Finally got some respite on the journey home after a long, hard day only to find that your earphones have tangled themselves up into half a billion knots? Tired of solving the da Vinci code every time you want to listen to some music?
Well you aren’t alone. The reason for all this, of course, is science. (It wouldn’t be on thefullapple otherwise). And no, there aren’t mini gnomes in your pocket dedicating their lives to making you wanting to stab yourself.
There is an entire mathematical discipline that specialises in how seemingly random tangles form called knot theory – which focuses solely on the “How in the world do things get tangled?” conundrum. In mathematics, knot theory has been an active field of research for more than a century.
First things first – Entropy
The second law of thermodynamics states that the entropy of an isolated system always increases. The entropy is a measure of the disorder of a system, so for example your college room is said to have a higher entropy if it’s in a state of complete mess compared to when you’ve just frantically tidied everything up. (Perhaps to avoid the judgement of Mom and Dad when they visit.) You can probably imagine a lot of ways of messing up your room, but only one way of having it perfectly tidy. Your room will therefore always tend to get messier over time, unless you put in an extra effort to counteract this trend. This is the basic idea of the second law. Nature seems to conspire to inconvenience us.
Similarly, there is only one way for a cord/wire/cable to be straight, but a massive number of ways it can get tangled.
Knot theory mathematicians have found literally hundreds of separate, unique types of individual knots, or “prime knots” and they can be combined in infinite ways. A prime knot is a knot that can’t be decomposed into a sum of simpler knots.
You could go your entire life and never see the same knot twice.
This means that any time you have a bunch of long, thin, flexible objects – your laptop chargers, the Christmas lights in your attic, your garden hose, you name it – the objects link in a number of places. When there’s enough contact points, and the objects are long and slim enough, the chances for these objects not getting into one of those trillions of knot states is downright miniscule.
So what causes these links in the first place?
To introduce the knots in the first place, the system need to be agitated, i.e. energy needs to be added. Even in the tiniest bit of motion such as jostling the box of Christmas lights when you move it, or the movement as you walk with your earphones in your pocket can makes these many knot states catastrophically accumulate, often within seconds.
If you think scientists wouldn’t want to spend their valuable time researching the knotting of a string, think again. This spontaneous knotting of strings was investigated in experiments by researchers at the University of California. The picture below shows the experimental setup. They put an unknotted string into a tumbling box and found that complex knots would form within seconds. They repeated this experiment a shocking number of 3,145 times (an unbelievable effort) and found 120 distinct prime knots.
Source: Raymer, Smith (2007) PNAS 104:16432
They also identified the dependence of the knotting on two key factors, the agitation time and the string length. As you would expect, the probability of knotting increases with increasing agitation time, which is the time of tumbling of the string. The dependence on the string length has some interesting features, shown in the figure below. No knots were observed for strings shorter than 46 cm, but for strings only slightly longer than that the probability of forming knots shot up dramatically.
An average pair of headphones is around 140 cm long, suggesting that they are in the regime of a knotting probability around 50%. So every time you agitate your earphones, e.g. when you drop them on the floor or fiddle with the earphones in your pocket, there is a 1 in 2 chance you’ll introduce a prime knot that will cost you another minute of your life to untangle.
Source: Raymer, Smith (2007) PNAS 104:16432
Are we all doomed, or is there is a way out of this?
As usual YouTube has a way around this:
So next time tangled up earphones make you want to punch a brick wall, don’t curse your luck or attempt any feng shui, buy a horseshoe or look for a nonexistent 4 leaved clover because it won’t stop it happening again. The reason your earphones get tangled is purely scientific. Follow the video above in order to ensure that your earphones are never knotted again.
Welcome to a life free of knotted chaos, you’re welcome.
Want to learn more about knot theory?
A comprehensive introduction to knot theory can be found here.
Journal of latest knot theory news can be found here.