for a given stiffness, a string has to be a certain length before a knot will form. Moreover, the longer and more flexible the string, the better a chance it has of becoming knotted, especially when the string is tumbled or shaken for a long time.

This illustration represents the knot experiment, in which knots form in a tumbled string. Dorian Raymer, UCSD.

The researchers did the experiment over and over again: 3,415 times in all. Knots formed in the strings about one-third of the time. The biggest surprise came when the physicists used mathematics to identify the types of knots that formed in the strings. Mathematicians called knot theorists have described many different kinds of knots, according to features such as crossings (the number of times a string crosses over itself). The physicists found that their real-life model produced 120 different knots. In fact, their strings formed all of the possible kinds of knots with up to seven crossings—and seven crossings is quite a tangle.

Digital photos of knots are combined here with computer-generated drawings based on mathematical calculations. Dorian Raymer, UCSD.

Raymer and Smith say that knotting tends to start from one end of a string. As a string tumbles, this free end braids itself with strands lying next to it, weaving over and under adjacent segments to create a knot. The more braiding, the more complex the knot.

So, if your headphone cord is long, thin, and flexible (if?!), the chances of it becoming knotted are annoyingly high. Sigh.

Muse, March 2008, p. 18.