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# Mystery of Tangled iPhone Earbuds Solved Through Mathematics

It doesn’t matter if earbuds are rolled neatly and stuffed into a pocket, haplessly tossed into a bag, or kept in a pretty little box, more often than not, there is a knot in them when retrieved, and just like a lab rat seeking its hourly bump of cocaine, you have to solve the puzzle at hand before you can get your fix. The practice goes on day after day after day unless you convert to using one of the countless ways to prevent it, which often take as much time and effort as does untangling the basic knots often formed with free-range earbuds.

Earbuds have been around for decades, but the popularity of the iPhone and iPod, combined with Apple’s insistence on providing the little white-corded, micro-speakers, as their standard listening device for them, have made them more common than ever. Aside from the earbud’s potential for damaging your hearing at high volumes, they also come with an even higher potential for frustrating self-knotting.

But why?

Why are these things so hard to keep unknotted?

A couple of guys from the University of California at San Diego decided to look at this issue more closely. Turns out there is mathematical explanation. The researchers used concepts of Knot Theory, yes Knot Theory is a thing, and it predates earbuds by about a century, and analyzed the probability of knot formation on a string, much like Apple’s earbuds, using factors like length, flexural rigidity, diameter, and factors of agitation. It is truly the kind of stuff that will make your head explode, if like me, you not only failed high school physics, but took three tries to pass college algebra before finally getting through it by scoring a “D” because the teacher liked you.

The results were published in a paper called Spontaneous Knotting of an Agitated String, and essentially concluded unless earbud cords are made in much shorter lengths, we’re all essentially screwed when it comes to knot avoidance. Not only are we screwed, be we the screwing takes place on the wrong side of the probability curve.

To complete the study, researchers dropped a string into a 1-foot square box, allowing it to fall naturally, and then agitated the box by rotating it in a specific way that allowed them to repeat the experiment thousands of times with varying lengths of string. After the tumble, the string was removed by its ends, photographed, and then the ends were connected together to assist with analysis of the knot, as knot theory primarily revolves around knots in a circle. At least that’s what I get out of it, you can check it out for yourself if this kind of specific detail is what gets you off. That’s the purpose of links like this one, which already appeared above.

Anyway, the results showed, as one might expect, shorter strings are less likely to become knotted during the process than longer ones.

Let’s say the average set of iPhone earbuds come with 55 inches, 139 cm, of cord, which includes the wire after the split to each ear piece. I use that number because it is one I saw in a report related to this particular study. It also seemed to agree with the numbers I saw on Apple’s website when I looked at iPhone accessories. (I should note, different models of earbuds come in different lengths, so being overly specific is kind of difficult without devoting way too much space to it.)

The probability that set of earbuds knotting during the agitation stage is right around 50 percent, according to the study’s results. That also happens to be the point where the curve comparing the results on a chart plotting probability and length begins to flatten out. This means cords longer than 55 inches, likely won’t knot with a significantly higher frequency. The curve up to that point, however, spikes dramatically.

Figure from Spontaneous knotting of an agitated string study.

The results can be affected by numerous other factors, including the size of the box, the stiffness of the string/wire being agitated, and the period for which it is actually under agitation, so applying this to a set of iPhone earbuds in a bag or pocket, is going to vary somewhat, but ultimately the principle of it all is still going to be the same.

So the odds are working against you when you put away your earbuds, even if you go to great lengths to coil them up before putting them away. If you drop them in a bag and carry them around for several days between use, or even several hours provided the bag is moved around, there is a good chance knots of some variety have formed. Therefore expecting anything less than a tangle when you retrieve them is just setting yourself up for disappointment.

I’m sure you didn’t need this article to point that out for you, but it’s always interesting to see mathematical evidence proving knot-tying gremlins do not live in your pocket, purse, backpack or briefcase.