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# Ever wondered why earphones get tangled?

A research paper titled Spontaneous knotting of an agitated string by Dorian M. Raymer and Douglas E. Smith of the University of California at San Diego Department of Physics, demonstrated this phenomenon
In the abstract, the researchers wrote: “It is well known that a jostled string tends to become knotted; yet the factors governing the “spontaneous” formation of various knots are unclear. We performed experiments in which a string was tumbled inside a box and found that complex knots often form within seconds.”

Knot theory is an area of mathematics that studies knots, as the name suggests, but not just the types of knots we are accustomed to — it has applications in biology, quantum computing, chemistry, and many other fields. In math, knots are always studied in closed loops and a knot is defined as a configuration that cannot be untangled into a simple loop. They are classified based on their number of crossings. Two knots are considered equivalent if one can be transformed into the other without detaching the ends.

The real question here is: How can the researchers’ experiments with string in boxes help us keep our headphones untangled? Here are some of their useful results:

“Tripling the agitation time caused a substantial increase in P, indicating that the knotting is kinetically limited.” This line means that the longer you store your headphones for, the more likely they are to get knotted. That makes sense — they have more time to move around and get tangled.

“Above a critical string length, the probability P of knotting at first increased sharply with length.” Thus, the longer your cord is, the more likely it is to get tangled.

“Increasing confinement of a stiff string in a box causes increased wedging of the string against the walls of the box, which reduces the tumbling motion that facilitates knotting.” Here, the researchers noted that a smaller box meant less room for the headphones to move around and get tangled in.

Similarly, when using a stiffer cord, “the tumbling motion was reduced because the finite stiffness of the coiled string tends to wedge it more firmly against the walls of the box.” This means that the stiffer a cord is, the less it bends and moves around — creating knots — in the box.

All this to say, stick with as short and stiff a headphone cord as possible, store them in a small pocket and don’t leave them there for too long. Then again, if — or should I say when — they do get tangled, why not take the time to study the knots? Who knows what you might discover!

It revealed that a cord of less than 46 centimetres in length (about 1 foot six inches) will almost never tangle itself when sealed inside a rotating box for a period.

But between 46 centimetres and 150 centimetres (about five feet), the probability of a knot forming rises dramatically. With a cord longer than that, the probability of a knot forming reaches a plateau of 50 percent.

It turns out that the odds of getting a knot do not go higher because a longer cord gets wedged inside the shape of the box and that prevents further tangles from forming. Raymer and Smith performed 3,415 trials to demonstrate this.

Here is what that curve looks like:

Apple’s iPhone earbuds are 139 centimetres (55 inches) long and thus right at the 50 percent tangle-rate-sweet-spot, at the top of the curve.

In other words, if you place your earbuds in a bag the odds of them tangling into a knot as you carry them around are 50 percent, at least.

“At least” because earbuds are, of course, a Y-shaped string, and thus the knotting frequency is compounded further. (Raymer and Smith didn’t look at strings with more than one branch, but anecdotally I can confirm that the tangle-rate is pretty high.)

Finally, here is a schematic showing how a cord that starts off neatly coiled – (you don’t just stuff them in there, do you?) – quickly becomes tangled inside a rotating box. It shows that one end of a cord only has to cross another part of the cord twice in order to start spontaneously knotting itself:

That last part is perhaps the most magical of all: The research shows that your earphones are indeed spontaneously knotting themselves. Sure, it’s because of their length and the agitation of the container they’re in.

But the knots really do form as a matter of physics, not because of your personal lack of neatness.